Well-posedness and Critical Index Set of the Cauchy Problem for the Coupled KdV-KdV Systems on $\mathbb{T}$

TL;DR

This paper investigates the well-posedness of coupled KdV-KdV systems on the torus, identifying critical index sets for different function spaces using number theory, revealing differences from the real line case.

Contribution

It determines the critical index sets for the well-posedness of coupled KdV systems on the torus, linking them to Diophantine approximation and contrasting with the real line case.

Findings

Critical index sets are explicitly characterized for various spaces.

Number theory techniques are used to identify these sets.

Differences from the real line case are highlighted.

Abstract

Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \quad v(x,0)=v_0(x)\] posed on the torus $\mathbb{T}$ in the spaces \[ {\cal H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_2:=H^s_0 (\mathbb{T})\times H^s(\mathbb{T}), \quad {\cal H}^s_3:=H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_4:=H^s (\mathbb{T})\times H^s (\mathbb{T}).\] For $k=1,2,3,4$, it is shown that for given $a_1$, $a_2$, $(c_{ij})$ and $(d_{ij})$, there exists a unique $s^*_k \in (-\infty, +\infty]$, called the critical index, such that the system is analytically well-posed in $\cal{H}^s_k$ for $s>s^*_k$ while the bilinear estimate, the key for the proof of the analytical well-posedness, fails if $s<s^{*}_k$. Viewing the critical index $s^*_k$ as a function of the coefficients $a_1$, $a_2$, $(c_{ij})$ and $(d_{ij})$, its range $\cal{C}_k$ is called the critical index set for the analytical well-posedness of the system in the space $\cal{H}^s_k$. Invoking some classical results of Diophantine approximation in number theory, we are able to identify that \[ \mbox{$ {\cal C}_1= \left \{ -\frac12, \infty \right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \}$ } \quad\text{and}\quad \mbox{${\cal C}_q= \left \{ -\frac12, -\frac14, \infty \right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \}$ $\quad$ for $\quad$ $q=2,3,4$.}\] This is in sharp contrast to the $R$ case in which the critical index set ${\cal C}$ for the analytical well-posedness of in the space $H^s (R)\times H^s (R)$ consists of exactly four numbers: $ {\cal C}=\left \{ -\frac{13}{12}, -\frac34, 0, \frac34 \right \}.$

Tables

Table 1. Table 1 : Sharp bilinear estimates on ℝ ℝ \mathbb{R} ( r = α 2 α 1 𝑟 subscript 𝛼 2 subscript 𝛼 1 r=\frac{\alpha_{2}}{\alpha_{1}} )

$s^{\ast }$			$r=\frac{1}{4}$	$r>\frac{1}{4}$, $r\ne 1$	$r=1$
(D1)	$-\frac{3}{4}$	$-\frac{3}{4}$	$\frac{3}{4}$	$0$	$-\frac{3}{4}$
(D2)	$-\frac{13}{12}$	$-\frac{3}{4}$	$\frac{3}{4}$	$0$	$-\frac{3}{4}$
(ND1)	$-\frac{3}{4}$	$-\frac{3}{4}$	$\frac{3}{4}$	$0$	$0$
(ND2)	$-\frac{3}{4}$	$-\frac{3}{4}$	$\frac{3}{4}$	$0$	$0$

Table 2. Table 2 : Sharp bilinear estimates on 𝕋 𝕋 \mathbb{T} by Oh [ 40 ]

$s^{\ast }$	$a_2\in (0,4]\setminus \left\{1\right\}$
(1.16)	$\min \{1,\frac{1}{2}+\frac{1}{2}{\nu }_c\}$
(1.17)	$\min \{1,\frac{1}{2}+\frac{1}{2}{\nu }_d\}$

Table 3. Table 3 : Sharp bilinear estimates on 𝕋 𝕋 \mathbb{T} in divergence form ( r = α 2 α 1 𝑟 subscript 𝛼 2 subscript 𝛼 1 r=\frac{\alpha_{2}}{\alpha_{1}} )

$s^{\ast }$		$r\ge \frac{1}{4}$, $r\ne 1$	$r=1$ ($\widehat{w_1}(0,\cdot )=\widehat{w_2}(0,\cdot )=0$)
(D1)	$-\frac{1}{2}$	$\min \{1,s_r\}$	$-\frac{1}{2}$
(D2) with $\widehat{w_2}(0,\cdot )=0$	$-\frac{1}{2}$	$\min \{1,s_r\}$	$-\frac{1}{2}$

Table 4. Table 4 : Sharp bilinear estimates on 𝕋 𝕋 \mathbb{T} in non-divergence form ( r = α 2 α 1 𝑟 subscript 𝛼 2 subscript 𝛼 1 r=\frac{\alpha_{2}}{\alpha_{1}} )

$s^{\ast }$		$r\ge \frac{1}{4}$, $r\ne 1$	$r=1$
(ND1) ($\widehat{w_2}(0,\cdot )=0$)	$-\frac{1}{4}$	$\min \{1,s_r\}$	$\frac{1}{2}$
(ND2)	$-\frac{1}{4}$	$\min \{1,s_r\}$	$\frac{1}{2}$ ($\widehat{w_1}(0,\cdot )=0$)

Equations

$$\left\{\begin{array}[]{rcl}u_{t}+a_{1}u_{xxx}&=&c_{11}uu_{x}+c_{12}vv_{x}+d_{11}u_{x}v+d_{12}uv_{x},\vspace{0.03in}\\ v_{t}+a_{2}v_{xxx}&=&c_{21}uu_{x}+c_{22}vv_{x}+d_{21}u_{x}v+d_{22}uv_{x},\vspace{0.03in}\\ \left.(u,v)\right|_{t=0}&=&(u_{0},v_{0})\end{array}\right.$$

$${\mathcal{H}}_1^s:=H_0^s(\mathbb{T})\times H_0^s(\mathbb{T}),{\mathcal{H}}_2^s:=H_0^s(\mathbb{T})\times H^s(\mathbb{T}),{\mathcal{H}}_3^s:=H^s(\mathbb{T})\times H_0^s(\mathbb{T}),{\mathcal{H}}_4^s:=H^s(\mathbb{T})\times H^s(\mathbb{T}).$$

$$\left\{\begin{array}[]{ll}\begin{pmatrix}u_{t}\\ v_{t}\end{pmatrix}+A_{1}\begin{pmatrix}u_{xxx}\\ v_{xxx}\end{pmatrix}=A_{2}\begin{pmatrix}uu_{x}\\ vv_{x}\end{pmatrix}+A_{3}\begin{pmatrix}u_{x}v\\ uv_{x}\end{pmatrix},&\vspace{0.1in}\\ \left.\begin{pmatrix}u\\ v\end{pmatrix}\right|_{t=0}=\begin{pmatrix}u_{0}\\ v_{0}\end{pmatrix},\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\begin{pmatrix}u_{t}\\ v_{t}\end{pmatrix}+\begin{pmatrix}a_{1}&0\\ 0&a_{2}\end{pmatrix}\begin{pmatrix}u_{xxx}\\ v_{xxx}\end{pmatrix}=C\begin{pmatrix}uu_{x}\\ vv_{x}\end{pmatrix}+D\begin{pmatrix}u_{x}v\\ uv_{x}\end{pmatrix},&\vspace{0.1in}\\ \left.\begin{pmatrix}u\\ v\end{pmatrix}\right|_{t=0}=\begin{pmatrix}u_{0}\\ v_{0}\end{pmatrix},\end{array}\right.$$

$$\left\{\begin{array}[]{rcll}u_{t}+a_{1}u_{xxx}&=&c_{11}uu_{x}+c_{12}vv_{x}+d_{11}u_{x}v+d_{12}uv_{x},&\vspace{0.03in}\\ v_{t}+a_{2}v_{xxx}&=&c_{21}uu_{x}+c_{22}vv_{x}+d_{21}u_{x}v+d_{22}uv_{x},&\vspace{0.03in}\\ \left.(u,v)\right|_{t=0}&=&(u_{0},v_{0}),\end{array}\right.$$

$$\left\{\begin{array}[]{rll}u_{t}+u_{xxx}&=&-vv_{x},\\ v_{t}+a_{2}v_{xxx}&=&-(uv)_{x},\\ \left.(u,v)\right|_{t=0}&=&(u_{0},v_{0}),\end{array}\right.$$

$$\left\{\begin{array}[]{rll}u_{t}+a_{1}u_{xxx}&=&-6a_{1}uu_{x}+c_{12}vv_{x},\vspace{0.03in}\\ v_{t}+v_{xxx}&=&-3uv_{x},\vspace{0.03in}\\ \left.(u,v)\right|_{t=0}&=&(u_{0},v_{0}),\end{array}\right.$$

$$\left\{\begin{array}[]{rcl}u_{t}+u_{xxx}+\sigma_{3}v_{xxx}&=&-uu_{x}+\sigma_{1}vv_{x}+\sigma_{2}(uv)_{x},\vspace{0.03in}\\ \rho_{1}v_{t}+\rho_{2}\sigma_{3}u_{xxx}+v_{xxx}+\sigma_{4}v_{x}&=&\rho_{2}\sigma_{2}uu_{x}-vv_{x}+\rho_{2}\sigma_{1}(uv)_{x},\vspace{0.03in}\\ \left.(u,v)\right|_{t=0}&=&(u_{0},v_{0}),\end{array}\right.$$

$$A_1=\left(\begin{array}{cc}1& {\sigma }_3\\ \frac{{\rho }_2{\sigma }_3}{{\rho }_1}& \frac{1}{{\rho }_1}\end{array}\right).$$

$$u_t+u_{xxx}+u{u}_x=0,u(x,0)=u_0(x)$$

$$H_{0}^{s}(\mathbb{T})=\Big{\{}f\in H^{s}(\mathbb{T}):\int_{0}^{2\pi}f(x)\,dx=0\Big{\}}.$$

$$\left\{\begin{array}[]{rcl}u_{t}+a_{1}u_{xxx}&=&d_{1}(uv)_{x},\vspace{0.03in}\\ v_{t}+a_{2}v_{xxx}&=&d_{2}(uv)_{x},\vspace{0.03in}\\ \left.(u,v)\right|_{t=0}&=&(u_{0},v_{0}).\end{array}\right.$$

$$\Vert w{\Vert }_{X_{s,b}^{\alpha }(G\times \mathbb{R})}:=\Vert ⟨k{⟩}^s⟨\tau -\alpha k^3{⟩}^b\hat{w}(k,\tau ){\Vert }_{L^2(G^{\ast }\times \mathbb{R})},$$

$$\Vert {\partial }_x(uv){\Vert }_{X_{s,b-1}^1}\lesssim \Vert u{\Vert }_{X_{s,b}^1}\Vert v{\Vert }_{X_{s,b}^1},\forall u,v\in X_{s,b}^1,$$

$$\Vert {\partial }_x(w_1{w}_2){\Vert }_{X_{s,b-1}^{{\alpha }_2}}\lesssim \Vert w_1{\Vert }_{X_{s,b}^{{\alpha }_1}}\Vert w_2{\Vert }_{X_{s,b}^{{\alpha }_1}},$$

$$\Vert {\partial }_x(w_1{w}_2){\Vert }_{X_{s,b-1}^{{\alpha }_1}}\lesssim \Vert w_1{\Vert }_{X_{s,b}^{{\alpha }_1}}\Vert w_2{\Vert }_{X_{s,b}^{{\alpha }_2}},$$

$$\Vert ({\partial }_x{w}_1)w_2{\Vert }_{X_{s,b-1}^{{\alpha }_1}}\lesssim \Vert w_1{\Vert }_{X_{s,b}^{{\alpha }_1}}\Vert w_2{\Vert }_{X_{s,b}^{{\alpha }_2}},$$

$$\Vert w_1({\partial }_x{w}_2){\Vert }_{X_{s,b-1}^{{\alpha }_1}}\lesssim \Vert w_1{\Vert }_{X_{s,b}^{{\alpha }_1}}\Vert w_2{\Vert }_{X_{s,b}^{{\alpha }_2}}.$$

$$\Vert {\partial }_x(v^2){\Vert }_{X_{s,b-1}^1}$$

$$\Vert {\partial }_x(vu){\Vert }_{X_{s,b-1}^{a_2}}$$

$$\begin{array}{rl}{c}_1=\frac{1}{2}-\frac{1}{6}\sqrt{\frac{12}{a_2}-3},& c_2=\frac{1}{2}+\frac{1}{6}\sqrt{\frac{12}{a_2}-3},\\ & \\ d_1=\frac{-3a_2-\sqrt{3a_2(4-a_2)}}{2(1-a_2)},& d_2=\frac{-3a_2+\sqrt{3a_2(4-a_2)}}{2(1-a_2)}.\end{array}$$

$${\nu }_c=\max \{\nu (c_1),\nu (c_2)\},{\nu }_d=\max \{\nu (d_1),\nu (d_2)\}.$$

$$s^{*}_{1}=\min\Big{\{}1,\,\frac{1}{2}+\frac{1}{2}\max\{\nu_{c},\nu_{d}\}\Big{\}}.$$

