Long time behavior of the NLS-Szeg{\"o} equation

TL;DR

This paper investigates how filtering positive Fourier modes affects the long-term behavior of the NLS-Szeg{"o} equation, revealing Sobolev estimates, stability properties, and wave turbulence phenomena.

Contribution

It provides new insights into the long-time dynamics, stability, and instability of solutions to the NLS-Szeg{"o} equation with dispersion effects.

Findings

Long time Sobolev estimates for small data

Orbital stability of plane wave solutions

Wave turbulence phenomena observed

Abstract

We are interested in the influence of filtering the positive Fourier modes to the integrable non linear Schr{\"o}dinger equation. Equivalently, we want to study the effect of dispersion added to the cubic Szeg{\"o} equation, leading to the NLS-Szeg{\"o} equation on the circle $\mathbb{S}^1$\begin{equation*}i \partial\_t u + \epsilon^{\alpha}\partial\_x^2 u = \Pi(|u|^2 u),\qquad 0<\epsilon<1, \qquad \alpha\geq 0.\end{equation*}There are two sets of results in this paper. The first result concerns the long time Sobolev estimates for small data. The second set of results concerns the orbital stability of plane wave solutions. Some instability results are also obtained, leading to the wave turbulence phenomenon.

Equations

$$i{\partial }_{t}u+{ϵ}^{\alpha }{\partial }_x^{2}u=\Pi (|u{|}^{2}u),0<ϵ<1,\alpha \ge 0.$$

$$i{\partial }_{t}u+{\partial }_x^{2}u=\Pi (|u{|}^{2}u),u(0,\cdot )=u_0.$$

$$\Pi :\sum _{k\in \mathbb{Z}}{u}_k{e}^{ikx}⟼\sum _{k\ge 0}{u}_k{e}^{ikx}.$$

$$i{\partial }_{t}u+{\partial }_x^{2}u=|u{|}^{2}u,(t,x)\in \mathbb{R}\times {\mathbb{S}}^1,$$

$$i{\partial }_{t}V=\Pi (|V{|}^{2}V),(t,x)\in \mathbb{R}\times {\mathbb{S}}^1.$$

$$\{\begin{array}{l}i{\partial }_{t}V=|V{|}^{2}V,\\ V(0,\cdot )=V_0.\end{array}$$

$$i{\partial }_{t}u+{ϵ}^{\alpha }{\partial }_x^{2}u=\Pi (|u{|}^{2}u),u(0,\cdot )=u_0,0<ϵ<1,\alpha \ge 0.$$

$$E^{\alpha ,ϵ}(u)=\frac{{ϵ}^{\alpha }}{2}\Vert {\partial }_{x}u{\Vert }_{L^2}^2+\frac{1}{4}\Vert u{\Vert }_{L^4}^4,u\in H_{+}^1.$$

$$\{\begin{array}{l}Q(u)=\Vert u{\Vert }_{L^2}^2,\\ I(u)=\mathrm{Im}{\int }_{{\mathbb{S}}^1}\overline{u}{\partial }_{x}u=\Vert u{\Vert }_{{\dot{H}}^{\frac{1}{2}}}^2,\end{array}$$

$$\underset{t\in \mathbb{R}}{\sup }\Vert u(t){\Vert }_{H^s}\le \Vert u_0{\Vert }_{L^2}^{1-2s}\Vert u_0{\Vert }_{H^{\frac{1}{2}}}^{2s},\forall s\in [0,\frac{1}{2}].$$

$$\{\begin{array}{l}{\sup }_{|t|\le \frac{a_s}{{ϵ}^{4-\alpha }}}\Vert u(t){\Vert }_{H^s}\le K_sϵ,\mathrm{if}\alpha \in [0,2];\\ {\sup }_{|t|\le \frac{a_s}{{ϵ}^2}}\Vert u(t){\Vert }_{H^s}\le K_sϵ,\mathrm{if}\alpha >2.\end{array}$$

$$\underset{|t|\le \frac{1}{{ϵ}^{2+\beta }}}{\sup }\Vert u(t){\Vert }_{H^s}\gtrsim ϵ|\ln ϵ{|}^{\frac{1}{2}}\gg ϵ,u(0)=u_0^{ϵ}.$$

$$\underset{t\in \mathbb{R}}{\sup }\Vert u(t){\Vert }_{H^1}{\lesssim }_{\Vert u_0{\Vert }_{H^1}}{ϵ}^{-\frac{\alpha }{2}},\forall 0<ϵ<1,\alpha \ge 0.$$

$$\underset{t\in \mathbb{R}}{\sup }\underset{\theta \in \mathbb{R}}{\inf }\Vert u(t)-e^{i\theta }{\mathbf{e}}_m{\Vert }_{H^1}\le {\mathcal{C}}_m{ϵ}^{1-\frac{\alpha }{2}}.$$

$$\underset{\begin{array}{c}0<ϵ<1\\ 0\le \alpha \le 2\end{array}}{\sup }\underset{\Vert \varphi -{\mathbf{e}}_m{\Vert }_{H^1}\le ϵ}{\sup }\underset{t\in \mathbb{R}}{\sup }\Vert S_{\alpha ,ϵ}(t)\varphi {\Vert }_{H^1}<\infty .$$

$$\underset{|t|\le \frac{b_{m,s}}{{ϵ}^{1-\frac{\alpha }{2}}}}{\sup }\underset{\theta \in \mathbb{R}}{\inf }\Vert u(t)-e^{i\theta }{\mathbf{e}}_m{\Vert }_{H^s}\le L_{m,s}{ϵ}^{1-\frac{\alpha }{2}}.$$

$$i{\partial }_{t}u+{\partial }_x^{2}u=\Pi (|u{|}^{2}u),u(0,\cdot )=u_0.$$

$$\underset{|t|\le \frac{d_{m,s}}{{ϵ}^2}}{\sup }\underset{\theta \in \mathbb{R}}{\inf }\Vert u(t)-e^{i\theta }{\mathbf{e}}_m{\Vert }_{H^s}\le K_{m,s}ϵ.$$

$$i{\partial }_{t}U+{\nu }^2{\partial }_x^{2}U=\Pi (|U{|}^{2}U),U(0,x)=e^{ix}+\delta ,$$

$$\Vert u(\frac{t^{\delta }}{{ϵ}^2}){\Vert }_{H^1}=ϵ\Vert U(t^{\delta }){\Vert }_{H^1}\simeq \frac{ϵ}{\delta }\simeq ϵ\sqrt{(\alpha -2)|\ln ϵ|}\gg ϵ,$$

$$i{\partial }_{t}u+{ϵ}^2{\partial }_x^{2}u=\Pi (|u{|}^{2}u),u(0,x)=ϵ(e^{ix}+\delta ),$$

$$\underset{t\in \mathbb{R}}{\sup }\Vert u(t){\Vert }_{H^1}=\mathcal{O}(ϵ),\underset{|t|\le \frac{d_{1,s}}{{ϵ}^2{\delta }^2}}{\sup }\Vert u(t){\Vert }_{H^s}=\mathcal{O}(ϵ),\forall 0<\delta \ll 1,\forall 0<ϵ<1,$$

$$i{\partial }_{t}V=\Pi (|V{|}^{2}V),V(0,\cdot )=V_0.$$

$$H_V(h)=\Pi (V\bar{h}).$$

$$T_b(h)=\Pi (bh).$$

$${\partial }_t{H}_V=[B_V,H_V].$$

$$\Vert V{\Vert }_{B_{1,1}^1}\simeq \mathrm{Tr}|H_V|.$$

$$\underset{t\in \mathbb{R}}{\sup }\Vert V(t){\Vert }_{L^{\infty }}{\lesssim }_s\Vert V_0{\Vert }_{H^s}$$

$$\{\begin{array}{l}{\lim }_{n\to +\infty }\frac{\Vert V(t_n){\Vert }_{H^s}}{|t_n{|}^p}=+\infty ,\forall s>\frac{1}{2},\forall p\ge 1,\\ {\lim }_{n\to +\infty }V(t_n^{\prime })=V_0.\end{array}$$

$$V^{\delta }(t,x)=\frac{a^{\delta }(t)e^{ix}+b^{\delta }(t)}{1-p^{\delta }(t)e^{ix}},$$

Full text

1. 1 Introduction

2. 1.1 Motivation

3. 1.2 Main results

4. 2 The cubic Szegő equation

5. 2.1 The Lax pair structure

6. 2.2 Wave turbulence

7. 2.3 A special case

8. 3 Long time behavior for small data

9. 3.1 The case $\alpha\geq 2$

10. 3.1.1 The bootstrap argument

11. 3.1.2 Optimality of the time interval if $\alpha>2$

12. 3.2 The case $0\leq\alpha<2$

13. 3.2.1 Identifying the resonances

14. 3.2.2 The Birkhoff normal form

15. 3.2.3 End of the proof of the case $0\leq\alpha<2$

16. 3.2.4 The open problem of optimality

17. 4 Orbital stability of the traveling plane wave $\mathbf{e}_{m}$

18. 4.1 The proof of $\mathbf{Theorem}$ 1.5

19. 4.2 Long time $H^{s}$-stability

20. 4.2.1 Proof of $\mathbf{Proposition}$ 1.7

21. 4.2.2 Homological equation

22. 4.3 Long time $H^{s}$-stability in the case $\alpha=0$

23. 4.3.1 The Birkhoff normal form

24. 4.3.2 End of the proof of $\mathbf{Theorem}$ 1.8

25. 5 Comparison to NLS equation

26. 6 Appendix

1 Introduction

We consider the NLS-Szegő equation defined on the circle $\mathbb{S}^{1}$

[TABLE]

Here $\Pi:L^{2}(\mathbb{S}^{1})\to L^{2}(\mathbb{S}^{1})$ denotes the orthogonal projector from $L^{2}(\mathbb{S}^{1})$ onto the space of $L^{2}$ boundary values of holomorphic functions on the unit disc,

[TABLE]

We denote by $L^{2}_{+}:=\Pi(L^{2}(\mathbb{S}^{1}))\subset L^{2}(\mathbb{S}^{1})$, $H^{s}_{+}:=H^{s}(\mathbb{S}^{1})\bigcap L^{2}_{+}$, for all $s\geq 0$, and $C^{\infty}_{+}:=C^{\infty}(\mathbb{S}^{1})\bigcap L^{2}_{+}$.

1.1 Motivation

The NLS-Szegő equation can be seen as the combination of two completely integrable systems: the defocusing cubic Schrödinger equation

[TABLE]

and the cubic Szegő equation

[TABLE]

They have both a Lax Pair structure and the action-angle coordinates, which can be used to obtain their explicit formulas with the inversed spectral method(see Zakharov–Shabat $[\mathbf{\ref{ZAKHAROV SHABAT}}]$, Faddeev–Takhtajan $[\mathbf{\ref{Faddeev-Takhtajan}}]$, Grébert–Kappeler $[\mathbf{\ref{Kappeler Grebert}}]$, Gérard $[\mathbf{\ref{Gerard defocusing NLS integrals}}]$, for the NLS equation and Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}},\mathbf{\ref{gerardgrellier2}},\mathbf{\ref{Gerard-grellier explicit formula szego equation}},\mathbf{\ref{Gerard grellier book cubic szego equation and hankel operators}}]$ for the cubic Szegő equation). However, these two Lax pairs cannot be combined in order to give a Lax pair for $(\ref{NLS-Szego})$. Moreover, the long time behaviors of these two equations are totally different.

The NLS equation $(\ref{cubic dNLS equation})$ has a sequence of conservation laws controlling every Sobolev norms(see Faddeev–Takhtajan $[\mathbf{\ref{Faddeev-Takhtajan}}]$, Grébert–Kappeler $[\mathbf{\ref{Kappeler Grebert}}]$, Gérard $[\mathbf{\ref{Gerard defocusing NLS integrals}}]$), so all the solutions are uniformly bounded in every $H^{s}$ space. Moreover, Grébert and Kappeler $[\mathbf{\ref{Kappeler Grebert}}]$ have proved the existence of the global Birkhoff coordinates for NLS equation. So the solutions of $(\ref{cubic dNLS equation})$ are actually almost periodic on $\mathbb{R}$ valued into $H^{s}(\mathbb{S}^{1})$.

Compared to $(\ref{cubic dNLS equation})$, the cubic Szegő equation, which stands for a non-dispersive model, has both the Lax pair structure and the wave turbulence phenomenon. Its long time behavior is extremely sensible according to the different initial data. P.Gérard and S.Grellier have shown that(in $[\mathbf{\ref{gerardgrellier growth of sobolev norm}},\mathbf{\ref{Gerard-grellier explicit formula szego equation}},\mathbf{\ref{Gerard grellier book cubic szego equation and hankel operators}}]$) for a $G_{\delta}$ dense subset of initial data in $C^{\infty}_{+}$, the solutions may blow up in $H^{s}$, for every $s>\frac{1}{2}$ with super–polynomial growth on some sequence of times, while they go back to their initial data on another sequence of times tending to infinity. However, all the $H^{\frac{1}{2}}$-solutions are almost periodic. (see also $\mathbf{Theorem}$ 2.3).

Remark 1.1**.**

Consider the following equation without the Szegő projector $\Pi$ on $\mathbb{S}^{1}$:

[TABLE]

Then $V(t,x)=e^{it|V_{0}|^{2}}V_{0}(x)$ and we have $\|V(t)\|_{H^{s}}\simeq|t|^{s}$, for all $s\geq 0$, if $|V_{0}|$ is not a constant function. Hence, the Szegő projector both accelerates the energy transfer to high frequencies, and facilitates the transition to low frequencies for $(\ref{ODE without szego projector})$.

One wonders about whether filtering the positive Fourier modes can change the long time Sobolev estimates of the cubic defocusing Schrödinger equation. So we introduce equation $(\ref{NLS-Szego})$. On the other hand, it can also be obtained from the cubic Szegő equation by adding the dispersive term $\partial_{x}^{2}$ to its linear part. In order to see the gradual change of the dispersion, we add the parameter $\epsilon^{\alpha}$ in front of the Laplacian $\partial_{x}^{2}$ to get a more general model, the NLS-Szegő equation (with small dispersion):

[TABLE]

Equation $(\ref{NLS-Szego})$ is the special case $\alpha=0$ for $(\ref{NLS-Szego epsilon^alpha})$.

We endow $L^{2}_{+}$ with the canonical symplectic form $\omega(u,v)=\mathrm{Im}\int_{\mathbb{S}^{1}}\frac{u\bar{v}}{2\pi}$. Equation $(\ref{NLS-Szego epsilon^alpha})$ has the Hamiltonian formalism with the energy functional

[TABLE]

Besides $E^{\alpha,\epsilon}$, equation $(\ref{NLS-Szego epsilon^alpha})$ has two other conservation laws,

[TABLE]

which give the estimate of the solution for low frequencies:

[TABLE]

Proceeding as in the case of equation $(\ref{cubic dNLS equation})$, one can prove the global existence and uniqueness of the solution of the NLS-Szegő equation in high frequency Sobolev spaces, by using the Brezis–Gallouët type estimate $[\mathbf{\ref{Brezis Gallouet inequality article}}]$, the Aubin–Lions–Simon theorem (see $\mathbf{Theorem}$ II.5.16 in Boyer–Fabrie $[\mathbf{\ref{Boyer}}]$) and the Trudinger type inequality (see Yudovich $[\mathbf{\ref{Yudovich trudinger}}]$, Vladimirov $[\mathbf{\ref{Vladimirov trudinger}}]$, Ogawa $[\mathbf{\ref{Ogawa trudinger}}]$ and Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}}]$). Its well-posedness problem in low frequency Sobolev spaces can be dealt with Strichartz’s inequality introduced in Bourgain $[\mathbf{\ref{bourgain strichartz inequality}}]$. Only the high frequency Sobolev estimates are considered in this paper.

Proposition 1.2**.**

For every $s\geq\frac{1}{2}$, given $u_{0}\in H^{s}_{+}$, there exists a unique solution $u\in C(\mathbb{R},H^{s}_{+})$ of $(\ref{NLS-Szego epsilon^alpha})$ such that $u(0)=u_{0}$. For every $T>0$, the mapping $u_{0}\in H^{s}_{+}\mapsto u\in C([-T,T],H^{s}_{+})$ is continuous.

1.2 Main results

The first result concerns the long time stability around the null solution of the NLS-Szegő equation $(\ref{NLS-Szego epsilon^alpha})$. If the initial data $u_{0}$ is bounded by $\epsilon$, we look for a time interval $I^{\alpha}_{\epsilon}$, in which the solution $u(t)$ is still bounded by $\mathcal{O}(\epsilon)$. Now we state the first result of this paper.

Theorem 1.3**.**

For every $s>\frac{1}{2}$, there exist two constants $a_{s}\in(0,1)$ and $K_{s}>0$ such that for all $0<\epsilon\ll 1$ and $u_{0}\in H^{s}_{+}$, if $\|u_{0}\|_{H^{s}}=\epsilon$ and $u$ denotes the solution of $(\ref{NLS-Szego epsilon^alpha})$, then

[TABLE]

Moreover, the time interval $I^{\alpha}_{\epsilon}=[-\frac{a_{s}}{\epsilon^{2}},\frac{a_{s}}{\epsilon^{2}}]$ is maximal for the case $\alpha>2$ and $s\geq 1$ in the following sense: for every $0<\epsilon\ll 1$, there exists $u_{0}^{\epsilon}\in C^{\infty}_{+}$ such that $\|u_{0}^{\epsilon}\|_{H^{s}}\simeq\epsilon$ and for every $\beta>0$, we have

[TABLE]

Remark 1.4**.**

In the case $\alpha\in[0,2)$, the proof is based on the Birkhoff normal form method, similarly to Bambusi $[\mathbf{\ref{Bambusi birkhoff normal form for nonlinear pde}}]$, Grébert $[\mathbf{\ref{GB birkoff normal form}}]$, Gérard–Grellier $[\mathbf{\ref{gerardgrellier effective dynamic}}]$ and Faou–Gauckler–Lubich $[\mathbf{\ref{faou--Gauckler--lubich Sobolev Stability of Plane Wave}}]$ for instance. However, the time interval $[-\frac{a_{s}}{\epsilon^{4-\alpha}},\frac{a_{s}}{\epsilon^{4-\alpha}}]$ may not be optimal. The resonant term of $6$ indices in the homological equation can not be cancelled by the Birkhoff normal form transform.(see $\mathbf{subsubsection}$ 3.2.4)

The second set of results concerns the long time $H^{s}$-estimates for the solutions of $(\ref{NLS-Szego epsilon^alpha})$, if its initial data is a perturbation of the plane wave $\mathbf{e}_{m}:x\mapsto e^{imx}$, for some $m\in\mathbb{N}$ and $s\geq 1$. Let $u=u(t,x)$ be the solution of equation $(\ref{NLS-Szego epsilon^alpha})$ such that $\|u(0)-\mathbf{e}_{m}\|_{H^{s}}=\epsilon$. Its energy functional $(\ref{energy functional of NLS Szego epsilon alpha})$ gives the following estimate:

[TABLE]

However, no information on the stability of the plane waves $\mathbf{e}_{m}$ is obtained from $(\ref{estimate of norm H1 de u by energy functional})$ during the process $\epsilon\to 0^{+}$. Consider the super-polynomial growth of Sobolev norms in the cubic Szegő equation case (see Gérard–Grellier $[\mathbf{\ref{gerardgrellier growth of sobolev norm}},\mathbf{\ref{Gerard grellier book cubic szego equation and hankel operators}}]$ and $\mathbf{Proposition}$ 2.4 in this paper), the occurence of wave turbulence phenomenon for $(\ref{NLS-Szego epsilon^alpha})$ depends on the level of its dispersion. We begin with three long time stability results for the polynomial dispersion $\epsilon^{\alpha}\partial_{x}^{2}$ case with $0\leq\alpha\leq 2$. The following theorem indicates $H^{1}$-orbital stability of the traveling waves $\mathbf{e}_{m}$ for equation $(\ref{NLS-Szego epsilon^alpha})$.