$$\mathcal{C}=\big{\{}-\frac{13}{12},\,-\frac{3}{4},\,0,\,\frac{3}{4}\big{\}}.$$

$$u_t+a{u}_{xxx}=cu{u}_x,u(x,0)=u_0(x),ac\ne 0.$$

$${\mathcal{H}}_1^s:=H_0^s(\mathbb{T})\times H_0^s(\mathbb{T}),{\mathcal{H}}_2^s:=H_0^s(\mathbb{T})\times H^s(\mathbb{T}),{\mathcal{H}}_3^s:=H^s(\mathbb{T})\times H_0^s(\mathbb{T}),{\mathcal{H}}_4^s:=H^s(\mathbb{T})\times H^s(\mathbb{T}).$$

$$s^{*}(a_{2})=\left\{\begin{array}[]{cll}-\frac{1}{2}&\text{if}&a_{2}\in(-\infty,0)\cup(4,\infty),\\ \infty&\text{if}&a_{2}=1,\\ \min\{1,\tilde{s}(a_{2})\}&\text{if}&a_{2}\in(0,4]\backslash\{1\}.\end{array}\right.$$

$$\big{\{}-\frac{1}{2},\infty\big{\}}\bigcup\Big{\{}\min\{1,\tilde{s}(a_{2})\}:a_{2}\in(0,4]\setminus\{1\}\Big{\}}.$$

$$0<\Big{|}\rho-\frac{m}{n}\Big{|}<\frac{1}{|n|^{\mu}}$$

$$\mu (\rho ):=\sup \{\mu \in \mathbb{R}:\text{ρ is approximable with power μ}\}$$

$$\Big{|}\rho-\frac{m}{n}\Big{|}\geq\frac{K}{|n|^{2+\nu}}$$

$$\nu (\rho ):=\inf \{\nu \in \mathbb{R}:\text{ρ is of type ν}\}$$

Full text

[TABLE]

posed on the periodic domain $\mathbb{T}$ in the following four spaces

[TABLE]

The coefficients are assumed to satisfy $a_{1}a_{2}\neq 0$ and $\sum\limits_{i,j}(c_{ij}^{2}+d_{ij}^{2})>0$.

Fix $k\in\{1,2,3,4\}$. Then for any coefficients $a_{1}$, $a_{2}$, $(c_{ij})$ and $(d_{ij})$, it is shown that there exists a critical index $s^{*}_{k}\in(-\infty,+\infty]$ such that the system (0.1) is analytically locally well-posed in $\mathcal{H}^{s}_{k}$ if $s>s^{*}_{k}$ but weakly analytically ill-posed if $s<s^{*}_{k}$. Viewing $s^{*}_{k}$ as a function of the coefficients, its range $\mathcal{C}_{k}$ is defined to be the critical index set for the analytical well-posedness of (0.1) in $\mathcal{H}^{s}_{k}$.

By investigating some properties of the irrationality exponents of the real numbers and by establishing some sharp bilinear estimates in non-divergence form, we manage to identify {\cal C}_{1}=\left\{-\frac{1}{2},\infty\right\}\bigcup\big{[}\frac{1}{2},1\big{]} and {\cal C}_{q}=\left\{-\frac{1}{2},-\frac{1}{4},\infty\right\}\bigcup\big{[}\frac{1}{2},1\big{]} for $q=2,3,4$. In particular, these sets contain an open interval $(\frac{1}{2},1)$. This is in sharp contrast to the $\mathbb{R}$ case in which the critical index set ${\cal C}$ for the analytical well-posedness of (0.1) in the space $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$ consists of exactly four numbers: ${\cal C}=\left\{-\frac{13}{12},-\frac{3}{4},0,\frac{3}{4}\right\}.$

††footnotetext: 2010 Mathematics Subject Classification. 35Q53; 35G55; 35L56; 35D30; 11J72; 11J04.††footnotetext: Key words and phrases. Well-posedness, KdV-KdV systems, Fourier restriction spaces, Bilinear estimates, Diophantine approximation, Irrationality exponents.

1 Introduction

1.1 cKdV systems and well-posedness

This paper studies the Cauchy problem for the coupled KdV-KdV (cKdV) systems posed on the periodic domain $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ in the following general form:

[TABLE]

where $\{A_{i}\}_{1\leq i\leq 3}$ are $2\times 2$ real constant matrices, $u=u(x,t)$ and $v=v(x,t)$ are real-valued functions of the variables $x\in\mathbb{T}$ and $t\in\mathbb{R}$. It is assumed that there exists an invertible real matrix $M$ such that $A_{1}=M\begin{pmatrix}a_{1}&0\\ 0&a_{2}\end{pmatrix}M^{-1}$ with $a_{1}a_{2}\neq 0$. By regarding $M^{-1}\begin{pmatrix}u\\ v\end{pmatrix}$ as the new unknown functions (still denoted by $u$ and $v$), it reduces to

[TABLE]

or equivalently,

[TABLE]

which is called in divergence form if $d_{11}=d_{12}$ and $d_{21}=d_{22}$, and in non-divergence form otherwise.

Listed below are a few specializations of (1.1) appeared in the literature.

• •

Majda-Biello system:

[TABLE]

where $a_{2}\neq 0$. This system was proposed by Majda and Biello in [36].

• •

Hirota-Satsuma system:

[TABLE]

where $a_{1}\neq 0$. This system is in non-divergence form and was proposed by Hirota-Satsuma in [18].

• •

Gear-Grimshaw system:

[TABLE]

where $\sigma_{i}\in\mathbb{R}(1\leq i\leq 4)$ and $\rho_{1},\,\rho_{2}>0$. When $\sigma_{4}=0$, this system is a special case of (1.1) with

[TABLE]

Note that $A_{1}$ in (1.7) is diagonalizable over $\mathbb{R}$ for any $\sigma_{3}\in\mathbb{R}$ and $\rho_{1},\,\rho_{2}>0$. Moreover, the eigenvalues of $A_{1}$ are nonzero unless $\rho_{2}\sigma_{3}^{2}=1$. So (1.6) can be reduced to the form (1.3) as long as $\rho_{2}\sigma_{3}^{2}\neq 1$. This system was derived by Gear-Grimshaw in [16] (also see [6] for the physical explanation).

We are mainly concerned with the well-posedness of the Cauchy problem for the system (1.3) in some scaled Banach spaces ${\cal X}^{s}(\mathbb{T})$.

Definition 1.1**.**

For any integer $l\geq 0$, the Cauchy problem of the system (1.3) is said to be $C^{l}$-locally well-posed (LWP) in the space ${\cal X}^{s}(\mathbb{T})$ if for any $\delta>0$, there exists $T>0$ such that

• (a)

for any $(u_{0},v_{0})\in{\cal X}^{s}(\mathbb{T})$ with $\|(u_{0},v_{0})\|_{{\cal X}^{s}(\mathbb{T})}\leq\delta$, (1.3) admits a unique solution $(u,v)$ in the space $C([0,T];{\cal X}^{s}(\mathbb{T}))$ satisfying the auxiliary condition $w\in\mathbb{Y}^{T}_{s}$, where $\mathbb{Y}^{T}_{s}$ is an auxiliary metric space;

• (b)

the corresponding solution map $K$ from the initial data $(u_{0},v_{0})$ to the solution $(u,v)$: $K(u_{0},v_{0})=(u,v)$, is $C^{l}$ smooth from $\{(u_{0},v_{0}):\ \|(u_{0},v_{0})\|_{{\cal X}^{s}(\mathbb{T})}\leq\delta\}$ to $C([0,T];{\cal X}^{s}(\mathbb{T}))\cap\mathbb{Y}^{T}_{s}$.

Similarly, if the solution map $K$ is real analytic (or uniformly continuous), then the system (1.3) is said to be analytically (or uniformly continuously) LWP in ${\cal X}^{s}(\mathbb{T})$.

In the literature, the well-posedness in Definition 1.1 is called conditional (cf. [26, 8] and the references therein). If the uniqueness holds in the space $C([0,T];{\cal X}^{s}(\mathbb{T}))$ without the auxiliary space $\mathbb{Y}^{T}_{s}$, then the well-posedness is said to be unconditional. On the other hand, the well-posedness in Definition 1.1 is called local, but if the lifespan $T$ can be specified independently of $\delta$, then the system (1.3) is said to be globally well-posed (GWP). On the other hand, if the local well-posedness fails, then it is called ill-posed.

Definition 1.2**.**

For any integer $l\geq 0$, the Cauchy problem of the systems (1.3) is said to be $C^{l}$ ill-posed (IP) in the space ${\cal X}^{s}(\mathbb{T})$ if there exists $\delta>0$ such that for any $T>0$, there does not exist a $C^{l}$ smooth solution map $K$ as defined in Definition 1.1. Similarly, if there does not exist a real analytic (or uniformly continuous) solution map, then (1.3) is said to be analytically (or uniformly continuously) IP in ${\cal X}^{s}(\mathbb{T})$.

In the rest of the paper, for the convenience of notations, the analytically LWP, GWP and IP will be written as A-LWP, A-GWP and A-IP respectively. Similarly, the uniformly continuously LWP, GWP and IP will be written as U-LWP, U-GWP and U-IP respectively.

1.2 Literature review

For the single KdV equation

[TABLE]

posed on either $\mathbb{R}$ or $\mathbb{T}$, the study of the its well-posedness began in the late 1960s with the work of Sjöberg [44, 45] and has culminated in the work of Killip and Visan[33]. Many significant breakthroughs were made in this study of more than fifty years (see e.g. Bona-Smith [7], Kato [22, 23, 24, 25], Constatin-Saut [15], Kenig-Ponce-Vega [27, 28, 29, 30, 31], Bourgain [9], Christ-Colliander-Tao [12], Colliander-Keel-Staffilani-Takaoka-Tao [13, 14], Kappeler -Topalov [21], Molinet [37, 38], Tao [46], Killip-Visan [33], to name a few).

Define $H^{s}(\mathbb{R})$ and $H^{s}(\mathbb{T})$ as the standard Sobolev spaces on $\mathbb{R}$ and $\mathbb{T}$ respectively. In addition, denote $H_{0}^{s}(\mathbb{T})$ to be the collection of the functions in $H^{s}(\mathbb{T})$ whose mean value is 0:

[TABLE]

The following theorem summarizes the known results on the well-posedness of (1.8).

Theorem 1.3** ([21, 33, 37, 38, 31, 17, 34, 13, 12, 9]).**

The Cauchy problem (1.8) is

• (1)

$C^{0}$-GWP in both spaces $H^{s}(\mathbb{R})$ and $H^{s}(\mathbb{T})$ for any $s\geq-1$, and $C^{0}$-IP for any $s<-1$;

• (2)

A-GWP in the space $H^{s}(\mathbb{R})$ for any $s\geq-\frac{3}{4}$, and U-IP for any $s<-\frac{3}{4}$;

• (3)

A-GWP in the space $H^{s}_{0}(\mathbb{T})$ for any $s\geq-\frac{1}{2}$, and U-IP for any $s<-\frac{1}{2}$;

• (4)

U-IP in the space $H^{s}(\mathbb{T})$ for any $s\in\mathbb{R}$.

For the cKdV systems (1.3) posed on $\mathbb{R}$, it has also been very well studied (see e.g. [1, 6, 40, 49] and the references therein). Of particular interest is a special class of (1.3) in the following form

[TABLE]

It was shown in [49] that it is A-LWP in the space $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s\geq-\frac{13}{12}$ if $a_{1}a_{2}<0$. This is rather surprising since even the single KdV equation (1.8) is $C^{0}$-IP in $H^{s}(\mathbb{R})$ for any $s<-1$ (see [37]). Similar phenomenon was also found in the coupled mKdV systems [11].

For the cKdV (1.3) posed on $\mathbb{T}$ in divergence form, its A-LWP has been thoroughly investigated by Oh [40]. As a specific application, [40] obtained sharp A-LWP results for the Majda-Biello system (1.4) for any $a_{2}\neq 0$. The most interesting case is when $a_{2}\in(0,4]\setminus\{1\}$. In this case, (1.4) is A-LWP in $H_{0}^{s}(\mathbb{T})\times H^{s}(\mathbb{T})$ for any $s>\min\{1,\,\tilde{s}(a_{2})\}$, where $\tilde{s}(a_{2})\geq\frac{1}{2}$ is some constant only depending on $a_{2}$. On the other hand, (1.4) is $C^{3}$-IP if $s<\min\{1,\,\tilde{s}(a_{2})\}$. So the regularity threshold for the A-LWP of (1.4) in the space $H_{0}^{s}(\mathbb{T})\times H^{s}(\mathbb{T})$ is $\min\{1,\tilde{s}(a_{2})\}$.