Theorem 1.5**.**

For all $\epsilon\in(0,1)$, $\alpha\in[0,2]$ and $m\in\mathbb{N}$, there exists $\mathcal{C}_{m}>0$ such that if $\|u(0)-\mathbf{e}_{m}\|_{H^{1}}=\epsilon$, then we have

[TABLE]

For each $t\in\mathbb{R}$, the infimum can be attained when $\theta=\arg u_{m}(t)$. A similar result is established by Zhidkov $[\mathbf{\ref{Zhidov qualitative theory book}}$, Sect. $3.3]$ and Gallay–Haragus $[\mathbf{\ref{Gallay--Haragus stability of small periodic waves}},\mathbf{\ref{Gallay--Haragus ORBITAL stability of periodic waves}}]$ for the 1D cubic Schrödinger equation. In small dispersion case, $\mathbf{Theorem}$ 1.5 gives a significant improvement of estimate $(\ref{estimate of norm H1 de u by energy functional})$. We denote by $S_{\alpha,\epsilon}$ the non linear evolution group defined by $(\ref{NLS-Szego epsilon^alpha})$ on $H^{\frac{1}{2}}_{+}$. In other words, for every $\phi\in H^{\frac{1}{2}}_{+}$, $t\mapsto S_{\alpha,\epsilon}(t)\phi$ is the solution $u\in C(\mathbb{R},H^{\frac{1}{2}}_{+})$ of equation $(\ref{NLS-Szego epsilon^alpha})$ such that $u(0)=\phi$.

Corollary 1.6**.**

For every $m\in\mathbb{N}$, we have

[TABLE]

Compared to $\mathbf{Proposition}$ 2.4 (see Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}},\mathbf{\ref{gerardgrellier effective dynamic}},\mathbf{\ref{Gerard-grellier explicit formula szego equation}}]$), the dispersive term $\epsilon^{\alpha}\partial_{x}^{2}$ counteracts the wave turbulence phenomenon in $H^{1}$ norm for equation $(\ref{NLS-Szego epsilon^alpha})$, if $0\leq\alpha\leq 2$. After the change of variable $u(t)=e^{i\arg u_{m}(t)}(\mathbf{e}_{m}+\epsilon^{1-\frac{\alpha}{2}}v(t))$, we use a bootstrap argument to get long time orbital stability of the traveling waves $\mathbf{e}_{m}$ with respect to higher Sobolev norms.

Proposition 1.7**.**

For all $s\geq 1$ and $m\in\mathbb{N}$, there exist two constants $b_{m,s}\in(0,1)$ and $L_{m,s}>0$ such that if $0\leq\alpha<2$ and $\|u(0)-\mathbf{e}_{m}\|_{H^{s}}=\epsilon\in(0,1)$, then we have

[TABLE]

We also look for a larger time interval in which the estimate $(\ref{bootstrap for orbital stability})$ holds, by using the Birkhoff normal form transformation. But the coefficients in front of the high frequency Fourier modes in the homological equation may be arbitrarily large, if $\alpha\in(0,2)$. For this reason, we return to the case $\alpha=0$ and consider equation $(\ref{NLS-Szego})$.

[TABLE]

Then the time interval can be enlarged as $[-\frac{d_{m,s}}{\epsilon^{2}},\frac{d_{m,s}}{\epsilon^{2}}]$ in this case.

Theorem 1.8**.**

In the case $\alpha=0$, for all $s\geq 1$ and $m\in\mathbb{N}$, there exist three constants $d_{m,s},\epsilon_{m,s}\in(0,1)$ and $K_{m,s}>0$ such that if $\|u(0)-\mathbf{e}_{m}\|_{H^{s}}=\epsilon\in(0,\epsilon_{m,s})$, then we have

[TABLE]

A similar result is obtained in Faou–Gauckler–Lubich $[\mathbf{\ref{faou--Gauckler--lubich Sobolev Stability of Plane Wave}}]$ for the focusing or defocusing cubic Schrödinger equation on the arbitrarily dimensional torus. (see $\mathbf{Section}$ 5 for the comparison between $(\ref{NLS-Szego})$ and $(\ref{cubic dNLS equation})$)

After stating the stability results, we turn to construct a large solution for $(\ref{NLS-Szego epsilon^alpha})$ with respect to the initial data, if the level of dispersion is exponentially small with respect to the level of perturbation of the plane wave $\mathbf{e}_{1}:x\mapsto e^{ix}$. We state the last result of this paper.

Theorem 1.9**.**

There exists a constant $K>0$ such that for all $0<\delta\ll 1$, we denote by $U$ the solution of the following NLS-Szegő equation with small dispersion

[TABLE]

where $\nu=e^{-\frac{\pi K}{2\delta^{2}}}$, then we have $\|U(t^{\delta})\|_{H^{1}}\simeq\frac{1}{\delta}$ with $t^{\delta}:=\frac{\pi}{\delta\sqrt{4+\delta^{2}}}$.

This $H^{1}$-instability result indicates that the support of the energy functional of equation $(\ref{rescaling NLS Szego equation with initial data exp(ix)+delta})$ is transferred to higher Fourier modes. This phenomenon is similar to the cubic Szegő equation case (see Gérard–Grellier $[\mathbf{\ref{gerardgrellier growth of sobolev norm}},\mathbf{\ref{Gerard-grellier explicit formula szego equation}}$, $\mathbf{\ref{Gerard grellier book cubic szego equation and hankel operators}}]$) and the 2D cubic NLS case (see Colliander–Keel–Staffilani–Takaoka–Tao $[\mathbf{\ref{Colliander J., Keel M., Staffilani G.,Takaoka H., Tao Transfer of energy }}]$). Compared to $\mathbf{Theorem}$ 1.5, adding the low-level dispersion $e^{-\frac{\pi K}{\delta^{2}}}\partial_{x}^{2}$ fails to change the quality of wave turbulence phenomenon ($\mathbf{Proposition}$ 2.4) for the cubic Szegő equation.

The second part of $\mathbf{Theorem}$ 1.3 is a consequence of $\mathbf{Theorem}$ 1.9. Indeed, if $\alpha>2$, we rescale $u(t,x)=\epsilon U(\epsilon^{2}t,x)$ with $e^{-\frac{\pi K}{2\delta^{2}}}=\nu=\epsilon^{\frac{\alpha-2}{2}}$. Then $u$ solves $(\ref{NLS-Szego epsilon^alpha})$ with $u(0,x)=\epsilon(e^{ix}+\delta)$ and

[TABLE]

while $\frac{t^{\delta}}{\epsilon^{2}}\simeq\frac{|\ln\epsilon|}{\epsilon^{2}}\ll\frac{1}{\epsilon^{2+\beta}}$, for all $\beta>0$. However, this method does not work in the critical case $\alpha=2$. If $u$ solves

[TABLE]

after rescaling $U(t,x)=\epsilon^{-1}u(\epsilon^{-2}t,x)$, we get equation $(\ref{rescaling NLS Szego equation with initial data exp(ix)+delta})$ with $\nu=1$, leading to $(\ref{NLS-Szego})$ with initial data $U(0,x)=e^{ix}+\delta$. $\mathbf{Theorem}$ 1.5 and $\mathbf{Theorem}$ 1.8 yield the following two estimates

[TABLE]

for every $s>\frac{1}{2}$. The problem of the optimal time interval in the case $\alpha=2$ of $\mathbf{Theorem}$ 1.3 remains open.

This paper is organized as follows. In $\mathbf{Section}$ 2, we recall some basic facts of the cubic Szegő equation and its consequences. In $\mathbf{Section}$ 3, we study long time behavior for $(\ref{NLS-Szego epsilon^alpha})$ with small data and prove $\mathbf{Theorem}$ 1.9 and $\mathbf{Theorem}$ 1.3. In $\mathbf{Section}$ 4, we study the orbital stability of the plane waves $\mathbf{e}_{m}$ for $(\ref{NLS-Szego epsilon^alpha})$ for every $m\in\mathbb{N}$ and give the proof of $\mathbf{Theorem}$ 1.5, $\mathbf{Proposition}$ 1.7 and $\mathbf{Theorem}$ 1.8. We compare the NLS equation and the NLS-Szegő equation in $\mathbf{Section}$ 5.

$Acknowledgments$

The author would like to express his gratitude towards Patrick Gérard for his deep insight, generous advice and continuous encouragement. He also would like to thank Jean-Marc Delort, Benoit Grébert, Sandrine Grellier and Thomas Kappeler for useful discussions.

2 The cubic Szegő equation

In this section, we recall some results of the cubic Szegő equation

[TABLE]

2.1 The Lax pair structure

Given $V\in H^{\frac{1}{2}}_{+}$, the Hankel operator $H_{V}:L^{2}_{+}\to L^{2}_{+}$ is defined by

[TABLE]

Given $b\in L^{\infty}(\mathbb{S}^{1})$, the Toeplitz operator $T_{b}:L^{2}_{+}\to L^{2}_{+}$ is defined by

[TABLE]

Theorem 2.1**.**

(Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}}]$) Set $V\in C(\mathbb{R};H^{s}_{+})$ for some $s>\frac{1}{2}$. Then $V$ solves the cubic Szegő equation if and only if $H_{V}$ satisfies the following evolutive equation

[TABLE]

where $B_{V}:=\frac{i}{2}H_{V}^{2}-iT_{|V|^{2}}$. In other words, $(L_{V},B_{V})$ is a Lax pair for the cubic Szegő equation.

The equation $(\ref{Lax equation for Szego})$ yields that the spectrum of the Hankel operator $H_{V}$ is invariant under the flow of the cubic Szegő equation. Thus the quantity $\mathrm{Tr}|H_{v}|$ is conserved. A theorem of Peller ($[\mathbf{\ref{Peller Hankel operators of class Sp}}]$ Theorem 2 p. 454) states that

[TABLE]

Using the embedding theorem $H^{s}\hookrightarrow B^{1}_{1,1}\hookrightarrow L^{\infty}$, for any $s>1$, we have the following $L^{\infty}$ estimate of the Szegő flow.

Corollary 2.2**.**

(Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}}]$) Assume $V_{0}\in H^{s}_{+}$ for some $s>1$, then we have

[TABLE]

2.2 Wave turbulence

The following theorem indicates its chaotic long time behavior with turbulence phenomenon for general initial data.

Theorem 2.3**.**

(Gérard–Grellier $[\mathbf{\ref{gerardgrellier growth of sobolev norm}},\mathbf{\ref{Gerard grellier book cubic szego equation and hankel operators}}]$) 1.There exists a $G_{\delta}-$dense set $U\subset C^{\infty}(\mathbb{S}^{1})\bigcap L^{2}_{+}$ such that if $V_{0}\in U$, then there exist two sequences $(t_{n})_{n\in\mathbb{N}}$ and $(t_{n}^{\prime})_{n\in\mathbb{N}}$ tending to infinity such that

[TABLE]

2.For every $V_{0}\in H^{\frac{1}{2}}_{+}$, the mapping $t\in\mathbb{R}\mapsto V(t)\in H^{\frac{1}{2}}_{+}$ is almost periodic.

2.3 A special case

In order to prove the optimality of the case $\alpha>2$ of $\mathbf{Theorem}$ 1.3, we compare the solution $u$ of the NLS-Szegő with small dispersion to the solution of the cubic Szegő equation with some special initial data. Set $V_{0}=V^{\delta}_{0}:=\delta+e^{ix}$, we denote $V^{\delta}$ the solution of $(\ref{cubic szego equation appendix})$. Refering to Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}}$ Sect. 6.1, 6.2; References Sect. 3; References Sect. $4]$, we have the following explicit formula

[TABLE]

where

[TABLE]

Proposition 2.4**.**

(Gérard–Grellier $[\mathbf{\ref{gerardgrellier1}},\mathbf{\ref{gerardgrellier effective dynamic}},\mathbf{\ref{Gerard-grellier explicit formula szego equation}}]$) For $0<\delta\ll 1$, set $t^{\delta}:=\frac{\pi}{2\omega}=\frac{\pi}{\delta\sqrt{4+\delta^{2}}}\sim\frac{\pi}{2\delta}$. Let $V^{\delta}$ be the solution of $(\ref{cubic szego equation appendix})$ with $V^{\delta}(0,x)=e^{ix}+\delta$, then we have the following estimate

[TABLE]

for every $s>\frac{1}{2}$.

Proof.

Expanding formula $(\ref{fractional explicit formula of cubic szego equation daisy effect})$ as Fourier series, we have

[TABLE]

with

[TABLE]

By the explicit formula of $p^{\delta}$, we have

[TABLE]

with $t^{\delta}:=\frac{\pi}{2\omega}=\frac{\pi}{\delta\sqrt{4+\delta^{2}}}$.

∎

3 Long time behavior for small data

3.1 The case $\alpha\geq 2$

For all $s>\frac{1}{2}$, consider the NLS-Szegő equation with small dispersion and small data.

[TABLE]

At first, we give the proof of the time interval $I^{\alpha}_{\epsilon}=[-\frac{a_{s}}{\epsilon^{2}},\frac{a_{s}}{\epsilon^{2}}]$ in the case $\alpha\geq 2$ of $\mathbf{Theorem}$ 1.3, which is based on a bootstrap argument. Then we prove the maximality of $I^{\alpha}_{\epsilon}$.

3.1.1 The bootstrap argument

Lemma 3.1**.**

Let $a,b,T>0$, $m>1$ and $M:[0,T]\longrightarrow\mathbb{R}_{+}$ be a continuous function satisfying

[TABLE]

Assume that $(mb)^{\frac{1}{m-1}}M(0)\leq 1$ and $(mb)^{\frac{1}{m-1}}a\leq\frac{m-1}{m}$. Then

[TABLE]

for all $\tau\in[0,T]$.

Proof.

The function $f_{m}:z\in\mathbb{R}_{+}\longmapsto z-bz^{m}$ attains its maximum at the critical point $z_{c}=(mb)^{-\frac{1}{m-1}}$. $f_{m}(z_{c})=\frac{m-1}{m}(mb)^{-\frac{1}{m-1}}$. Since $a\leq\max_{z\geq 0}f_{m}(z)=f_{m}(z_{c})$, there exists $z_{-}\leq z_{c}\leq z_{+}$ such that

[TABLE]

and $f_{m}(z_{\pm})=a$. Since $f_{m}(M(\tau))\leq a$, $\forall 0\leq\tau\leq T$ and $M(0)\leq z_{c}$, we have $M([0,T])\subset[0,z_{-}]$. By the concavity of $f_{m}$ on $[0,+\infty[$, we have $f_{m}(z)\geq\frac{f_{m}(z_{c})}{z_{c}}z$ for all $z\in[0,z_{c}]$. Consequently, $M(t)\leq z_{-}\leq\frac{m}{m-1}a$, for all $0\leq t\leq T$. ∎

Proof of of estimate $(\ref{epsilon^(4-alpha) estimates formula})$ in the case $\alpha>2$ .

For all $\alpha\geq 0$ and $\epsilon\in(0,1)$ fixed, we rescale $u$ as $u=\epsilon\mu$, equation $(\ref{NLS Szego small initial data epsilon alpha})$ becomes

[TABLE]

Duhamel’s formula of equation $(\ref{equation NLS-Szego alpha epsilon after rescaling})$ gives the following estimate:

[TABLE]

Here $C_{s}$ denotes the Sobolev constant in the inequality $\||\mu|^{2}\mu\|_{H^{s}}\leq C_{s}\|\mu\|_{H^{s}}^{3}$. We choose $a_{s}=\frac{4}{27C_{s}}$ and the following estimate holds

[TABLE]

by using $\mathbf{Lemma}$ 3.1 with $m=3$, $T=\frac{a_{s}}{\epsilon^{2}}$, $a=M(0)=1$, $b=C_{s}\epsilon^{2}T$ and $M(t)=\sup_{0\leq\tau\leq t}\|\mu(\tau)\|_{H^{s}}$.

∎

3.1.2 Optimality of the time interval if $\alpha>2$

In order to prove the optimality of $I^{\alpha}_{\epsilon}$ in which estimate $(\ref{estimate for alpha strickly bigger than 2})$ holds, we set $u(0,x)=\epsilon(e^{ix}+\delta)$ and rescale $u(t,x)=\epsilon U(\epsilon^{2}t,x)$. Then, we have

[TABLE]

where $\nu:=\epsilon^{\frac{\alpha-2}{2}}$. Since the optimality is a consequence of $\mathbf{Theorem}$ 1.9, we prove at first $\mathbf{Theorem}$ 1.9 by comparing $U$ to the solution of the cubic Szegő equation with the same initial data,

[TABLE]

Proof of $\mathbf{Theorem}$ 1.9.

We shall estimate their difference $r(t,x):=U(t,x)-V(t,x)$, which satisfies the following equation

[TABLE]

with $Q(r):=\Pi(\overline{V}r^{2}+2V|r|^{2}+|r|^{2}r)$. Thus, we can calculate the derivative of $\|r(t)\|_{H^{1}}^{2}$,

[TABLE]

Then, we have

[TABLE]

The $L^{\infty}$ estimate of $V$ is given by $\mathbf{Corollary}$ 2.2 and the $H^{s}$ estimate of $V$ is given by $\mathbf{Proposition}$ 2.4, for all $s>\frac{1}{2}$. Thus, we have

[TABLE]

and there exist $C_{1},C_{3}>0$ such that

[TABLE]

for all $0\leq t\leq t^{\delta}=\frac{\pi}{\delta\sqrt{4+\delta^{2}}}$. We use a bootstrap argument to deal with the term $\mathcal{O}(\|r\|_{H^{1}}^{3})$. Set

[TABLE]

then we have

[TABLE]

where $C$ denotes the Sobolev constant. Consequently, for all $0\leq t\leq\min(T,t^{\delta})$, we have

[TABLE]

with $K:=\max(C_{1}^{2}+C_{3}^{2},2+2M_{\infty}^{2}+12CM_{\infty}+2C^{2}+(12CM_{\infty}+6C^{2})C_{1})$. We set

[TABLE]

Using Grönwall’s inequality, we deduce that

[TABLE]

Since $\|V(t^{\delta})\|_{H^{1}}\simeq\frac{1}{\delta}$ by $\mathbf{Theorem}$ 2.4, we have $\|U(t^{\delta})\|_{H^{1}}=\|V(t^{\delta})+r(t^{\delta})\|_{H^{1}}\simeq\frac{1}{\delta}$.

∎

Fix $\alpha>2$, for every $0<\epsilon\ll 1$, we set

[TABLE]

Then we have $\|u(T^{\alpha,\epsilon})\|_{H^{1}}\simeq\epsilon\sqrt{(\alpha-2)|\ln\epsilon|}\gg\epsilon$, while $u(0,x)=\epsilon(e^{ix}+\delta)$. Then the optimality of $I^{\alpha}_{\epsilon}=[-\frac{a_{s}}{\epsilon^{2}},\frac{a_{s}}{\epsilon^{2}}]$ is obtained.

3.2 The case $0\leq\alpha<2$

We assume at first that $u(0)\in C^{\infty}_{+}$ so that the energy functional of $(\ref{NLS Szego small initial data epsilon alpha})$

[TABLE]

is well defined. For general initial data $u(0)\in H^{s}_{+}$, if $s\in(\frac{1}{2},1)$, we use the density argument $\overline{C^{\infty}_{+}}=H^{s}_{+}$.

We rescale $u(t,x)\mapsto\epsilon^{-\frac{\alpha}{2}}u(-\epsilon^{\alpha}t,x)$, then the equation $(\ref{NLS Szego small initial data epsilon alpha})$ is reduced to the case $\alpha=0$. It suffices to prove the following estimate

[TABLE]

if $u$ solves $i\partial_{t}u+\partial_{x}^{2}u=\Pi(|u|^{2}u)$ with $\|u(0)\|_{H^{s}}=\epsilon$.

3.2.1 Identifying the resonances

The study of the resonant set of the NLS-Szegő equation is necessary before Birkhoff normal form transformation. We refer to Eliasson–Kuksin $[\mathbf{\ref{eliasson-kuksin kam for nls}}]$ to see the analysis of the resonances for a more general non linear term and KAM theorem for the NLS equation.