By contrast, the study for the cKdV (1.3) posed on $\mathbb{T}$ in non-divergence form is far from complete. Angulo [2, 3] studied the Hirota-Satsuma system (1.5) and obtained the following results. When $a_{1}\neq 0,1$, (1.5) is A-LWP in $H^{s}_{0}(\mathbb{T})\times H^{s}_{0}(\mathbb{T})$ for any $s\geq 1$ and when $a_{1}=1$, (1.5) is A-LWP in $L^{2}_{0}(\mathbb{T})\times H^{1}_{0}(\mathbb{T})$. ***Actually, the mean-zero condition on $v$ is not needed, although it is needed on $u$. In fact, the mean-zero condition on $v$ is not applicable since the Hirota-Satsuma system (1.5) does not preserve the mean value of $v$. As a result, it makes more sense to adjust the spaces $H^{s}_{0}(\mathbb{T})\times H^{s}_{0}(\mathbb{T})$ and $L^{2}_{0}(\mathbb{T})\times H^{1}_{0}(\mathbb{T})$ in [2, 3] to be $H^{s}_{0}(\mathbb{T})\times H^{s}(\mathbb{T})$ and $L^{2}_{0}(\mathbb{T})\times H^{1}(\mathbb{T})$ respectively. Comparing to the known results on the cKdV systems in divergence form, there still exist a lot of room to explore and to optimize the regularity thresholds for the non-divergence cases.

1.3 Bilinear estimates

In Theorem 1.3, the optimal regularity index -1 for either the $\mathbb{R}$ case or the $\mathbb{T}$ case was obtained by the inverse scattering method [21, 33]. This method relies on the complete integrability of the single KdV equation and usually leads to $C^{0}$ well-posedness. Another method, called the bilinear estimate approach, does not require the complete integrability property and usually gives the sharp analytical well-posedness. For example, by using this method, [31] discovered the best regularity index for the A-LWP in the $\mathbb{R}$ case (and the $\mathbb{T}$ case resp.) to be $-\frac{3}{4}$ (and $-\frac{1}{2}$ resp.).

Since most cKdV systems do not possess the complete integrability property, the bilinear estimate approach seems to be the most powerful tool to use. This method is based on the Fourier restriction spaces $X^{\alpha}_{s,b}$ which was first introduced by Bourgain in [9].

Definition 1.4** ([9, 31]).**

Let $G=\mathbb{R}$ (or $\mathbb{T}$) and denote $G^{*}=\mathbb{R}$ (or $\mathbb{Z}$) to be the dual group of $G$. For any $\alpha,\,s,\,b\in\mathbb{R}$ with $\alpha\neq 0$, the Fourier restriction space $X^{\alpha}_{s,b}(G\times\mathbb{R})$ is defined to be the completion of the Schwartz space $\mathscr{S}(G\times\mathbb{R})$ with respect to the norm

[TABLE]

where $\langle\cdot\rangle:=1+|\cdot|$ and $\widehat{w}$ refers to the space-time Fourier transform of $w$.

Then the following is the so-called bilinear estimate for the KdV equation (1.8).

[TABLE]

where in the $\mathbb{T}$ case, $u$ and $v$ are also required to have zero means, i.e. $u(\cdot,t),\,v(\cdot,t)\in H^{s}_{0}(\mathbb{T})$ for any $t\in\mathbb{R}$.

Bourgain [9] first proved (1.11) for $s=0$ and $b=\frac{1}{2}$ in both $\mathbb{R}$ and $\mathbb{T}$ cases, which lead to the A-LWP of (1.8) in $H^{s}(\mathbb{R})$ or in $H^{s}_{0}(\mathbb{T})$ for any $s\geq 0$. Later, Kenig, Ponce and Vega[31] optimized this estimate.

Lemma 1.5** ([31]).**

• •

In the $\mathbb{R}$ case, (1.11) holds for any $s>-\frac{3}{4}$ with some $b=b(s)>\frac{1}{2}$, but fails for any $s<-\frac{3}{4}$ and $b\in\mathbb{R}$.

• •

In the $\mathbb{T}$ case, (1.11) holds for any $s\geq-\frac{1}{2}$ with $b=\frac{1}{2}$, but fails if $s<-\frac{1}{2}$ or $b\neq\frac{1}{2}$.

For the cKdV systems (1.3), the situations are more complicated since four types of bilinear estimates, (1.12)–(1.15), need to be studied.

Divergence forms:

[TABLE]

Non-divergence forms:

[TABLE]

where $w_{1}$ (or $w_{2}$) refers to $u$ or $v$, and $\alpha_{1}$ (or $\alpha_{2}$) represents $a_{1}$ or $a_{2}$ in (1.3). ††† Strictly speaking, in the $\mathbb{T}$ case, the above bilinear estimates need to be considered in more complicated forms and some zero-mean conditions should be imposed on $w_{1}$ and $w_{2}$. But in order to illustrate the idea more clearly, we drop these technical details and just take the forms which are consistent with the $\mathbb{R}$ case. For precise bilinear estimates in the $\mathbb{T}$ case, please see Corollary 1.16, Theorem 1.18 and Section 5.

There are various ways to write out these estimates, what we adopted here is to fix the first term on the right hand side to be $\|w_{1}\|_{X^{\alpha_{1}}_{s,b}}$.

In the $\mathbb{R}$ case, all the sharp regularity indices have been found for the bilinear estimates (1.12)–(1.15).

Lemma 1.6** ([1, 40, 49]).**

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}\setminus\{0\}$ and denote $r=\frac{\alpha_{2}}{\alpha_{1}}$. Then for any bilinear estimate among (1.12)–(1.15), there exists a critical index $s^{*}$, as shown in Table 1, such that this bilinear estimate holds for any $s>s^{*}$ with some $b=b(s)>\frac{1}{2}$, but fails for any $s<s^{*}$ and $b\in\mathbb{R}$.

From the above table, we can see there are four critical indices \big{\{}-\frac{13}{12},\,-\frac{3}{4},\,0,\,\frac{3}{4}\big{\}} for the bilinear estimates associated to the cKdV systems posed on $\mathbb{R}$.

In the $\mathbb{T}$ case, Oh[40] studied the Majda-Biello system[36] whose associated bilinear estimates are (1.16) and (1.17).

[TABLE]

Note that the above two estimates are special (essentially equivalent) cases of (1.12) and (1.13). In fact,

• •

(1.16) corresponds to (1.12) by setting $w_{1}=w_{2}=v$, $\alpha_{1}=a_{2}$ and $\alpha_{2}=1$.

• •

(1.17) corresponds to (1.13) by setting $w_{1}=v$, $w_{2}=u$, $\alpha_{1}=a_{2}$, and $\alpha_{2}=1$.

In both cases, the ratio between $\alpha_{2}$ and $\alpha_{1}$ is $r=\frac{1}{a_{2}}$. The most interesting case is when $a_{2}\in(0,4]\setminus\{1\}$, i.e. when r\in\big{[}\frac{1}{4},\infty\big{)}\setminus\{1\}. In this case, the key technical issue is to know how close the rational numbers can approximate a given real number. Oh[40] adopted the so-called minimal type index $\nu(\rho)$ for any real number $\rho$ (see Definition 1.14) to capture this approximation. Let $c_{1},c_{2}$ be the roots of the quadratic equation $3a_{2}x^{2}-3a_{2}x+a_{2}-1=0$ and let $d_{1},d_{2}$ be the roots of the quadratic equation $(1-a_{2})x^{2}+3a_{2}x-3a_{2}=0$. More specifically, let

[TABLE]

Denote

[TABLE]

Lemma 1.7** ([40]).**

Let $a_{2}\in(0,4]\setminus\{1\}$. Define $\nu_{c}$ and $\nu_{d}$ as in (1.19). Then for the bilinear estimate (1.16) or (1.17), there exists a critical index $s^{*}$, as shown in Table 2, such that this bilinear estimate holds for any $s>s^{*}$ with $b=\frac{1}{2}$, but fails for any $s<s^{*}$ and $b\in\mathbb{R}$.

Based on this discovery, Oh further concludes the sharp A-LWP for the Majda-Biello system (1.4).

Theorem 1.8** ([40]).**

Let $a_{2}\in(0,4]\setminus\{1\}$. Define

[TABLE]

Then (1.4) is A-LWP for any $s>s^{*}_{1}$, but $C^{3}$-IP for any $s<s^{*}_{1}$, in the space $H^{s}_{0}(\mathbb{T})\times H^{s}(\mathbb{T})$.

1.4 Weakly analytical ill-posedness

For the single KdV equation (1.8), it follows from Theorem 1.3 and Lemma 1.5 that the critical indices for the bilinear estimate (1.11) match those for the A-LWP of (1.8) in both $\mathbb{R}$ and $\mathbb{T}$ cases. For Majda-Biello system (1.4), we can also see from Lemma 1.7 and Theorem 1.8 that (1.4) is A-LWP if and only if both the corresponding bilinear estimates (1.16) and (1.17) hold.

Inspired by these observations, it is conjectured that for any cKdV system (1.3), its regularity threshold for the A-LWP is equivalent to that for the associating bilinear estimates. In fact, by the standard argument in [9, 31, 13], once the associated bilinear estimates are justified for some index $s$, then the A-LWP can also be established for the same $s$. In other words, if a cKdV system is A-IP, then at least one of the corresponding bilinear estimates must fail. But whether the failure of the bilinear estimates implies the A-IP is not known in general. This motivates the following definition.

Definition 1.9**.**

The Cauchy problem of the single KdV equation (1.8) is said to be weakly A-IP if the bilinear estimate (1.11) fails. Similarly, the cKdV systems (1.3) are said to be weakly A-IP if at least one of the corresponding bilinear estimates fails.

Based on the above definition, the aforementioned conjecture is translated to the equivalence between A-IP and weakly A-IP.

Conjecture 1.10**.**

For any cKdV system (1.3), it is A-IP if and only if it is weakly A-IP.

Since A-IP always implies weakly A-IP, the conjecture further reduces to “weakly A-IP impies A-IP”. For the single KdV and the Majda-Biello system, this conjecture has been confirmed. But for some special cKdV, say (1.9) posed on $\mathbb{R}$, although the critical index for the corresponding bilinear estimate has been found to be $-\frac{13}{12}$ in Lemma 1.6, it is still unknown if it is A-IP in $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s<-\frac{13}{12}$.

1.5 Critical index set

Consider the cKdV systems (1.3) posed on $\mathbb{R}$. Denote $\mathcal{H}^{s}(\mathbb{R})=H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$.

Definition 1.11**.**

For any cKdV system (1.3) posed on $\mathbb{R}$ with $a_{1}a_{2}\neq 0$ and either $C=(c_{ij})$ or $D=(d_{ij})$ does not vanish.

• •

If there exists $s^{*}\in\mathbb{R}$, depending on $a_{1}$, $a_{2}$, $C$ and $D$, such that (1.3) is A-LWP in the space $\mathcal{H}^{s}(\mathbb{R})$ for any $s>s^{*}$, but (weakly) A-IP for any $s<s^{*}$ in $\mathcal{H}^{s}(\mathbb{R})$, then this $s^{*}$ is called the (weakly) analytically critical index. If there does not exist such a critical index, then we define $s^{*}=\infty$.

• •

Regarding the above critical index $s^{*}$ as a map: $s^{*}={\cal G}(a_{1},a_{2},C,D)$, then the range of this map, denoted as $\mathcal{C}$, is called the (weakly) analytically critical index set of (1.3) in $\mathcal{H}^{s}(\mathbb{R})$.

In the following, we will write ”analytically critical index“ to be ”A-critical index“. Based on Definition 1.11, it follows from Lemma 1.6 that the weakly A-critical index set of (1.3) in $\mathcal{H}^{s}(\mathbb{R})$ is

[TABLE]

Now we consider the following (1.20) which is a variant of the single KdV equation.

[TABLE]

Similar to Definition 1.11, we can define the A-critical index set of (1.20) posed on $\mathbb{R}$ or $\mathbb{T}$. For any $l\geq 0$, we can also define the $C^{l}$-critical index set analogously. By some simple scaling computation, one can see that the values of $a$ and $c$ in (1.20) do not affect its well-posedness. So it follows from Theorem 1.3 that

• •

The $C^{0}$-critical index set of (1.20) in $H^{s}(\mathbb{R})$ or $H^{s}(\mathbb{T})$ is $\{-1\}$.

• •

The A-critical index set of (1.20) is \big{\{}-\frac{3}{4}\big{\}} in the space $H^{s}(\mathbb{R})$, \big{\{}-\frac{1}{2}\big{\}} in the space $H^{s}_{0}(\mathbb{T})$, and $\{\infty\}$ in the space $H^{s}(\mathbb{T})$.

Two observations can be drawn from the above results. Firstly, unlike (1.20), the coefficients of the cKdV systems (1.3) do have an effect on the A-critical index. Secondly, if the problem is posed on $\mathbb{T}$, then the critical index can be different if the underlying spaces are different (e.g. $H^{s}_{0}(\mathbb{T})$ cf. $H^{s}(\mathbb{T})$).