We use again the change of variable $u=\epsilon\mu$ and we rewrite Duhamel’s formula of $\mu$ with $\eta(t)=\sum_{k\geq 0}\eta_{k}(t)e^{ikx}:=e^{-it\partial_{x}^{2}}\mu(t)$. Then we have

[TABLE]

for all $k\geq 0$. Recall the classical identification of the resonant set

[TABLE]

In order to cancel all the resonances, we apply the transformation $v(t):=e^{2it\epsilon^{2}\|\mu(0)\|_{L^{2}}^{2}}\mu(t)$. As $\|\mu\|_{L^{2}}$ is a conservation law, we have

[TABLE]

The equation $(\ref{NLSF scaling kill resonance})$ can be seen as the Hamiltonian system with respect to the energy function

[TABLE]

Then we have $R(v)=\frac{1}{4}\left(\sum_{\begin{smallmatrix}k_{1}-k_{2}+k_{3}-k_{4}=0\\ k_{1}^{2}-k_{2}^{2}+k_{3}^{2}-k_{4}^{2}\neq 0\end{smallmatrix}}v_{k_{1}}\overline{v}_{k_{2}}v_{k_{3}}\overline{v}_{k_{4}}-\sum_{k\geq 0}|v_{k}|^{4}\right)$.

3.2.2 The Birkhoff normal form

Equation $(\ref{NLSF scaling kill resonance})$ is transferred to another Hamiltonian equation which is closer to the linear Schrödinger equation by Birkhoff normal form method. We try to find a symplectomorphism $\Psi_{\epsilon}$ such that the energy functional $H^{\epsilon}$ is reduced to the Hamiltonian

[TABLE]

where $\tilde{R}(v)=-\frac{1}{4}\sum_{k\geq 0}|v_{k}|^{4}$. $\Psi_{\epsilon}$ is chosen as the value at time $1$ of the Hamiltonian flow of some energy $\epsilon^{2}F$.

We fix the value $s>\frac{1}{2}$. Recall that, given a smooth real valued function $H$, we denote $X_{H}$ the Hamiltonian vector field, i.e,

[TABLE]

Given two smooth real-valued functions $F$ and $G$ on $H^{s}_{+}$, their Poisson bracket $\{F,G\}$ is defined by

[TABLE]

for all $v=\sum_{k\geq 0}v_{k}e^{ikx}\in H^{s}_{+}$. In particular, if $F$ and $G$ are respectively homogeneous of order $p$ and $q$, then their Poisson bracket is homogeneous of order $p+q-2$.

Lemma 3.2**.**

Set $F(v):=\sum_{k_{1}-k_{2}+k_{3}-k_{4}=0}f_{k_{1},k_{2},k_{3},k_{4}}v_{k_{1}}\overline{v}_{k_{2}}v_{k_{3}}\overline{v}_{k_{4}}$, with the coefficients

[TABLE]

Thus, $F$ is real-valued and its Hamiltonian field $X_{F}$ is smooth on $H^{s}_{+}$ such that $\{F,H_{0}\}+R=\tilde{R}$ and the following estimates hold.

[TABLE]

for all $v\in H^{s}_{+}$.

Proof.

$F$ well defined because $\sup_{(k_{1},k_{2},k_{3},k_{4})\in\mathbb{Z}^{4}}|f_{k_{1},k_{2},k_{3},k_{4}}|\leq\frac{1}{4}$, the Sobolev embedding yields that $|F(v)|\leq\frac{1}{4}\left(\sum_{k\geq 0}|\hat{u}(k)|\right)^{4}\lesssim_{s}\|u\|^{4}_{H^{s}}$. The Young’s convolution inequality $l^{1}*l^{1}*l^{2}\hookrightarrow l^{2}$ implies that $\|X_{F}(v)\|_{H^{s}}\lesssim_{s}\|v\|_{H^{s}}^{3}$ and $\|\mathrm{d}X_{F}(v)\|_{B(H^{s})}\lesssim_{s}\|v\|_{H^{s}}^{2}$. Using $(\ref{Poisson bracket})$ and the definition of $f_{k_{1},k_{2},k_{3},k_{4}}$, we have

[TABLE]

∎

Set $\chi_{\sigma}:=\exp(\epsilon^{2}\sigma X_{F})$ the Hamiltonian flow of $\epsilon^{2}F$, i.e.,

[TABLE]

We perform the canonical transformation $\Psi_{\epsilon}:=\chi_{1}=\exp(\epsilon^{2}X_{F})$. The next lemma will prove the local existence of $\chi_{\sigma}$, for $|\sigma|\leq 1$ and give the estimate of the difference between $v$ and $\Psi_{\epsilon}^{-1}(v)$

Lemma 3.3**.**

For $s>\frac{1}{2}$, there exist two constants $\rho_{s},C_{s}>0$ such that for all $v\in H^{s}_{+}$, if $\epsilon\|v\|_{H^{s}}\leq\rho_{s}$, then $\chi_{\sigma}(v)$ is well defined on the interval $[-1,1]$ and the following estimates hold:

[TABLE]

Proof.

The inequality $\|\mathrm{d}X_{F}(v)\|_{B(H^{s})}\leq C_{s}\|v\|_{H^{s}}^{2}$ implies that the Lipschitz coefficient of the mapping $v\longmapsto\epsilon^{2}X_{F}(v)$ is bounded by $C_{s}\epsilon^{2}\|v\|_{H^{s}}^{2}\leq C_{s}\rho_{s}^{2}$. If $\rho_{s}$ is sufficiently small, then the Hamiltionian flow $(\sigma,v)\mapsto\chi_{\sigma}(v)$ exists on the maximal interval $(-\sigma^{*},\sigma^{*})$, by the Picard-Lindelöf theorem. Assume that $\sigma^{*}<1$, then $\mathbf{Lemma}$ 3.2 and the following integral formula

[TABLE]

yield that

[TABLE]

By $\mathbf{Lemma}$ 3.1 with $M(t)=\sup_{0\leq\tau\leq t}\|\chi_{\tau}(v)\|_{H^{s}}$, $m=3$, $a=M(0)=\|v\|_{H^{s}}$ and $b=C_{s}\epsilon^{2}$, we have

[TABLE]

if $\epsilon\|v\|_{H^{s}}\leq\frac{2}{3\sqrt{3C_{s}}}$. This is a contradiction to the blow-up criterion. Hence $\sigma^{*}\geq 1$, and we have $\sup_{|\sigma|\leq 1}\|\chi_{\sigma}(v)\|_{H^{s}}\leq\frac{3}{2}\|v\|_{H^{s}}$, if $\epsilon\|v\|_{H^{s}}\leq\rho_{s}:=\frac{2}{3\sqrt{3C_{s}}}$.

Since $\|X_{F}(v)\|_{H^{s}}\leq C_{s}\|v\|_{H^{s}}^{3}$, for all $\sigma\in[-1,1]$, we have

[TABLE]

if $\epsilon\|v\|_{H^{s}}\leq\rho_{s}$. We differentiate equation $(\ref{intergal equation for chi})$ and use again $\mathbf{Lemma}$ 3.2 to obtain

[TABLE]

Here we use the Gronwall inequality in the last step.

∎

Recall that $\Psi_{\epsilon}=\chi_{1}$. The normal form of the energy $H^{\epsilon}$ is given below.

Lemma 3.4**.**

For $s>\frac{1}{2}$, there exists a smooth mapping $Y:H^{s}_{+}\longrightarrow H^{s}_{+}$ and a constant $C^{\prime}_{s}>0$ such that

[TABLE]

for all $v\in H^{s}_{+}$ such that $\epsilon\|v\|_{H^{s}}\leq\rho_{s}$. Let us set $w(t):=\Psi_{\epsilon}^{-1}(v(t))$, then we have

[TABLE]

if $\epsilon\|w(t)\|_{H^{s}}\leq\rho_{s}$.

Proof.

We expand the energy $H^{\epsilon}\circ\Psi_{\epsilon}=H^{\epsilon}\circ\chi_{1}$ with Taylor’s formula at time $\sigma=1$ around [math]. Since $\chi_{0}=\mathrm{Id}_{H^{s}_{+}}$, one gets

[TABLE]

We set $G(\sigma):=(1-\sigma)\{F,\tilde{R}\}+\sigma\{F,R\}$, $\forall\sigma\in[0,1]$. Since $X_{\{F,R\}}$ and $X_{\{F,\tilde{R}\}}(u)$ are homogeneous of degree $5$ with uniformly bounded coefficients, we have

[TABLE]

By the chain rule of Hamiltonian vector fields:

[TABLE]

and $\mathbf{Lemma}$ 3.3, we have

[TABLE]

for all $v\in H^{s}_{+}$ such that $\epsilon\|v\|_{H^{s}}\leq\rho_{s}$. Thus we define $Y:=\int_{0}^{1}X_{G(\sigma)\circ\chi_{\sigma}}\mathrm{d}\sigma$ and we have

[TABLE]

If $\epsilon\|v\|_{H^{s}}\leq\rho_{s}$, then $\|Y(v)\|_{H^{s}}\lesssim_{s}\|v\|_{H^{s}}^{5}$.

Since $\widehat{X_{\tilde{R}}(w)}(k)=-i|w_{k}|^{2}w_{k}$, $\forall k\geq 0$ and $w(t)=\Psi_{\epsilon}^{-1}(v(t))$, we have the following infinite dimensional Hamiltonian system on the Fourier modes:

[TABLE]

If $\epsilon\|w(t)\|_{H^{s}}\leq\rho_{s}$, then we have

[TABLE]

∎

3.2.3 End of the proof of the case $0\leq\alpha<2$

Proof.