The main purpose of this paper is to investigate the weakly A-critical index set of the cKdV systems (1.3) posed on $\mathbb{T}$. We will consider the following four spaces:

[TABLE]

Definition 1.12**.**

Let $k\in\{1,2,3,4\}$. We define $\mathcal{C}_{k}$ to be the weakly A-critical index set of the cKdV systems (1.3) in the space $\mathcal{H}^{s}_{k}$.

From the results in Oh[40] on the Majda-Biello system (1.4), the weakly A-critical index $s^{*}(a_{2})$ in the space $\mathcal{H}^{s}_{2}$ is

[TABLE]

where $\tilde{s}(a_{2})\geq\frac{1}{2}$ is some constant only depending on $a_{2}$. So the weakly A-critical index set for (1.4) in the space $\mathcal{H}^{s}_{2}$ is

[TABLE]

But what is the precise range of $\tilde{s}(a_{2})$ over the region $(0,4]\backslash\{1\}$? This question was not answered in [40], we will find an answer in this paper.

1.6 Irrationality exponent

In order to find out the critical index set for (1.3) posed on $\mathbb{T}$, as we mentioned before, it is crucial to estimate how well the real number can be approximated by rational functions. In the number theory, such estimate is called the Diophantine approximation. ‡‡‡The interested readers are referred to S. Lang [35] and Y. Bugeaud [10] for a nice introduction to this subject.

One standard characterization is via the irrationality exponent.

Definition 1.13** (see e.g. [5]).**

A real number $\rho$ is said to be approximable with power $\mu$ if the inequality

[TABLE]

holds for infinitely many $(m,n)\in\mathbb{Z}\times\mathbb{Z}^{*}$, and

[TABLE]

is called the irrationality exponent of $\rho$.

For any rational number $\rho$, $\mu(\rho)=1$. For any irrational number $\rho$, $\mu(\rho)\geq 2$, but exact value of $\mu(\rho)$ is difficult to find in general. The basic properties of $\mu(\rho)$ are collected in Proposition 3.1. Recalling the minimal type index $\nu(\rho)$ in Oh’s paper [40], it is closely related to $\mu(\rho)$ but is defined slightly different.

Definition 1.14** (see e.g. [4, 40]).**

A real number $\rho$ is said to be of type $\nu$ if there exists a positive constant $K=K(\rho,\nu)$ such that the inequality

[TABLE]

holds for any $(m,n)\in\mathbb{Z}\times\mathbb{Z}^{*}$, and

[TABLE]

is called the minimal type index of $\rho$, where the infimum is understood as $\infty$ if the set \{\nu\in\mathbb{R}:\text{\rho$is of type$\nu}\} is empty.

If $\rho\in\mathbb{Q}$, then it is easy to see $\nu(\rho)=\infty$ and $\mu(\rho)=1$. If $\rho\in\mathbb{R}\setminus\mathbb{Q}$, then it will be shown in Proposition 3.5 that $\nu(\rho)=\mu(\rho)-2$.

1.7 Main results

Recall the irrationality exponent $\mu(\cdot)$ in Definition 1.13. For $r\geq\frac{1}{4}$, define

[TABLE]

The basic properties of $s_{r}$ are collected in Proposition 3.6.

For the Majda-Biello system (1.4) with $a_{2}\in(0,4]\setminus\{1\}$, Oh[40] showed the critical regularity indexes for the associated bilinear estimates to be \min\big{\{}1,\frac{1}{2}+\frac{1}{2}\nu_{c}\big{\}} and \min\big{\{}1,\frac{1}{2}+\frac{1}{2}\nu_{d}\big{\}}, see Table 2. The first main result of this paper is to demonstrate these two indexes are actually the same.

Theorem 1.15**.**

Let $a_{2}\in(0,4]\setminus\{1\}$ and define $\nu_{c}$ and $\nu_{d}$ as in (1.19). Then $\nu_{c}=\nu_{d}$ and

[TABLE]

where $s_{\frac{1}{a_{2}}}$ is defined as in (1.26).

The key observation in the proof of this theorem is the invariance of the irrationality exponent under the reciprocal operation, that is $\mu(\rho)=\mu(1/\rho)$, see Proposition 3.4.

Combining Theorem 1.15 with Lemma 1.7 (also see Oh[40]), we are able to write out the sharp regularity indexes for the bilinear estimates (1.12) and (1.13) in a more unified way.

Corollary 1.16**.**

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}\setminus\{0\}$ and denote $r=\frac{\alpha_{2}}{\alpha_{1}}$. Define $s_{r}$ as in (1.26) for $r\geq\frac{1}{4}$. Then for the bilinear estimate (1.12) or (1.13), there exists a critical index $s^{*}$, as shown in Table 3, such that this bilinear estimate holds for any $s>s^{*}$ with $b=\frac{1}{2}$, but fails for any $s<s^{*}$ and $b\in\mathbb{R}$.

Let $\mathcal{U}$ be the range of $s_{r}$: ${\cal U}=\left\{s_{r}:\frac{1}{4}\leq r<\infty\right\}$. Then it will be shown in Proposition 3.6 that \mathcal{U}=\big{[}\frac{1}{2},1\big{]}. As a result, we obtain the following conclusion (Recall that $H^{s}_{2}:=H^{s}_{0}(\mathbb{T})\times H^{s}(\mathbb{T})$ in (1.21)).

Corollary 1.17**.**

For the Majda-Biello systems (1.4) posed on $\mathbb{T}$ with $a_{2}\neq 0$, its weakly A-critical index set in the space $H^{s}_{2}$ is \big{\{}-\frac{1}{2},\infty\big{\}}\bigcup\big{[}\frac{1}{2},1\big{]}.

Next, we study the cKdV systems (1.3) in non-divergence form. The main result is the following sharp bilinear estimates.

Theorem 1.18**.**

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}\setminus\{0\}$ and denote $r=\frac{\alpha_{2}}{\alpha_{1}}$. Define $s_{r}$ as in (1.26) for $r\geq\frac{1}{4}$. Then for the bilinear estimate (1.14) or (1.15), there exists a critical index $s^{*}$, as shown in Table 4, such that this bilinear estimate holds for any $s>s^{*}$ with $b=\frac{1}{2}$, but fails for any $s<s^{*}$ and $b\in\mathbb{R}$.

As a corollary, consider the Hirota-Satusma systems (1.5).

Corollary 1.19**.**

For the Hirota-Satsuma systems (1.5) posed on $\mathbb{T}$ with $a_{1}\neq 0$ and $c_{12}\in\mathbb{R}$, its weakly A-critical index set in the space $H^{s}_{2}$ is \big{\{}-\frac{1}{4},\infty\big{\}}\bigcup\big{[}\frac{1}{2},1\big{]}.

The more detailed well-posedness results on the Hirota-Satsuma system are shown in the appendix. Based on the results in Table 3 and 4, we can actually study the well-posedness of cKdV systems (1.3) in the space $\mathcal{H}^{s}_{k}$ for any $k\in\{1,2,3,4\}$. The following is a summary of their weakly A-critical index sets.

Theorem 1.20**.**

The weakly A-critical index sets $\mathcal{C}_{k}\,(k=1,2,3,4)$ in Definition 1.12 are

[TABLE]

Fix any $k\in\{1,2,3,4\}$. The above result enables us to provide a complete classification of the cKdV systems (1.3) in $H^{s}_{k}$.

Theorem 1.21** (Classification of the systems (1.3)).**

Assume $a_{1}a_{2}\neq 0$ and either $C$ or $D$ does not vanish. Fix $k\in\{1,2,3,4\}$ and define $\mathcal{H}^{s}_{k}$ as in (1.21). Then the systems (1.3) are completely classified into a family of classes, each of which corresponds to a unique index $s^{*}\in{\cal C}_{k}$ such that any system in this class is A-LWP in the space ${\cal H}^{s}_{k}$ if $s>s^{*}$ while it is weakly A-IP if $s<s^{*}$.

1.8 Organization of the paper

The organization of the rest of the paper is as follows. In Section 2, we will present the definitions of the Fourier restriction spaces and the resonance functions. Section 3 is devoted to explore properties of the irrationality exponents and prove Theorem 1.15. Some linear estimates will be introduced in Section 4. Theorem 1.18 will be broken into Lemma 5.1, and 5.2 and Proposition 5.3 and 5.4 in Section 5. We will justify Lemma 5.1 and 5.2 in Section 6 and prove Proposition 5.3 and 5.4 in Section 7. Finally, Appendix A includes the analytical well-posedness results about the Hirota-Satsuma systems (1.5) which can be obtained as a corollary of Theorem 1.18.

2 Fourier restriction spaces on $\mathbb{T}$

To study the LWP of (1.3), we adopt the similar treatment as in [13] to deal with more general periodic problem posed on $\mathbb{T}_{\lambda}=\mathbb{R}/(2\pi\lambda\mathbb{Z})$ for $\lambda\geq 1$ and thus consider the system

[TABLE]

We use $\mathcal{F}_{x}$, $\mathcal{F}_{t}$ and $\mathcal{F}$ to denote the spatial, temporal and space-time Fourier transform respectively. However, when there is no confusion, we simply use $\,\hat{}\,$ to denote any of these three types of Fourier transforms. On the other hand, $\mathcal{F}_{x}^{-1}$, $\mathcal{F}_{t}^{-1}$ and $\mathcal{F}^{-1}$ represent the corresponding inverse Fourier transforms.

The temporal Fourier transform and its inverse are defined standardly. The definitions of the spatial Fourier transform and its inverse are more complicated. Denote the frequency space corresponding to $\mathbb{T}_{\lambda}$ by \mathbb{Z}_{\lambda}=\big{\{}k\,\big{|}\,k=n/\lambda\,\,\,\text{for some n\in\mathbb{Z}}\big{\}}. The normalized counting measure $dk^{\lambda}$ on $\mathbb{Z}_{\lambda}$ is defined by

[TABLE]

For any function $f(x)$ on $\mathbb{T}_{\lambda}$,

[TABLE]

On the other hand, for any functon $g(k)$ on $\mathbb{Z}_{\lambda}$,

[TABLE]

Consequently, the norm for the Sobolev space $H^{s}(\mathbb{T}_{\lambda})$ is defined as

[TABLE]

where $\langle\cdot\rangle:=1+|\cdot|$. The homogeneous subspace $H_{0}^{s}(\mathbb{T}_{\lambda})$ is defined as $H_{0}^{s}(\mathbb{T}_{\lambda})=\{f\in H^{s}(\mathbb{T}_{\lambda}):\hat{f}(0)=0\}$. Finally, the space-time Fourier transform and its inverse are defined as $\mathcal{F}=\mathcal{F}_{t}\mathcal{F}_{x}$ and $\mathcal{F}^{-1}=\mathcal{F}_{x}^{-1}\mathcal{F}_{t}^{-1}$.

For any $\alpha\neq 0$ and $\lambda\geq 1$, consider

[TABLE]

The solution to (2.6) is given explicitly by

[TABLE]

with

[TABLE]

The following is a generalized version of (1.4) for the definition of the Fourier restriction spaces on $\mathbb{T}$.

Definition 2.1** ([9, 31, 13]).**

For any $\alpha,\,s,\,b,\,\lambda\in\mathbb{R}$ with $\alpha\neq 0$ and $\lambda\geq 1$, the Fourier restriction space $X^{\alpha}_{s,b,\lambda}$ is defined to be the completion of the Schwartz space $\mathscr{S}(\mathbb{T}_{\lambda}\times\mathbb{R})$ with respect to the norm

[TABLE]

where $\widehat{w}$ refers to the space-time Fourier transform of $w$.

It has been pointed out in [31] that one needs to take $b=\frac{1}{2}$ for the periodic case. However, this space barely fails to be in C\big{(}\mathbb{R}_{t};H^{s}_{x}\big{)}. To ensure the continuity of the time flow of the solution, a smaller space $Y^{\alpha}_{s,\lambda}$ will be used via the norm

[TABLE]

Since the second term $\|\langle k\rangle^{s}\widehat{w}(k,\tau)\|_{L^{2}(\mathbb{Z}_{\lambda};L^{1}(\mathbb{R}))}$ has already dominated the $L^{\infty}_{t}H^{s}_{x}$ norm of $w$, it follows that Y^{\alpha}_{s,\lambda}\subseteq C\big{(}\mathbb{R}_{t};H^{s}_{x}\big{)}. The companion spaces $Z^{\alpha}_{s,\lambda}$ via the norm (2.11) is then introduced to control the $Y^{\alpha}_{s,\lambda}$ norm of the integral term from the Duhamel principle (see Lemma 4.1).

[TABLE]

For convenience, we will drop $\lambda$ when it equals $1$. That is, $X^{\alpha}_{s,b}:=X^{\alpha}_{s,b,1}$, $Y^{\alpha}_{s}:=Y^{\alpha}_{s,1}$ and $Z^{\alpha}_{s}:=Z^{\alpha}_{s,1}$. Throughout this paper,

[TABLE]

Definition 2.2** ([46]).**

Let $(\alpha_{1},\alpha_{2},\alpha_{3})$ be a triple in $(\mathbb{R}^{*})^{3}$. Define the resonance function $H$ associated to this triple by

[TABLE]

where $\phi^{\alpha_{i}}(k_{i})$ is as defined in (2.8). The resonance set of $H$ is defined to be the zero set of $H$, that is

[TABLE]

3 Proof of Theorem 1.15

We first collect some classical results about the irrationality exponent $\mu(\rho)$.

Proposition 3.1**.**

1. (a)

If $\rho\in\mathbb{Q}$, then $\mu(\rho)=1$; 2. (b)

If $\rho\in\mathbb{R}\setminus\mathbb{Q}$, then $\mu(\rho)\geq 2$; 3. (c)

If $\rho$ is an irrational algebraic number, then $\mu(\rho)=2$; 4. (d)

For almost every $\rho\in\mathbb{R}$, $\mu(\rho)=2$; 5. (e)

The function $\mu$ maps $\mathbb{R}$ onto $\{1\}\cup[2,\infty]$.

Proof.

Part (a) and (b) are standard. Part (c) is the famous Thue-Siegel-Roth theorem [48, 43, 42]. Part (d) is the Khintchine theorem [32]. Part (e) was proved by Jarnik[19, 20] using the theory of continued fractions. ∎

Next, we present two technical lemmas concerning the irrationality exponent function $\mu$.

Lemma 3.2**.**

If a real number $\rho$ is approximable with power $\mu$, then there exist $\{(m_{j},n_{j})\}_{j=1}^{\infty}\subset\mathbb{Z}\times\mathbb{Z}^{*}$ with the following two properties.

1. (1)

$\{n_{j}\}_{j=1}^{\infty}$* is an increasing positive sequence and $\lim\limits_{j\rightarrow\infty}n_{j}=\infty$;* 2. (2)

For each $j\geq 1$, $(m_{j},n_{j})$ satisfies

[TABLE]

Proof.

The conclusion follows from Definition 1.13 and the observation that for any fixed $n\in\mathbb{Z}^{*}$, there are at most finitely many $m\in\mathbb{Z}$ such that

[TABLE]

∎

Lemma 3.3**.**

Let $\rho\in\mathbb{R}\setminus\mathbb{Q}$ with $\mu(\rho)<\infty$. Then for any $\epsilon>0$, there exists a constant $K=K(\rho,\epsilon)>0$ such that the inequality

[TABLE]

holds for any $(m,n)\in\mathbb{Z}\times\mathbb{Z}^{*}$, where $\mu_{\epsilon}=\mu(\rho)+\epsilon$.

Proof.

Since $\mu_{\epsilon}=\mu(\rho)+\epsilon>\mu(\rho)$, it follows from Definition 1.13 that there exist at most finitely many $(m,n)\in\mathbb{Z}\times\mathbb{Z}^{*}$ such that

[TABLE]

In addition, since $\rho$ is an irrational number, \big{|}\rho-\frac{m}{n}\big{|} is never zero. Therefore, by choosing a sufficiently small constant $K=K(\rho,\epsilon)$, (3.1) holds for any $(m,n)\in\mathbb{Z}\times\mathbb{Z}^{*}$. ∎

Now some invariant properties of the irrationality exponent $\mu$ will be justified.

Proposition 3.4**.**

1. (a)

For any $\sigma\in\mathbb{Q}^{*}$ and $\rho\in\mathbb{R}$, $\mu(\sigma\rho)=\mu(\rho)$. 2. (b)

For any $\sigma\in\mathbb{Q}$ and $\rho\in\mathbb{R}$, $\mu(\rho+\sigma)=\mu(\rho)$. 3. (c)

For any $\rho\in\mathbb{R}^{*}$, \mu\big{(}\frac{1}{\rho}\big{)}=\mu(\rho).

Proof.

As part (a) and (b) are obvious, we will only prove part (c). First, by taking advantage of (a), we may assume $\rho>0$. Then due to symmetry, it reduces to prove

[TABLE]

If $\rho\in\mathbb{Q}$, then it is trivial. So we further assume $\rho$ is an irrational number, which implies \mu\big{(}\frac{1}{\rho}\big{)}\geq 2. Then it suffices to show that if $\rho$ is approximable with some power $\mu\geq 2$, then $\frac{1}{\rho}$ is approximable with the power $\mu-\epsilon$ for any $\epsilon>0$.

Let $\rho$ be approximable with some power $\mu\geq 2$ and fix any $\epsilon>0$. It follows from Lemma 3.2 that there exists a sequence $\{(m_{j},n_{j})\}_{j=1}^{\infty}\subset\mathbb{Z}\times\mathbb{Z}^{*}$ such that $n_{j}>0$, $\lim\limits_{j\rightarrow\infty}n_{j}=\infty$ and

[TABLE]

Since $\rho>0$ and $\mu\geq 2$, it immediately yields that $\lim\limits_{j\rightarrow\infty}m_{j}=\infty$. In addition, for sufficiently large $j$, we have $0<\frac{m_{j}}{n_{j}}<2\rho$. Hence, for any such $j$, noticing

[TABLE]

we obtain

[TABLE]

Finally, since $\lim\limits_{j\rightarrow\infty}m_{j}=\infty$, when $j$ is large enough, $m_{j}^{\epsilon}>(2\rho)^{\mu-1}/\rho$. Therefore,

[TABLE]

Since the above inequality is valid for any large $j$, $\frac{1}{\rho}$ is approximable with the power $\mu-\epsilon$. ∎

Finally, we discuss the relation between the irrationality exponent $\mu(\rho)$ in Definition 1.13 and the minimal type index $\nu(\rho)$ in Definition 1.14.

Proposition 3.5**.**

1. (a)

If $\rho\in\mathbb{Q}$, then $\nu(\rho)=\infty$ and $\mu(\rho)=1$; 2. (b)

If $\rho\in\mathbb{R}\setminus\mathbb{Q}$, then $\nu(\rho)=\mu(\rho)-2$.

Proof.

Part (a) is obvious, so we will only prove part (b). Let $\rho\in\mathbb{R}\setminus\mathbb{Q}$. We will first show $\nu(\rho)\geq\mu(\rho)-2$. If $\rho$ is approximable with power $\mu$, then it follows from Lemma 3.2 that there exists a sequence $\{(m_{j},n_{j})\}_{j=1}^{\infty}\subset\mathbb{Z}\times\mathbb{Z}^{*}$ such that $n_{j}>0$, $\lim\limits_{j\rightarrow\infty}n_{j}=\infty$ and

[TABLE]

As a result, for any $\epsilon>0$, there does not exist $K=K(\rho,\epsilon)$ such that

[TABLE]

holds for all $j\geq 1$. In other words, $\rho$ is not of type $\mu-2-\epsilon$ according to Definition 1.14. Therefore, $\nu(\rho)\geq\mu-2-\epsilon$. Sending $\epsilon\rightarrow 0^{+}$ and then taking supremum with respect to $\mu$ leads to $\nu(\rho)\geq\mu(\rho)-2$.

Now if $\mu(\rho)=\infty$, then it follows from $\nu(\rho)\geq\mu(\rho)-2$ that $\nu(\rho)=\infty$. So in the following, we just assume $\mu(\rho)<\infty$ and intend to show $\nu(\rho)\leq\mu(\rho)-2$. According to Lemma 3.3, for any $\epsilon>0$, there exists a constant $K=K(\rho,\epsilon)>0$ such that

[TABLE]

holds for any $(m,n)\in\mathbb{Z}\times\mathbb{Z}^{*}$. Hence, $\rho$ is of type $\mu(\rho)+\epsilon-2$, which implies $\nu(\rho)\leq\mu(\rho)+\epsilon-2$. Sending $\epsilon\rightarrow 0^{+}$ yields $\nu(\rho)\leq\mu(\rho)-2$. ∎

For $r\geq\frac{1}{4}$, denote $\sigma_{r}$ and $s_{r}$ as in (1.26). That is,

[TABLE]

The properties of $s_{r}$ can be derived from Proposition 3.1.

Proposition 3.6**.**

Let $r\geq\frac{1}{4}$ be given. Then

1. (a)

$s_{r}=1$* if $\sqrt{12r-3}\in\mathbb{Q}$;* 2. (b)

$\frac{1}{2}\leq s_{r}\leq 1$* for any $r\geq\frac{1}{4}$;* 3. (c)

$s_{r}=\frac{1}{2}$* if $r$ is an algebraic number and $\sqrt{12r-3}\notin\mathbb{Q}$.* 4. (d)

$s_{r}=\frac{1}{2}$* for almost every $r\geq\frac{1}{4}$;* 5. (e)

The range of $s_{r}$ over $r\in[\frac{1}{4},\infty)$ is $[\frac{1}{2},1]$.

Proof.

1. (a)

When $\sqrt{12r-3}\in\mathbb{Q}$, it follows from Proposition 3.1(a) that $\sigma_{r}=1$. Therefore, $s_{r}=1$. 2. (b)

This part is obviously due to the definition (1.26) for $s_{r}$. 3. (c)

When $r$ is an algebraic number, $\sqrt{12r-3}$ is also an algebraic number. Now if $\sqrt{12r-3}\notin\mathbb{Q}$, then it follows from Proposition 3.1(c) that $\sigma_{r}=2$. Hence, $s_{r}=\frac{1}{2}$. 4. (d)

By Proposition 3.1(d), $\sigma_{r}=2$ for almost every $r\geq\frac{1}{4}$. Thus, $s_{r}=\frac{1}{2}$ for almost every $r\geq\frac{1}{4}$. 5. (e)

Combining Proposition 3.1(e) with Proposition 3.4(a), we conclude the range of $\sigma_{r}$ is $\{1\}\cup[2,\infty]$. As a result, the range of $s_{r}$ is $[\frac{1}{2},1]$.

∎

Now we are ready to prove Theorem 1.15.

Proof of Theorem 1.15.

Let $a_{2}\in(0,4]\setminus\{1\}$. Recall $\nu_{c}=\max\{\nu(c_{1}),\nu(c_{2})\}$, where

[TABLE]

and recall $\nu_{d}=\max\{\nu(d_{1}),\nu(d_{2})\}$, where

[TABLE]

Denote $r=\frac{1}{a_{2}}$ and $\rho_{r}=\sqrt{12r-3}$. Then $r\geq\frac{1}{4}$ and it follows from (3.2) that $\sigma_{r}=\mu(\rho_{r})$. In addition, based on Proposition 3.1,

[TABLE]

Similarly, $\mu(c_{2})=\sigma_{r}$. On the other hand, noticing that $d_{1}=\frac{1}{c_{1}}$ and $d_{2}=\frac{1}{c_{2}}$, so it follows from Proposition 3.4 that $\mu(d_{1})=\mu(c_{1})$ and $\mu(d_{2})=\mu(c_{2})$. Hence,

[TABLE]

Then according to Proposition 3.5, $\nu(d_{1})=\nu(d_{2})=\nu(c_{1})=\nu(c_{2})$. In particular, $\nu_{d}=\nu_{c}=\nu(c_{1})$.

Next, we will justify (1.27). Denote

[TABLE]

Then it suffices to prove $\widetilde{s}=s_{r}$.

• •

Case 1: $\rho_{r}\in\mathbb{Q}$. In this case, $c_{1}\in\mathbb{Q}$, so it follows from Proposition 3.5 that $\nu(c_{1})=\infty$. Therefore, $\widetilde{s}=1$. On the other hand, by Proposition 3.6(a), we also have $s_{r}=1$.

• •

Case 2: $\rho_{r}\in\mathbb{R}\setminus\mathbb{Q}$. In this case, $c_{1}\in\mathbb{R}\setminus\mathbb{Q}$, so it follows from Proposition 3.5 that $\nu(c_{1})=\mu(c_{1})-2$, which implies $\nu(c_{1})=\sigma_{r}-2$. Putting this into (3.4) yields

[TABLE]

Since $\rho_{r}\in\mathbb{R}\setminus\mathbb{Q}$, then Proposition 3.1(b) implies $\sigma_{r}=\mu(\rho_{r})\geq 2$. Thus, we conclude from (3.2) that

[TABLE]

∎

4 Linear estimates

Let $\psi\in C_{0}^{\infty}(\mathbb{R})$ be a bump function supported on $[-2,2]$ and $\psi=1$ on $[-1,1]$. We first present two linear estimates, one is for the solution to the homogeneous linear KdV equation (2.6), and another one is for the solution to the forced linear KdV equation (2.6) with the right hand side being $F$ instead of 0. Recall the notation $S^{\alpha}_{\lambda}$ as defined in (2.7).

Lemma 4.1**.**

There exists a constant $C$ which only depends on the bump function $\psi$ such that for any $\alpha,s,\lambda\in\mathbb{R}$ with $\alpha\neq 0$ and $\lambda\geq 1$,

[TABLE]

and

[TABLE]

The proof of the above lemma is almost the same as those for Lemma 7.1 and Lemma 7.2 in [13], so we omit it. Next, we provide two well-known embedding results.