Recall that $w(t)=\chi_{-1}(v(t))$ and $\|v(0)\|_{H^{s}}=1$. $\mathbf{Lemma}$ 3.3 yields that

[TABLE]

Set $K_{s}:=3(C_{s}+1)$. Then $\|w(0)\|_{H^{s}}\leq\frac{K_{s}}{3}$. We define

[TABLE]

and

[TABLE]

For all $\epsilon\in(0,\epsilon_{s})$ and $t\in[0,T]$, we have $\epsilon\|v(t)\|_{H^{s}}\leq m_{s}$. Hence $\mathbf{Lemma}$ 3.3 gives the following estimate

[TABLE]

So we have $\epsilon\sup_{0\leq t\leq T}\|w(t)\|_{H^{s}}\leq m_{s}$ and \Big{|}\frac{\mathrm{d}}{\mathrm{d}t}\|w(t)\|_{H^{s}}^{2}\Big{|}\leq C^{\prime}_{s}\epsilon^{4}\|w(t)\|^{6}_{H^{s}}, by $\mathbf{Lemma}$ 3.4. Set $a_{s}:=\frac{1}{3^{7}K_{s}^{4}C^{\prime}_{s}}$. We can precise the estimate of $\|w(t)\|_{H^{s}}$ by limiting $|t|\leq a_{s}\epsilon^{-4}$:

[TABLE]

for all $0\leq t\leq\min(T,\frac{a_{s}}{\epsilon^{4}})$. Then we have

[TABLE]

for all $t\in[0,\frac{a_{s}}{\epsilon^{4}}]$. Consequently, we have

[TABLE]

In the case $t<0$, we use the same procedure and we replace $t$ by $-t$. ∎

3.2.4 The open problem of optimality

Let $u$ be the solution of the NLS-Szegő equation

[TABLE]

Recall that $H^{\epsilon}=H_{0}+\epsilon^{2}R$ is the energy functional of the equation $(\ref{NLS Szego with small data for the higher order time interval})$ with

[TABLE]

$\chi_{\sigma}=\exp(\epsilon^{2}\sigma X_{F})$, with $F(v):=\sum_{k_{1}-k_{2}+k_{3}-k_{4}=0}f_{k_{1},k_{2},k_{3},k_{4}}v_{k_{1}}\overline{v}_{k_{2}}v_{k_{3}}\overline{v}_{k_{4}}$, with the coefficients

[TABLE]

We recall also that $\tilde{R}(v)=\{F,H_{0}\}(v)+R(v)=-\frac{1}{4}\sum_{k\geq 0}|v_{k}|^{4}$.

In order to get a longer time interval in which the solution is uniformly bounded by $\mathcal{O}(\epsilon)$, we expand the Hamiltonian $H^{\epsilon}\circ{\chi_{1}}$ by using the Taylor expansion of higher order to see whether the resonances can be cancelled by the Birkhoff normal form method.

[TABLE]

We try to cancel the term $\frac{\epsilon^{4}}{2}\{F,R+\tilde{R}\}$ by using a canonical transform to $H_{1}^{\epsilon}=H^{\epsilon}\circ{\chi_{1}}$ with the following functional

[TABLE]

We want to solve the homological equation $\{G,H_{0}\}+\frac{1}{2}\{F,R+\tilde{R}\}=0$. We can calculate that

[TABLE]

In the first term of the preceding formula, there is a resonance set $k_{1}^{2}-k_{2}^{2}+k_{3}^{2}-k_{4}^{2}+k_{5}^{2}-k_{6}^{2}=0$ that cannot be cancelled by the other two terms. We can see Grébert–Thomann $[\mathbf{\ref{Grebert Thomann Resonant dynamics for the quintic NLS}}]$ and Haus–Procesi $[\mathbf{\ref{Haus Procesi beating solution for quintic nls}}]$ for instance to analyse the resonant set for $6$ indices for the quintic NLS equation.

[TABLE]

Thus the resonant subset

[TABLE]

should be cancelled before the Birkhoff normal form transform, just like the step $\mu\mapsto v=e^{2it\epsilon^{2}\|\mu(0)\|_{L^{2}}^{2}}\mu(t)$, which can cancel all the resonances

[TABLE]

before we do the canonical transformation $H^{\epsilon}\circ\chi_{1}$. We only know that $f_{k_{1},k_{2},k_{3},k_{1}-k_{2}+k_{3}}=f_{k_{4},k_{5},k_{6},k_{4}-k_{5}+k_{6}}$ if

[TABLE]

This resonant subset contains the case $k_{5}\neq k_{6}$. The optimality of the time interval for the case $0\leq\alpha<2$ remains open.

4 Orbital stability of the traveling plane wave $\mathbf{e}_{m}$

Consider the following NLS-Szegő equation

[TABLE]

We shall prove at first $H^{1}$-orbital stability of the traveling waves $\mathbf{e}_{m}$, for all $m\in\mathbb{N}$. Then, we study their long time $H^{s}$-stability, for all $s\geq 1$.

4.1 The proof of $\mathbf{Theorem}$ 1.5

We follow the idea of using conserved quantities mentioned in Gallay–Haragus $[\mathbf{\ref{Gallay--Haragus stability of small periodic waves}}]$ for equation $(\ref{NLS-Szego epsilon^alpha orbital stability})$.

Proof.

For all $m\in\mathbb{N}$, $0<\epsilon<1$ and $0\leq\alpha\leq 2$, we denote $u(0,x)=e^{imx}+\epsilon f(x)$ with $\|f\|_{H^{1}}\leq 1$. The NLS-Szegő equation has three conservation laws:

[TABLE]

with $D=-i\partial_{x}$ and $(u,v):=\mathrm{Re}\int_{\mathbb{S}^{1}}\bar{u}v$. Thus the following quantity is conserved,

[TABLE]

Then, we have $\sup_{t\in\mathbb{R}}\|Du(t)-mu(t)\|_{L^{2}}\lesssim_{m}\epsilon^{1-\frac{\alpha}{2}}$. Recall that $\mathbf{e}_{m}(x)=e^{imx}$, then the following estimate holds,

[TABLE]

We have

[TABLE]

and by the conservation of $\|u(t)\|_{L^{2}}$, we have

[TABLE]

Thus $\sup_{t\in\mathbb{R}}\|u(t)-u_{m}(0)e^{i(\arg u_{m}(t)-\arg u_{m}(0))}\mathbf{e}_{m}\|_{H^{1}}\lesssim_{m}\epsilon^{1-\frac{\alpha}{2}}$. The proof can be finished by $u_{m}(0)=1+\epsilon f_{m}=1+\mathcal{O}(\epsilon)$. ∎

The preceding theorem also holds for the defocusing NLS equation on $\mathbb{T}^{d}$, for $d=1,2,3$ (in the energy sub-critical case) with $\mathbb{T}=\mathbb{R}/\mathbb{Z}\simeq\mathbb{S}^{1}$.(see Gallay–Haragus $[\mathbf{\ref{Gallay--Haragus stability of small periodic waves}},\mathbf{\ref{Gallay--Haragus ORBITAL stability of periodic waves}}]$) We refer to Zhidkov $[\mathbf{\ref{Zhidov qualitative theory book}}$ Sect. $3.3]$ for a detailed analysis of the stability of plane waves.

Remark 4.1**.**

Obtaining the estimate $\sup_{t\in\mathbb{R}}\|u(t)-u_{m}(t)\mathbf{e}_{m}\|_{L^{\infty}}\lesssim_{m}\epsilon^{1-\frac{\alpha}{2}}$ by the Sobolev embedding $H^{1}(\mathbb{S}^{1})\hookrightarrow L^{\infty}$, we can also proceed by using the following estimate, which is uniform on $x$ and $t$,

[TABLE]

Integrating the preceding term with respect to $x$, we have

[TABLE]

Thus $\sup_{t\in\mathbb{R}}|u_{m}(t)|=1+\mathcal{O}_{m}(\epsilon^{1-\frac{\alpha}{2}})$ and $u_{m}(t)=e^{i\arg u_{m}(t)}+\mathcal{O}_{m}(\epsilon^{1-\frac{\alpha}{2}})$. Then we have

[TABLE]

Recall that if $z=1+\mathcal{O}(\epsilon)$ then $e^{i\arg z}=1+\mathcal{O}(\epsilon)$.

4.2 Long time $H^{s}$-stability

For every $s\geq 1$, we suppose that $\|u(0)-\mathbf{e}_{m}\|_{H^{s}}\leq\epsilon$. Thanks to the estimate

[TABLE]

we change the variable $u\mapsto v=v^{m,\alpha,\epsilon}(t,x)=\sum_{n\geq 0}v_{n}(t)e^{inx}\in C^{\infty}(\mathbb{R}\times\mathbb{S}^{1})$ such that

[TABLE]

to study $H^{s}$-stability of plane waves $\mathbf{e}_{m}$ and we have

[TABLE]

Proposition 4.2**.**

For every $s\geq 1$, $m\in\mathbb{N}$, $\epsilon\in(0,1)$ and $\alpha\in[0,2)$, if $u$ is smooth and solves $(\ref{NLS-Szego epsilon^alpha orbital stability})$ with $u(0,x)=e^{imx}+\epsilon f(x)$ and $\|f\|_{H^{s}}\leq 1$, $v$ is defined by formula $(\ref{change of variable for orbital stability u to v})$, then we have

[TABLE]

Moreover, there exists a smooth function $\varphi=\varphi_{m}:\mathbb{R}\to\mathbb{R}/2\pi\mathbb{Z}\simeq\mathbb{S}^{1}$ and $\epsilon^{*}_{m}\in(0,1)$ such that for every $1<2^{-}<2$, we have

[TABLE]

The parameter $v$ satisfies the following equation

[TABLE]

where $H_{e^{2imx}}(v):=\Pi[e^{2imx}\bar{v}]$ denotes the Hankel operator.

Proof.