Lemma 4.2**.**

There exists a universal constant $C$ such that for any $\alpha\in\mathbb{R}^{*}$, $\lambda\geq 1$ and for any function $g$ on $\mathbb{T}_{\lambda}\times\mathbb{R}$,

[TABLE]

The proof of this estimate can be found in ([39], Lemma 2.3.2).

Lemma 4.3**.**

Let $\alpha\in\mathbb{R}^{*}$. Then there exists a constant $C=C(\alpha)$ such that for any $\lambda\geq 1$ and for any function $g$ on $\mathbb{T}_{\lambda}\times\mathbb{R}$,

[TABLE]

When $\alpha=1$ and $\lambda=1$, Lemma 4.3 was first proved by Bourgain in [9] for a version when the left hand side of (4.4) is localized in time. Then Tao removed such a restriction in ([46], Proposition 6.4). Later, a more elementary proof was provided by Oh in his online note [41]. Actually, similar method had been applied earlier to the Schrödinger equation (see e.g. [47], Proposition 2.13).

5 Bilinear estimates in non-divergence form

This section will present the rigorous version of the bilinear estimates (1.14) and (1.15) in non-divergence form. Meanwhile, Theorem 1.18 will be broken into Lemma 5.1, 5.2 and Proposition 5.3, 5.4 in more general settings. We denote $s_{r}$ as in (1.26) for any $r\geq\frac{1}{4}$.

Lemma 5.1**.**

Let $\lambda\geq 1$ and $\alpha_{1},\alpha_{2}\in\mathbb{R}^{*}$ with $r:=\frac{\alpha_{2}}{\alpha_{1}}$. Assume one of the conditions below is satisfied.

1. (a)

$r<\frac{1}{4}$, $s\geq-\frac{1}{4}$ and $p>0$; 2. (b)

$r=1$, $s\geq\frac{1}{2}$ and $p>0$; 3. (c)

$r\in[\frac{1}{4},\infty)\setminus\{1\}$, $s\geq 1$ and $p>0$; 4. (d)

$r\in[\frac{1}{4},\infty)\setminus\{1\}$* with $s_{r}<1$, $s>s_{r}$ and $p>s_{r}$.*

Then there exist constants $\epsilon=\epsilon(\alpha_{1},\alpha_{2})$ and $C=C(\alpha_{1},\alpha_{2},s,p)$ such that

[TABLE]

for any $w_{1}\in X_{s,\frac{1}{2},\lambda}^{\alpha_{1}}$ and $w_{2}\in X_{s,\frac{1}{2},\lambda}^{\alpha_{2}}$ with $\widehat{w_{2}}(0,\cdot)=0$.

Lemma 5.2**.**

Let $\lambda\geq 1$ and $\alpha_{1},\alpha_{2}\in\mathbb{R}^{*}$ with $r:=\frac{\alpha_{2}}{\alpha_{1}}$. Assume one of the conditions below is satisfied.

1. (a)

$r<\frac{1}{4}$* and $s\geq-\frac{1}{4}$ and $p>0$;* 2. (b)

$r=1$, $s\geq\frac{1}{2}$, $p>0$ and $\widehat{w_{1}}(0,\cdot)=0$; 3. (c)

$r\in[\frac{1}{4},\infty)\setminus\{1\}$, $s\geq 1$ and $p>0$; 4. (d)

$r\in[\frac{1}{4},\infty)\setminus\{1\}$* with $s_{r}<1$, $s>s_{r}$ and $p>s_{r}$.*

Then there exist constants $\epsilon=\epsilon(\alpha_{1},\alpha_{2})$ and $C=C(\alpha_{1},\alpha_{2},s,p)$ such that

[TABLE]

for any $w_{1}\in X_{s,\frac{1}{2},\lambda}^{\alpha_{1}}$ and $w_{2}\in X_{s,\frac{1}{2},\lambda}^{\alpha_{2}}$.

Next, we will address the sharpness of Lemma 5.1 and Lemma 5.2. Without loss of generality, we take $\lambda=1$.

Proposition 5.3**.**

The bilinear estimate

[TABLE]

fails for any $b\in\mathbb{R}$ (and hence (5.1) fails by taking $b=\frac{1}{2}$) under any of the following conditions.

• (a)

$r<\frac{1}{4}$* and $s<-\frac{1}{4}$;*

• (b)

$r=1$* and $s<\frac{1}{2}$;*

• (c)

$r\in[\frac{1}{4},\infty)\setminus\{1\}$* and $s<s_{r}$;*

• (d)

$r\in\mathbb{R}^{*}$, $s\in\mathbb{R}$, but without the restriction $\widehat{w_{2}}(0,\cdot)=0$ in (5.3).

Proposition 5.4**.**

The bilinear estimate

[TABLE]

fails for any $b\in\mathbb{R}$ (and hence (5.2) fails by taking $b=\frac{1}{2}$) under any of the following conditions.

• (a)

$r<\frac{1}{4}$* and $s<-\frac{1}{4}$;*

• (b)

$r=1$, $s<\frac{1}{2}$ and $\widehat{w_{1}}(0,\cdot)=0$;

• (c)

$r=1$, $s\in\mathbb{R}$, but without the restriction $\widehat{w_{1}}(0,\cdot)=0$;

• (d)

$r\in[\frac{1}{4},\infty)\setminus\{1\}$* and $s<s_{r}$.*

6 Proofs of the bilinear estimates

The goal of this section is to prove Lemma 5.1 and Lemma 5.2. Since their proofs are similar, we will only justify Lemma 5.2.

6.1 Idea of the proof

Without loss of generality, we consider the following simpler version of (5.2) with $\lambda=1$,

[TABLE]

By duality and Plancherel identity, in order to verify (6.1), it suffices to prove (see [46] or Lemma 6.1)

[TABLE]

where $\langle\cdot\rangle=1+|\cdot|$, A=\Big{\{}(k_{1},k_{2},k_{3},\tau_{1},\tau_{2},\tau_{3})\in\mathbb{Z}^{3}\times\mathbb{R}^{3}:\sum\limits_{i=1}^{3}k_{i}=\sum\limits_{i=1}^{3}\tau_{i}=0\Big{\}} and

[TABLE]

where $\phi^{\alpha}(k)=\alpha k^{3}$ is as defined in (2.8).

In (6.2), the loss of the spatial derivative in the bilinear estimate (6.1) is reflected via the term $\dfrac{|k_{2}|\langle k_{3}\rangle^{s}}{\langle k_{1}\rangle^{s}\langle k_{2}\rangle^{s}}$ and the gain of the time derivative is reflected via the term $\langle L_{1}\rangle^{\frac{1}{2}}\langle L_{2}\rangle^{\frac{1}{2}}\langle L_{3}\rangle^{\frac{1}{2}}$. How to compensate the loss of the spatial derivative from the gain of the time derivative is the key point. Denote

[TABLE]

Then we need to control $R_{1}$ by $R_{2}$. Since $\sum\limits_{i=1}^{3}k_{i}=0$, then $\langle k_{3}\rangle\leq\langle k_{1}\rangle\langle k_{2}\rangle$. As a result, $R_{1}$ is decreasing in $s$, which means the larger $s$ is, the more likely the bilinear estimate will hold. So the interest lies in the search for the smallest $s$ such that the bilinear estimate holds. Noticing that $L_{i}$ contains the time variable $\tau_{i}$, a single $L_{i}$ alone can barely have any contributions to control $R_{1}$. On the other hand, as $\sum\limits_{i=1}^{3}\tau_{i}=0$,

[TABLE]

is a function of $\{k_{i}\}_{i=1,2,3}$ only. Because of this, we define the resonance function $H_{2}$ as in Definition 2.2:

[TABLE]

Since $R_{2}\gtrsim\langle H_{2}\rangle^{\frac{1}{2}}$, then it is easier to control $R_{1}$ by $R_{2}$ in the region where $H_{2}$ is large. The situation becomes more complicated near the region where $H_{2}$ vanishes. The zero set of $H_{2}$ is called the resonance set as in Definition 2.2.

By writing $k_{3}=-k_{1}-k_{2}$ in (6.3) and simplifying,

[TABLE]

If $k_{2}=0$, then $H_{2}(k_{1},k_{2},k_{3})=0$. If $k_{2}\neq 0$, then $H_{2}$ can be rewritten as

[TABLE]

where

[TABLE]

The following is the classification of the roots of $h_{r}$ depending on the values of $r$.

1. (1)

$r<\frac{1}{4}$: $h_{r}(x)$ does not have real roots; 2. (2)

$r=\frac{1}{4}$: $h_{r}(x)$ has a unique root $-\frac{1}{2}$; 3. (3)

$r>\frac{1}{4}$ but $r\neq 1$: $h_{r}(x)$ has two roots, neither of which equals -1 or 0; 4. (4)

$r=1$: $h_{r}(x)$ has two roots -1 and 0.

Among the above four situations, Case (3) is the most interesting one, so we will first focus on this case. Assume r\in\big{(}\frac{1}{4},\infty\big{)}\setminus\{1\} and denote the two roots of $h_{r}(x)$ as $x_{1r}$ and $x_{2r}$, i.e.

[TABLE]

Define $\sigma_{r}$ and $s_{r}$ as in (1.26). Then it follows from Proposition 3.4 that

[TABLE]

In addition, the resonance function $H_{2}$ can be written as

[TABLE]

As a result, the resonance set consists of three lines: $k_{2}=0$, $k_{1}=x_{1r}k_{2}$ and $k_{1}=x_{2r}k_{2}$. The most difficult estimate is near the line $k_{1}=x_{1r}k_{2}$ or $k_{1}=x_{2r}k_{2}$. Without loss of generality, let us consider the region near the line $k_{1}=x_{1r}k_{2}$. In this situation, \big{|}\frac{k_{1}}{k_{2}}-x_{2r}\big{|}\approx|x_{1r}-x_{2r}| which is a positive constant. Hence,

[TABLE]

In addition, when $\frac{k_{1}}{k_{2}}$ is very close to $x_{1r}$, we have $|k_{1}|\sim|k_{2}|\sim|k_{3}|$ . Consequently,

[TABLE]

On the other hand, noticing that both $k_{1}$ and $k_{2}$ are integers, the estimate of \big{|}x_{1r}-\frac{k_{1}}{k_{2}}\big{|} reduces to the problem of the Diophantine approximation of $x_{1r}$.

• •

If $x_{1r}\in\mathbb{Q}$ or if its irrationality exponent $\sigma_{r}=\mu(x_{1r})\geq 3$, then $s_{r}=1$ and there exist infinitely many $(k_{1},k_{2})$ such that

[TABLE]

So $\langle H_{2}(k_{1},k_{2},k_{3})\rangle\sim 1$ due to (6.9). Then in order to have $|R_{1}|\lesssim\langle H_{2}\rangle^{\frac{1}{2}}$, it follows from (6.10) that $s\geq 1=s_{r}$.

• •

If $2\leq\sigma_{r}<3$, then $s_{r}=\frac{\sigma_{r}-1}{2}$ and it follows from Lemma 3.3 that for any $\epsilon>0$ and for any integers $k_{1}$ and $k_{2}$,

[TABLE]

So it follows from (6.9) that $|H_{2}(k_{1},k_{2},k_{3})|\gtrsim|k_{2}|^{3-\sigma_{r}-\epsilon}$. Then in order to have $|R_{1}|\lesssim\langle H_{2}\rangle^{\frac{1}{2}}$, we need $1-s\leq(3-\sigma_{r}-\epsilon)/2$, that is

[TABLE]

The above argument explains why the critical index is $s_{r}$ when $r\in(\frac{1}{4},\infty)\setminus\{1\}$. Next, we will briefly discuss the rest cases (1), (2) and (4).

• •

When $r=\frac{1}{4}$, the argument is similar to the above. The two roots of $h_{r}(x)$ are the same: $x_{1r}=x_{2r}=-\frac{1}{2}$. So $s_{r}=\sigma_{r}=1$. Meanwhile, H_{2}(k_{1},k_{2},k_{3})=-3\alpha_{1}k_{2}^{3}\big{(}\frac{k_{1}}{k_{2}}+\frac{1}{2}\big{)}^{2}. So there exist infinitely many integer pairs $(k_{1},k_{2})$ such that $\frac{k_{1}}{k_{2}}+\frac{1}{2}=0$, which implies $|H_{2}(k_{1},k_{2},k_{3})|=0$. In order to have $|R_{1}|\lesssim\langle H_{2}\rangle^{\frac{1}{2}}$, it follows from (6.10) that $s\geq 1=s_{r}$.

• •

When $r<\frac{1}{4}$, $|h_{r}(x)|\gtrsim 1+x^{2}$, so it follows from (6.5) that

[TABLE]

For $s<0$, the worst situation is when $|k_{1}|\sim|k_{2}|\gg 1$ and $|k_{3}|\lesssim 1$. In this situation, $R_{1}\sim|k_{2}|^{1-2s}$ and $|H_{2}|\sim|k_{2}|^{3}$. So in order to ensure $R_{1}\lesssim\langle H_{2}\rangle^{\frac{1}{2}}$, $s$ needs to be at least $-\frac{1}{4}$.

• •

When $r=1$, $|H_{2}(k_{1},k_{2},k_{3})|\sim|k_{1}k_{2}k_{3}|$. The worst situation occurs when $k_{1}=-k_{2}$ and $k_{3}=0$, which implies $H_{2}=0$ and $R_{1}\sim|k_{2}|^{1-2s}$. So in order to have $R_{1}\lesssim\langle H_{2}\rangle^{\frac{1}{2}}=1$, $s$ should be at least $\frac{1}{2}$.

6.2 Auxiliary results

For any vector $(\vec{k},\vec{\tau})\in\mathbb{Z}_{\lambda}^{3}\times\mathbb{R}^{3}$, we denote it as $(\vec{k},\vec{\tau})=(k_{1},k_{2},k_{3},\tau_{1},\tau_{2},\tau_{3})$. For any $\lambda\geq 1$,

[TABLE]

and for any given $(k_{3},\tau_{3})\in\mathbb{Z}_{\lambda}\times\mathbb{R}$,

[TABLE]

Lemma 6.1**.**

Let $\lambda\geq 1$, $s\in\mathbb{R}$ and $\alpha_{i}\in\mathbb{R}^{*}$ for $1\leq i\leq 3$. The bilinear estimate

[TABLE]

holds if and only if the following two estimates hold,

[TABLE]

and

[TABLE]

where $L_{i}=\tau_{i}-\phi^{\alpha_{i}}(k_{i})$.

This lemma is essentially proved in [46, 13] by using duality, Plancherel identity and definition (2.11), so we omit the details here.

Lemma 6.2**.**

Let $\lambda\geq 1$, $q\geq\frac{1}{3}$ and $\alpha_{i}\in\mathbb{R}^{*}$ for $1\leq i\leq 3$. Then there exists a constant $C=C(\alpha_{1},\alpha_{2},\alpha_{3},p)$ such that for any functions $\{f_{i}\}_{i=1}^{3}$ on $\mathbb{Z}_{\lambda}\times\mathbb{R}$,

[TABLE]

where $L_{i}=\tau_{i}-\phi^{\alpha_{i}}(k_{i})$ and M:=\max\big{\{}\langle L_{1}\rangle,\langle L_{2}\rangle,\langle L_{3}\rangle\big{\}}. In particular, if $H(k_{1},k_{2},k_{3})$ denotes the resonance function as defined in Definition 2.2, then

[TABLE]

This lemma is also standard (e.g. see [9, 13, 40]), but the statement is a little bit different from that in the literature, so we will briefly include a proof here for the convenience of the readers.

Proof of Lemma 6.2.

Since $H(k_{1},k_{2},k_{3})=-\sum\limits_{i=1}^{3}L_{i}$, it is obvious that $\langle H\rangle\lesssim M$, so it suffices to prove (6.17).

• •

Let’s first focus on the region where $M=\langle L_{1}\rangle$. In this region, $\langle L_{1}\rangle^{q}\gtrsim M^{q}$. Define g_{1}=\mathcal{F}^{-1}\big{(}f_{1}(k_{1},\tau_{1})\big{)} and

[TABLE]

Then

[TABLE]

As a result, it follows from Lemma 4.3 and $q\geq\frac{1}{3}$ that

[TABLE]

• •

In the regions where $M=\langle L_{2}\rangle$ or $\langle L_{3}\rangle$, the argument is similar, so (6.17) is justified.

∎

6.3 Proof of Lemma 5.2

As we have seen in Section 6.1, the main ideas in the proofs for part (a)–part (d) in Lemma 5.2 are analogous. So we will only carry out the detailed proof for part (d) which is the most technical case. The framework of the following proof is similar to that in [40].

Proof of Part (d).

First, we recall from Proposition 3.6(b) that $s_{r}\geq\frac{1}{2}$. In addition, it follows from the assumption $s_{r}<1$ and Proposition 3.6(a) that $r\neq\frac{1}{4}$. The case of $\frac{1}{4}<r<1$ is analogous to the case of $r>1$, so we will just assume $\frac{1}{4}<r<1$ in the rest of the proof.

Fix $s>s_{r}$ and $p>s_{r}$, according to Lemma 6.1, it remains to prove

[TABLE]

and

[TABLE]

where

[TABLE]

Since $\frac{\langle k_{3}\rangle}{\langle k_{1}\rangle\langle k_{2}\rangle}\leq 1$, it suffices to consider the case when $s$ is sufficiently close to $s_{r}$. So we can just assume

[TABLE]

Next, we will prove (6.18) first and then briefly mention the proof for (6.19).

Proof of (6.18). First, in the region where $k_{2}=0$, the integrand on the LHS of (6.18) vanishes, so we only need to focus on the region where $k_{2}\neq 0$. Recall the resonance function $H_{2}$ in (6.5):

[TABLE]

where $h_{r}$ is defined in (6.6). When $\frac{1}{4}<r<1$, it follows from (6.7) that the two roots $x_{1r}$ and $x_{2r}$ of $h_{r}$ satisfy $-1<x_{1r}<x_{2r}<0$. Hence, there exists a constant d_{r}\in\big{(}0,\frac{1}{8}\big{)}, which only depends on $r$, such that (see Figure 1)

[TABLE]

On the other hand, there exists a constant $\epsilon_{r}$, which also only depends on $r$, such that

[TABLE]

In the following, we denote

[TABLE]

Since |H_{2}(k_{1},k_{2},k_{3})|=\big{|}\sum\limits_{i=1}^{3}L_{i}\big{|}, then $\langle H_{2}(k_{1},k_{2},k_{3})\rangle\lesssim\text{MAX}$.

Case 1. \Big{|}\dfrac{k_{1}}{k_{2}}-x_{1r}\Big{|}\geq d_{r} and \Big{|}\dfrac{k_{1}}{k_{2}}-x_{2r}\Big{|}\geq d_{r}. In this case, it follows from (6.21) and (6.23) that

[TABLE]

So $|k_{2}|^{3}\lesssim|H_{2}|\lesssim\text{MAX}$. In addition, $k_{2}\neq 0$ implies that $|k_{2}|\geq\frac{1}{\lambda}$. Hence, $|k_{2}|^{2}\lesssim\lambda(\text{MAX})$. Recalling $\dfrac{\langle k_{3}\rangle^{s}}{\langle k_{1}\rangle^{s}\langle k_{2}\rangle^{s}}\leq 1$, so

[TABLE]

where the last inequality is due to Lemma 6.2. This verifies (6.18) since $p>s_{r}\geq\frac{1}{2}$.

Case 2. \Big{|}\dfrac{k_{1}}{k_{2}}-x_{1r}\Big{|}<d_{r} or \Big{|}\dfrac{k_{1}}{k_{2}}-x_{2r}\Big{|}<d_{r}. Without loss of generality, it is assumed that \Big{|}\dfrac{k_{1}}{k_{2}}-x_{1r}\Big{|}<d_{r}. Then it can be seen from (6.22) or Figure 1 that $|k_{1}|\sim|k_{2}|\sim|k_{3}|$ and \Big{|}\dfrac{k_{1}}{k_{2}}-x_{2r}\Big{|}>3d_{r}. Meanwhile, if $|k_{2}|\lesssim 1$, then (6.18) is obviously true. So we will just assume $|k_{2}|\gg 1$. Thus, (6.18) reduces to

[TABLE]

Denote

[TABLE]

then $\epsilon_{s}>0$. Next, we will split the range of \big{|}\frac{k_{1}}{k_{2}}-x_{1r}\big{|} further.

Case 2.1. \frac{d_{r}}{\lambda|k_{2}|}\leq\big{|}\frac{k_{1}}{k_{2}}-x_{1r}\big{|}<d_{r}. In this case,

[TABLE]

Then it follows from $|H_{2}|\lesssim\text{MAX}$ and $s>s_{r}\geq\frac{1}{2}$ that $|k_{2}|^{1-s}\lesssim(\lambda|H_{2}|)^{\frac{1-s}{2}}\lesssim\lambda^{\frac{1}{4}}(\text{MAX})^{\frac{1}{4}}$. Hence,

[TABLE]

Case 2.2. \big{|}\frac{k_{1}}{k_{2}}-x_{1r}\big{|}<\frac{d_{r}}{\lambda|k_{2}|}.

Since $s_{r}<1$, it follows from (1.26) that $s_{r}=\frac{\sigma_{r}-1}{2}$. In addition, by taking advantage of (6.8) and Lemma 3.3, there exists a constant $K=K(r,s)$ such that

[TABLE]

Combining $s_{r}=\frac{\sigma_{r}-1}{2}$ and (6.25) yields $\sigma_{r}+\epsilon_{s}=2s+1$. Therefore,

[TABLE]

which implies

[TABLE]

Then we obtain

[TABLE]

So (6.24) reduces to

[TABLE]

• •

Case 2.2.1. $\langle L_{1}\rangle=\text{MAX}$. Then (6.28) reduces to

[TABLE]

For fixed $k_{1}$, it follows from \big{|}\frac{k_{1}}{k_{2}}-x_{1r}\big{|}<\frac{d_{r}}{\lambda|k_{2}|} that $k_{2}\in E_{r}(k_{1})$ which is defined as follows.

[TABLE]

It is easily seen that the size $|E_{r}(k_{1})|$ is at most $\frac{2d_{r}}{|x_{1r}|}$. Define

[TABLE]

Since $(k_{3},\tau_{3})=-(k_{1}+k_{2},\tau_{1}+\tau_{2})$, the LHS of (6.29) is bounded above by

[TABLE]

By the change of variable $(k_{1},\tau_{1})\rightarrow(-k_{1},-\tau_{1})$ and the definition (2.2) for the measure $dk_{2}^{\lambda}$, the above quantity is seen to be dominated by

[TABLE]

Since the size of the set $E_{r}(-k_{1})$ is bounded by the constant $\frac{2d_{r}}{|x_{1r}|}$, then by applying the Plancherel identity, we obtain

[TABLE]

Noticing $\mathcal{F}_{t}^{-1}=\mathcal{F}_{x}\mathcal{F}^{-1}$, then it follows from the definition of $\mathcal{F}_{x}$ and Holder’s inequality that

[TABLE]

In addition, it is not difficult to show that for any fixed $t$,

[TABLE]

Then plugging (6.34) and (6.35) into (6.33) yields

[TABLE]

Finally, by Holder’s inequality and Lemma 4.2,

[TABLE]

where the last inequality is due to (6.31).

• •

Case 2.2.2. $\text{MAX}=\langle L_{2}\rangle$ or $\langle L_{3}\rangle$. The argument is similar to that in Case 2.2.1, therefore omitted.

Proof of (6.19). We basically reduce the proof of (6.19) to the following two cases. In different regions, we have the flexibility to choose which case to estimate.

• •

First, by Cauchy-Schwartz inequality and duality, the proof of (6.19) reduces to establish

[TABLE]

• •

Secondly, for fixed $k_{3}$, if $L_{3}$ is restricted to some set $\Omega(k_{3})$, then it follows from Holder’s inequality in $\tau_{3}$ that

[TABLE]

So if we can prove

[TABLE]

then (6.19) will follow from (6.37), duality and (6.18).

If $\langle L_{1}\rangle=\text{MAX}$ or $\langle L_{2}\rangle=\text{MAX}$ or $\langle L_{3}\rangle=\text{MAX}$ with $\langle L_{1}\rangle\langle L_{2}\rangle\gtrsim\langle L_{3}\rangle^{\frac{1}{100}}$, then we can apply the similar argument as that in the proof of (6.18) to justify (6.36).

If $\langle L_{3}\rangle=\text{MAX}$ with $\langle L_{1}\rangle\langle L_{2}\rangle\ll\langle L_{3}\rangle^{\frac{1}{100}}$, then we will justify (6.37). Similar to the proof of (6.18), the most delicate situation is when $\frac{k_{1}}{k_{2}}$ is very near the root $x_{1r}$ or $x_{2r}$. Without loss of generality, we will only investigate the region where \big{|}\frac{k_{1}}{k_{2}}-x_{1r}\big{|}<\frac{d_{r}}{\lambda|k_{2}|} in the rest argument (see the above Case 2.2).