Since $u_{m}(t)=e^{i\arg u_{m}(t)}(1+\epsilon^{1-\frac{\alpha}{2}}v_{m}(t))$, we have $1+\epsilon^{1-\frac{\alpha}{2}}v_{m}(t)=|u_{m}(t)|\in\mathbb{R}$. So $v_{m}(t)\in\mathbb{R}$, for all $t\in\mathbb{R}$. By using the conservation law $\|\cdot\|_{L^{2}}$ and estimate $(\ref{uniform estimate of v})$, we have

[TABLE]

which yields that $\sup_{t\in\mathbb{R}}|v_{m}(t)|\lesssim_{m}\epsilon^{\min(\frac{\alpha}{2},1-\frac{\alpha}{2})}$. Recall that

[TABLE]

Then we have $u_{m}(0)=1+\epsilon f_{m}=1+\mathcal{O}(\epsilon)$ and $|e^{i\arg u_{m}(0)}-1|\lesssim\epsilon$. Thus we have

[TABLE]

We define $\theta(t):=\mathrm{arg}(u_{m}(t))$. Combing the following two formulas

[TABLE]

[TABLE]

we obtain that

[TABLE]

where $H_{e^{2imx}}(v):=\Pi[e^{2imx}\bar{v}]$ denotes the Hankel operator. The Fourier mode $v_{m}(t)$ satisfies the following equation

[TABLE]

Estimate $(\ref{uniform estimate of v})$ yields that

[TABLE]

Thus, we have

[TABLE]

The imaginary part and the real part of $(\ref{equation v_1 with O(epsilon2) alpha})$ give respectively the two following estimates:

[TABLE]

for all $0<\epsilon\ll 1$. Then we define $\varphi(t):=\frac{(1+m^{2}\epsilon^{\alpha})t+\theta(t)}{\epsilon^{\min(1,2-\alpha)}}$. Consequently, there exists $\epsilon^{*}_{m}\in(0,1)$ such that

[TABLE]

We replace $\theta^{\prime}(t)$ by $-1-m^{2}\epsilon^{\alpha}+\epsilon^{\min(1,2-\alpha)}\varphi^{\prime}(t)$ in $(\ref{equation with epsilon is not divided alpha})$ and we obtain $(\ref{NLSF with u0=epsilon + exp(ix) equation v alpha <<1})$. ∎

4.2.1 Proof of $\mathbf{Proposition}$ 1.7

For every $n\in\mathbb{N}$, we define the projector $\mathbb{P}_{n}:L^{2}_{+}\to L^{2}_{+}$ such that

[TABLE]

Now we prove $\mathbf{Proposition}$ 1.7 by using a bootstrap argument.

Proof.

At the beginning, we suppose that $u(0)\in C^{\infty}_{+}$. In the general case $u(0)\in H^{s}_{+}$, the proof can be completed by using the continuity of the flow $u(0)\mapsto u$ from $H^{s}_{+}$ to $C([-\frac{b_{s,m}}{\epsilon^{1-\frac{\alpha}{2}}},\frac{b_{s,m}}{\epsilon^{1-\frac{\alpha}{2}}}],H^{s}_{+})$. We use the same transformation $u\mapsto v$ as $(\ref{change of variable for orbital stability u to v})$. $\mathbf{Proposition}$ 4.2 yields that there exists $A_{m,s}\geq 1$ such that $\|v(0)\|_{H^{s}}\lesssim_{m,s}\epsilon^{\frac{\alpha}{2}}\leq A_{m,s}$. By using estimate $(\ref{uniform estimate of v})$, we have

[TABLE]

We define that $L_{m,s}:=\max(2(1+4m^{2})^{\frac{s}{2}}I_{m},2A_{m,s}+1)$ and

[TABLE]

Rewrite equation $(\ref{NLSF with u0=epsilon + exp(ix) equation v alpha <<1})$ on Fourier modes and we have

[TABLE]

with $Z(v)=\sum_{n\geq 0}[Z(v)]_{n}e^{inx}=\Pi(e^{-imx}v^{2}+2e^{imx}|v|^{2})+\epsilon^{1-\frac{\alpha}{2}}\Pi(|v|^{2}v)$. Then we have

[TABLE]

Then we have

[TABLE]

For all $t\in[0,T]$, we have

[TABLE]

Define $b_{m,s}=\frac{1}{64C_{s}L_{m,s}^{2}}$ and we have $\|v(t)\|_{H^{s}}\leq L_{m,s}$, for all $t\in[0,\frac{b_{s,m}}{\epsilon^{1-\frac{\alpha}{2}}}]$. The case $t<0$ is similar. ∎

4.2.2 Homological equation

We try to improve $\mathbf{Proposition}$ 1.7 and get a longer time interval in which the solution $v$ is still bounded by $\mathcal{O}(1)$, by using The Birkhoff normal form method. Recall the symplectic form $\omega(u,v)=\mathrm{Im}\int_{\mathbb{S}^{1}}u\overline{v}\frac{\mathrm{d}\theta}{2\pi}$ on the energy space $H^{1}_{+}$ and the Poisson bracket for two smooth real-valued functionals $F,G:C^{\infty}_{+}\to\mathbb{R}$

[TABLE]

for all $v=\sum_{k\geq 0}v_{k}e^{ikx}\in C^{\infty}_{+}$. For all $0\leq\alpha<2$ and $0<\epsilon\ll 1$, equation $(\ref{NLSF with u0=epsilon + exp(ix) equation v alpha <<1})$ has also the Hamiltonian formalism, which is non autonomous. Its energy functional is

[TABLE]

with

[TABLE]

We want to cancel all the high frequencies in the term $\mathcal{H}_{1}^{m}(v)$ by composing $\mathcal{H}^{m,\alpha,\epsilon}$ and the Hamiltonian flow of some auxiliary functional $\mathcal{F}_{m}$. In order to get the appropriate $\mathcal{F}_{m}$, we need to solve the homological system

[TABLE]

such that $\mathcal{R}_{m}$ depends only on finitely many Fourier modes of $v$. The remaining coefficient in front of $\epsilon^{1-\frac{\alpha}{2}}$ would be $\mathcal{R}_{m}+\varphi^{\prime}(t)\epsilon^{\min(\frac{\alpha}{2},1-\frac{\alpha}{2})}(-\tilde{\mathcal{N}}_{2}+\frac{\mathcal{N}_{2}}{2})$. One can prove the following proposition.(see also $\mathbf{Proposition}$ 4.4 and $\mathbf{Appendix}$ for the proof in the special case $\alpha=0$)

Proposition 4.3**.**

For every $m\in\mathbb{N}$, we define the following homogenous functional $\mathcal{F}_{m}$ of degree $3$:

[TABLE]

for some $a_{k,l,j}=a_{j,l,k}\in\mathbb{C}$. Then we have the following formula

[TABLE]

where

[TABLE]

for some $c_{j,j+k-m,k}=c_{k,j+k-m,j}\in\mathbb{C}$ and

[TABLE]

The term $\mathrm{Reson}^{low}$ depends only on the small Fourier modes $v_{1},v_{2},\cdots,v_{3m}$. We try to find a bounded sequence $(a_{j,l,k})_{j-l+k=m}$ such that $\mathrm{Reson}^{\geq 2m+1}=0$ in order to cancel the term $\mathcal{H}^{m}_{1}$. However, the coefficient $(1-2(j-m)(k-m)\epsilon^{\alpha})$ in front of the parameter $a_{k,j+k-m,j}$ may have an arbitrarily small absolute value if $\alpha>0$. Such sequence does not exist if $\epsilon^{-\alpha}\in 2\mathbb{N}\bigcap[2(m+1)^{2},+\infty)$.

We suppose that $\epsilon^{-\alpha}\notin\mathbb{Q}$, then $\mathrm{Reson}^{\geq 2m+1}=0$ is equivalent to a linear system, which has a unique solution

[TABLE]

for all $k\geq 2m+1$. In the case $m=0$, $(\ref{Solution of linear system of all a _kln})$ has only the last two formulas. When $\alpha>0$, the sequence $(a_{j,l,k})_{j-l+k=m}$ can be arbitrarily large, for $0<\epsilon\ll 1$. We suppose that $\epsilon^{\alpha}$ is an irrational algebraic number of degree $d\geq 2$. Then we have the Liouville estimate $[\mathbf{\ref{Liouville diophantine approximation}}]$

[TABLE]

which loses the regularity of $v$ in the estimate of $X_{F_{m}}(v)$. It is difficult to find the same kind of estimate for the transcendental numbers, which can preserve the regularity. So we return to the case $\alpha=0$.

4.3 Long time $H^{s}$-stability in the case $\alpha=0$

For $\alpha=0$ and every $m\in\mathbb{N}$ and $s\geq 1$, assume that $u$ is the smooth solution of the NLS-Szegő equation

[TABLE]

and $u(t,x)=e^{i\arg u_{m}(t)}(e^{imx}+\epsilon v(t,x))$. Then $v$ is the solution of the following Hamiltonian equation

[TABLE]

Its energy functional is

[TABLE]

with

[TABLE]

We define $\tilde{\mathcal{N}}_{2}(v):=\|v-\mathbb{P}_{2m}v\|_{L^{2}}^{2}=\sum_{n\geq 2m+1}|v_{n}|^{2}$ and the following proposition holds.

Proposition 4.4**.**

For every $s>\frac{1}{2}$ and $m\in\mathbb{N}$, there exists a sequence $(a_{j,l,k})_{j-l+k=m}$ such that $a_{j,l,k}=a_{k,l,j}$, $\sup_{m\geq 1}\sup_{j-l+k=m}|a_{j,l,k}|=\frac{1}{2}$ and the functional ${\mathcal{F}_{m}}:H^{s}_{+}\to\mathbb{R}$, defined by

[TABLE]

satisfy that $\{\mathcal{F}_{m},\mathcal{L}_{m}\}=-\tilde{\mathcal{N}}_{2}$ and $\mathcal{R}_{m}:=\{\mathcal{F}_{m},\mathcal{H}_{0}^{m}\}+\mathcal{H}^{m}_{1}$ is a finite sum of the Fourier modes $v_{1},\cdots,v_{3m}$. Moreover, for all $v,h\in H^{s}_{+}$, we have

[TABLE]

Proof.

For the convenience of the reader, the detailed calculus for $\mathcal{R}_{m}=\{\mathcal{F}_{m},\mathcal{H}_{0}^{m}\}+\mathcal{H}^{m}_{1}$ and formula $(\ref{Solution of linear system of all a _kln})$ in the case $\alpha=0$ are given in $\mathbf{Appendix}$. We define $a_{j,j+k-m,k}=0$, for all $0\leq j,k\leq 2m$ and $a_{n,m+1+n,2m+1}=a_{2m+1,m+1+n,n}=0$, for all $0\leq n\leq m-1$. Combing $\mathbf{Proposition}$ 4.3 and $(\ref{Solution of linear system of all a _kln})$ with $\alpha=0$, we have

[TABLE]

By $(\ref{Solution of linear system of all a _kln})$ with $\alpha=0$, we have $|a_{j,j+k-m,k}|\leq\frac{1}{2}$, for all $j,k\geq 0$. By the definition of $\mathcal{F}_{m}$, we have

[TABLE]

The last two estimates are obtained by Young’s convolution inequality for $l^{1}*l^{2}\hookrightarrow l^{2}$.