Firstly, it follows from \big{|}\frac{k_{1}}{k_{2}}-x_{1r}\big{|}<\frac{d_{r}}{\lambda|k_{2}|} that

[TABLE]

So when $k_{3}$ is fixed, the choice of $k_{1}$ is at most $\frac{2d_{r}}{1+x_{1r}}$. Secondly,

[TABLE]

As a result, $|H_{2}|\sim|L_{3}|\gg 1$ and $\langle L_{3}+H_{2}\rangle\ll|H_{2}|^{\frac{1}{100}}$. So there exists a small constant \delta\in\big{(}0,\frac{1}{10}\big{)} such that $\langle L_{3}+H_{2}\rangle\leq\delta\,|H_{2}|^{\frac{1}{100}}$. Fix this $\delta$. For any $k_{3}\in\mathbb{Z}_{\lambda}$, define

[TABLE]

Then $\tau_{3}-\phi^{\alpha_{1}}(k_{3})=L_{3}\in\Omega^{\delta}(k_{3})$. For any $M\geq 1$, define $\Omega^{\delta}_{M}(k_{3})=\{\eta\in\Omega^{\delta}(k_{3}):M/2\leq|\eta|\leq 2M\}$. For any $\eta\in\Omega^{\delta}_{M}(k_{3})$, the choices of $k_{1}$, as mentioned above, is at most $\frac{2d_{r}}{1+x_{1r}}$. On the other hand, it follows from

[TABLE]

that $|H_{2}|\sim\eta\sim M$. So \big{\langle}\eta+H_{2}(k_{1},k_{2},k_{3})\big{\rangle}\leq\delta M^{\frac{1}{100}}. As a result, \big{|}\Omega_{M}^{\delta}(k_{3})\big{|}\lesssim M^{\frac{1}{100}}. By the change of variable $\eta=\tau_{3}-\phi_{3}(k_{3})=L_{3}$,

[TABLE]

which verifies (6.37). ∎

7 Sharpness of the bilinear estimates

In this section we will prove Propositions 5.3 and 5.4 which justify the sharpness of the bilinear estimates in Lemmas 5.1 and 5.2. Since their proofs are similar, we will only prove Proposition 5.4. Throughout this section,

[TABLE]

The following lemma will be frequently used in this section.

Lemma 7.1**.**

Let $E_{i}\subseteq\mathbb{Z}\times\mathbb{R}\,(1\leq i\leq 3)$ be bounded regions such that $E_{1}+E_{2}\subseteq-E_{3}$, i.e.,

[TABLE]

Then

[TABLE]

Proof.

Rewriting the left hand side as

[TABLE]

then the conclusion follows from (7.1). ∎

7.1 Proof of Proposition 5.4

Similar to Lemma 6.1, (5.4) is equivalent to

[TABLE]

where

[TABLE]

The corresponding resonance function $H_{2}$ is given by (6.4):

[TABLE]

where $r=\frac{\alpha_{2}}{\alpha_{1}}$. If $k_{2}\neq 0$, then $H_{2}$ can be rewritten as in (6.5):

[TABLE]

where $h_{r}(x)=x^{2}+x+\frac{1-r}{3}$.

Proof of Part (a).

Suppose there exist $r<\frac{1}{4}$, $s<-\frac{1}{4}$ and $b\in\mathbb{R}$ such that (7.2) holds.

• •

For large $N$, define $f_{i}=-\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{2}\subseteq-B_{3}$. In addition, for any $(k_{i},\tau_{i})\in B_{i}$, $1\leq i\leq 3$, $\langle L_{1}\rangle\sim 1$,$\langle L_{2}\rangle\sim 1$ and it follows from (7.4) that $|H_{2}(k_{1},k_{2},k_{3})|\sim N^{3}$. Therefore, $\langle L_{3}\rangle\sim N^{3}$ and we conclude from (7.2) that

[TABLE]

Noticing $|B_{i}|\sim 1$, applying Lemma 7.1 leads to $N^{3b-2s-2}\lesssim C$. In other words,

[TABLE]

• •

Similarly, define $f_{i}=-\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{3}\subseteq-B_{2}$ and by similar argument, we conclude

[TABLE]

(7.5) and (7.6) together yields $s\geq-\frac{1}{4}$, which contradicts to the assumption $s<-\frac{1}{4}$. In addition, when $s=-\frac{1}{4}$, $b$ has to be exactly $\frac{1}{2}$. ∎

Proof of Part (b).

Under the additional assumption $\widehat{w_{1}}(0,\cdot)=0$, (5.4) is equivalent to (7.2) with the additional restriction $f_{1}(0,\cdot)=0$. When $r=1$, writing $\alpha_{1}=\alpha_{2}=\alpha$, then $H_{2}(k_{1},k_{2},k_{3})=3\alpha k_{1}k_{2}k_{3}$. Define $f_{i}=-\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{2}\subseteq-B_{3}$. In addition, for any $(k_{i},\tau_{i})\in B_{i}$, $1\leq i\leq 3$, $\langle L_{1}\rangle\sim 1$, $\langle L_{2}\rangle\sim 1$ and $H_{2}(k_{1},k_{2},k_{3})=0$, which implies $\langle L_{3}\rangle\sim 1$. Then it follows from (7.2) that

[TABLE]

Noticing $|B_{i}|\sim 1$, applying Lemma 7.1 leads to $s\geq\frac{1}{2}$. ∎

Proof of Part (c).

Analogous to part (b) but without the restriction $\widehat{w_{1}}(0,\cdot)$, so $f_{1}(0,\cdot)$ is not required to be zero anymore. So it is valid to define $f_{i}=-\mathbbm{1}_{B_{i}}$, where

[TABLE]

Then $B_{1}+B_{2}\subseteq-B_{3}$. In addition, for any $(k_{i},\tau_{i})\in B_{i}$, $1\leq i\leq 3$, $\langle L_{1}\rangle\sim 1$, $\langle L_{2}\rangle\sim 1$ and $H_{2}(k_{1},k_{2},k_{3})=0$, which implies $\langle L_{3}\rangle\sim 1$. Then it follows from (7.2) that

[TABLE]

which implies $N\lesssim C$ due to Lemma 7.1. But this is impossible since $N$ can be arbitrarily large. ∎

Proof of Part (d).

The following proof is in the similar spirit as that in ([40], Proposition 3.9).

The proofs for the case $r\in[\frac{1}{4},1)$ and the case $r\in(1,\infty)$ are analogous, so we will just assume $r\in[\frac{1}{4},1)$. Suppose there exist $s<s_{r}$ and $b\in\mathbb{R}$ such that (5.4) holds. Then (7.2) holds for some constant $C=C(\alpha_{1},\alpha_{2},s,b)$. Since $r\in[\frac{1}{4},1)$, the function $h_{r}$ has two roots

[TABLE]

In addition, $-1<x_{1r}\leq x_{2r}<0$ and $\mu(x_{1r})=\mu(x_{2r})=\mu(\sqrt{12r-3})=\sigma_{r}$. When $k_{2}\neq 0$, the resonance function $H_{2}$ can be written as

[TABLE]

In the following, we will first show that $\frac{1}{3}\leq b\leq\frac{2}{3}$ and then derive a contradiction to $s<s_{r}$.

For large positive $N$, define $f_{i}=\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{2}\subseteq-B_{3}$. In addition, for any $(k_{i},\tau_{i})\in B_{i}$, $1\leq i\leq 3$, $\langle L_{1}\rangle\sim 1$,$\langle L_{2}\rangle\sim 1$ and

[TABLE]

which implies $\langle L_{3}\rangle\sim N^{3}$. Then it follows from (7.2) that

[TABLE]

Noticing $|B_{i}|\sim 1$, applying Lemma 7.1 leads to $b\leq\frac{2}{3}$. On the other hand, define $f_{i}=\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{3}\subseteq-B_{2}$ and by applying similar arguments, we conclude that $b\geq\frac{1}{3}$. Thus, $\frac{1}{3}\leq b\leq\frac{2}{3}$.

Next, we discuss two situations depending on whether $\sqrt{12r-3}$ is a rational number or not.

• •

Case 1. $\sqrt{12r-3}\in\mathbb{Q}$.

In this case, $x_{1r}\in\mathbb{Q}$, $\sigma_{r}=1$ and $s_{r}=1$. In addition, since $-1<x_{1r}<0$, we can assume $x_{1r}=\frac{p_{1}}{q_{1}}$ for some $p_{1},\,q_{1}\in\mathbb{Z}$ with $p_{1}<0$, $q_{1}>0$ and $p_{1}+q_{1}>0$. For any large positive $N$, define $f_{i}=-\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{2}\subseteq-B_{3}$. In addition, for any $(k_{i},\tau_{i})\in B_{i}$, $1\leq i\leq 3$, $\langle L_{1}\rangle\sim 1$, $\langle L_{2}\rangle\sim 1$ and $\frac{k_{1}}{k_{2}}=\frac{p_{1}}{q_{1}}=x_{1r}$. As a result,

[TABLE]

which implies $\langle L_{3}\rangle\sim 1$. Then it follows from (7.2) that

[TABLE]

Noticing $|B_{i}|\sim 1$, applying Lemma 7.1 leads to $s\geq 1$, which contradicts to $s<s_{r}=1$.

• •

Case 2. $\sqrt{12r-3}\notin\mathbb{Q}$.

Since $s<s_{r}$ by the assumption, there exists $\epsilon_{0}>0$ such that $s+\epsilon_{0}<s_{r}$. Recalling $\mu(x_{1r})=\sigma_{r}$, so $x_{1r}$ is approximable with power $\sigma_{r}-\epsilon_{0}$. Hence, it follows from Lemma 3.2 that there exists a sequence $\{(m_{j},\,n_{j})\}_{j=1}^{\infty}\subset\mathbb{Z}\times\mathbb{Z}^{*}$ such that $n_{j}>0$, $\lim\limits_{j\rightarrow\infty}n_{j}=\infty$ and

[TABLE]

When $j$ is large enough, $\frac{m_{j}}{n_{j}}$ will be very close to $x_{1r}$. Since $-1<x_{1r}<0$, it implies $m_{j}<0$, $m_{j}+n_{j}>0$ and $|m_{j}|\sim m_{j}+n_{j}\sim n_{j}$. In addition, $|\frac{m_{j}}{n_{j}}-x_{2r}|\approx|x_{1r}-x_{2r}|$ which is a positive constant only depending on $r$. As a result,

[TABLE]

Denote

[TABLE]

Then no matter $\sigma_{r}<3$ or $\sigma_{r}\geq 3$, it always holds that

[TABLE]

• –

For any large $j$ as in the above discussion, define $f_{i}=\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{2}\subseteq-B_{3}$. In addition, for any $(k_{i},\tau_{i})\in B_{i}$, $1\leq i\leq 3$, $\langle L_{1}\rangle\sim 1$, $\langle L_{2}\rangle\sim 1$ and

[TABLE]

which implies $\langle L_{3}\rangle\lesssim n_{j}^{\zeta_{r}}$. Then it follows from (7.2) and $|m_{j}|\sim m_{j}+n_{j}\sim n_{j}$ that

[TABLE]

Noticing $|B_{i}|\sim 1$, applying Lemma 7.1 leads to

[TABLE]

• –

On the other hand, define $f_{i}=\mathbbm{1}_{B_{i}}$ for $1\leq i\leq 3$, where

[TABLE]

Then $B_{1}+B_{3}\subseteq-B_{2}$. In addition, by applying similar arguments, we conclude that

[TABLE]

Adding (7.9) and (7.10) together yields $s\geq 1-\frac{\zeta_{r}}{2}$. Recalling $s+\epsilon_{0}<s_{r}$, so

[TABLE]

If $\sigma_{r}\geq 3$, then $s_{r}=1$ and $\zeta_{r}=\epsilon_{0}$, which contradicts to (7.11). If $2\leq\sigma_{r}<3$, then $s_{r}=\frac{\sigma_{r}-1}{2}$ and $\zeta_{r}=3-\sigma_{r}+\epsilon_{0}$, which also contradicts to (7.11).

∎

Appendix A Well-posedness for the Hirota-Satsuma systems (1.5)

In this appendix, we summarize the analytical well-posedness results on the Hirota-Satsuma systems (1.5).

Theorem A.1**.**

The Hirota-Satsuma system (1.5) is A-LWP in $\mathcal{H}_{2}^{s}$ if one of the following conditions is satisfied.

1. (1)

$a_{1}\in(-\infty,\frac{1}{4})\setminus\{0\}$* and $s\geq-\frac{1}{4}$;* 2. (2)

$a_{1}=1$, $c_{12}=0$ and $s\geq\frac{1}{2}$; 3. (3)

$a_{1}\in[\frac{1}{4},\infty)\setminus\{1\}$, and $s\geq 1$ or $s>s_{a_{1}}$ (equivalently $s\geq\min\{1,\,s_{a_{1}}+\}$).

Due to the following conserved energies for (1.5), the A-GWP follows directly from Theorem A.1.

[TABLE]

Theorem A.2**.**

Let $c_{12}>0$. Then the Hirota-Satsuma system (1.5) is A-GWP in $\mathcal{H}^{s}_{2}$ if one of the following conditions is satisfied.

1. (1)

$a_{1}\in(-\infty,\frac{1}{4})\setminus\{0\}$* and $s\geq 0$;* 2. (2)

$a_{1}\in[\frac{1}{4},1)$* and $s\geq 1$.*