∎

4.3.1 The Birkhoff normal form

Set $\chi^{m}_{\sigma}:=\exp(\epsilon\sigma X_{\mathcal{F}_{m}})$ the Hamiltonian flow of $\epsilon{\mathcal{F}_{m}}$, i.e.,

[TABLE]

We perform the canonical transformation $\Psi_{m,\epsilon}:=\chi^{m}_{1}=\exp(\epsilon X_{\mathcal{F}_{m}})$. We want to reduce the energy functional $\mathcal{H}^{m,\epsilon}$ to the following norm

[TABLE]

Since $\mathcal{R}_{m}$ depends only on low frequency Fourier modes $v_{1},\cdots,v_{3m}$, the high-frequency filtering $H^{s}$ norm of the solution of $\partial_{t}w(t)=X_{\mathcal{H}^{m,\epsilon}(t)\circ\Psi_{m,\epsilon}}(w(t))$ is handled by the Birkhoff normal form transformation. The estimate of $\|\mathbb{P}_{3m}(v)\|_{H^{s}}$ is given by $(\ref{uniform estimate of v})$. The next lemma will give the local existence of $\chi^{m}_{\sigma}$, for $|\sigma|\leq 1$ and estimate the difference between $v$ and $\Psi_{m,\epsilon}^{-1}(v)$.

Lemma 4.5**.**

For every $s>\frac{1}{2}$ and $m\in\mathbb{N}$, there exist two constants $\gamma_{m,s},C_{m,s}>0$ such that for all $v\in H^{s}_{+}$, if $\epsilon\|v\|_{H^{s}}\leq\gamma_{m,s}$, then $\chi^{m}_{\sigma}(v)$ is well defined on the interval $[-1,1]$ and the following estimates hold:

[TABLE]

The proof is based on a bootstrap argument, which is similar to $\mathbf{Lemma}$ 3.3, given by $\mathbf{Lemma}$ 3.1 with $m=2$. We shall perform the canonical transform below. Recall that $\Psi_{m,\epsilon}=\chi_{1}^{m}$.

Lemma 4.6**.**

For all $s>\frac{1}{2}$, $m\in\mathbb{N}$ and $0<\epsilon<\epsilon^{*}_{m}$, there exists a smooth mapping $\mathcal{Y}_{m}:\mathbb{R}\times H^{s}_{+}\longrightarrow H^{s}_{+}$ and a constant $C^{\prime}_{m,s}>0$ such that for all $t\in\mathbb{R}$, we have

[TABLE]

and $\sup_{t\in\mathbb{R}}\|\mathcal{Y}_{m}(t,v)\|_{H^{s}}\leq C^{\prime}_{m,s}\|v\|^{2}_{H^{s}}(1+\|v\|_{H^{s}})$, for all $v\in H^{s}_{+}$ such that $\epsilon\|v\|_{H^{s}}\leq\gamma_{m,s}$. Set $w(t):=\Psi_{m,\epsilon}^{-1}(v(t))$, then we have $\partial_{t}w(t)=X_{\mathcal{H}^{m,\epsilon}(t)\circ\Psi_{m,\epsilon}}(w(t))$ and

[TABLE]

if $\epsilon\|w(t)\|_{H^{s}}\leq\gamma_{m,s}$.

Proof.

For every $t\in\mathbb{R}$, we expand the energy $\mathcal{H}^{m,\epsilon}(t)\circ\Psi_{m,\epsilon}=\mathcal{H}^{m,\epsilon}(t)\circ\chi^{m}_{1}$ with Taylor’s formula at time $\sigma=1$ around [math]. Since $\chi^{m}_{0}=\mathrm{Id}_{H^{s}_{+}}$, we have

[TABLE]

and

[TABLE]

Since we have the homological system $\begin{cases}\{\mathcal{F}_{m},\mathcal{H}_{0}^{m}\}+\mathcal{H}^{m}_{1}=\mathcal{R}_{m}\\ \{\mathcal{F}_{m},\mathcal{L}_{m}\}+\tilde{\mathcal{N}}_{2}=0\\ \end{cases}$ in $\mathbf{Proposition}$ 4.4, we have

[TABLE]

where $\mathcal{G}_{m}(t,\sigma)=\{\mathcal{F}_{m},(1-\sigma)\mathcal{R}_{m}+\sigma\mathcal{H}^{m}_{1}+\varphi^{\prime}(t)((\sigma-1)\tilde{\mathcal{N}}_{2}+\frac{1}{2}\mathcal{N}_{2})\}$. We set

[TABLE]

then we get $X_{\mathcal{H}^{m,\epsilon}(t)\circ\chi^{m}_{1}}=X_{\mathcal{H}^{m}_{0}}+\varphi^{\prime}(t)X_{\mathcal{L}_{m}}+\epsilon\left(X_{\mathcal{R}_{m}}+\varphi^{\prime}(t)(-X_{\tilde{\mathcal{N}}_{2}}+\frac{1}{2}X_{\mathcal{N}_{2}})\right)+\epsilon^{2}\mathcal{Y}_{m}(t)$.

Since $\mathcal{F}_{m}$, $\mathcal{H}^{1}_{m}$ and $\mathcal{R}_{m}$ are homogeneous series of order $3$ with uniformly bounded coefficients, $\mathcal{N}_{2}$ and $\tilde{\mathcal{N}}_{2}$ are homogeneous series of order $2$ with uniformly bounded coefficients, we have

[TABLE]

because for $\mathcal{J}_{m}(v)=\sum_{j-l+k=m}\mathrm{Re}(b_{j,l,k}v_{j}\overline{v}_{l}v_{k})$ with $\sup_{j-l+k=m}|b_{j,l,k}|<+\infty$, we have

[TABLE]

and $\{\mathcal{F}_{m},\mathcal{N}_{2}\}(v)=-2\mathrm{Im}(\sum_{j-l+k=m}a_{j,l,k}v_{j}\overline{v}_{l}v_{k})$. Recall that $\sup_{0<\epsilon<\epsilon^{*}_{m}}\sup_{t\in\mathbb{R}}|\varphi^{\prime}(t)|<+\infty$ and $X_{\mathcal{N}_{4}}(v)=-4i\Pi(|v|^{2}v)$, then we have

[TABLE]

By using $\mathbf{Lemma}$ 4.5 and $(\ref{composition symplectic gradient})$, for all $v\in H^{s}_{+}$ such that $\epsilon\|v\|_{H^{s}}\leq\gamma_{m,s}$, we have

[TABLE]

and $\sup_{t\in\mathbb{R}}\|X_{\mathcal{N}_{4}\circ\chi^{m}_{1}}(v)\|_{H^{s}}\lesssim_{m,s}\|v\|^{2}_{H^{s}}(1+\|v\|_{H^{s}})$. Then we obtain the estimate of $\mathcal{Y}_{m}$.

Since $w(t)=\chi_{-1}^{m}(v(t))$, we have the following infinite dimensional Hamiltonian system on the Fourier modes:

[TABLE]

because for all $n\geq 3m+1$, we have

[TABLE]

Consequently, if $\epsilon\|w(t)\|_{H^{s}}\leq\gamma_{m,s}$, then we have

[TABLE]

∎

4.3.2 End of the proof of $\mathbf{Theorem}$ 1.8

The proof is completed by a bootstrap argument and estimate $(\ref{estimates of w(t) epsilon alpha=0})$, obtained by the Birkhoff normal form transformation. It suffices to prove the case $u(0)\in C^{\infty}_{+}$ by the same density argument in the proof of $\mathbf{Proposition}$ 1.7.

Proof.

For all $m\in\mathbb{N}$ and $s\geq 1$, we recall that $u(t,x)=e^{i\arg u_{m}(t)}(e^{imx}+\epsilon v(t,x))$, $\partial_{t}v(t)=X_{\mathcal{H}^{m,\epsilon}(t)}(v(t))$ and $w(t)=\chi^{m}_{-1}(v(t))$. In $\mathbf{Proposition}$ 4.2, we have shown that there exists $A_{m,s}\geq 1$ such that $\sup_{0<\epsilon<1}\|v(0)\|_{H^{s}}\leq A_{m,s}$. By using $(\ref{uniform estimate of v})$, we have

[TABLE]

Set $K_{m,s}:=\max(6A_{m,s},6(1+9m^{2})^{\frac{s}{2}}I_{m})$, $\epsilon_{m,s}:=\min(\epsilon^{*}_{m},\frac{\gamma_{m,s}}{3K_{m,s}},\frac{1}{48C_{m,s}K_{m,s}})$ and

[TABLE]

We choose $\epsilon\in(0,\epsilon_{m,s})$. Since $\epsilon=\epsilon\|v(0)\|_{H^{s}}\leq\epsilon A_{m,s}\leq\gamma_{m,s}$, $\mathbf{Lemma}$ 4.5 yields that

[TABLE]

So we have $\|w(0)\|_{H^{s}}\leq\frac{K_{m,s}}{3}$. For all $t\in[0,T_{m,s}]$, we have $\epsilon\|v(t)\|_{H^{s}}\leq{\gamma_{m,s}}$. Hence $\mathbf{Lemma}$ 4.5 gives the following estimate

[TABLE]

So we have $\epsilon\sup_{0\leq t\leq T_{m,s}}\|w(t)\|_{H^{s}}\leq\gamma_{m,s}$, which implies that

[TABLE]

in $\mathbf{Lemma}$ 4.6. Set $d_{m,s}:=\frac{1}{486K_{m,s}^{2}C^{\prime}_{m,s}}$. We can obtain the following estimate:

[TABLE]

for all $0\leq t\leq\min(T_{m,s},\frac{d_{m,s}}{\epsilon^{2}})$. We use $\mathbf{Lemma}$ 4.5 again to obtain that

[TABLE]

for all $t\in[0,\frac{d_{m,s}}{\epsilon^{2}}]$. In the case $t<0$, we use the same procedure and we replace $t$ by $-t$. Consequently, we have

[TABLE]

∎

5 Comparison to NLS equation

Although we have some similar results for the NLS equation, there are still some differences between the NLS equation and the NLS-Szegő equation. We denote by $u=u(t,x)=\sum_{n\geq 0}u_{n}(t)e^{inx}$ the solution of the NLS equation

[TABLE]

In Fourier modes, we have $i\partial_{t}u_{n}=n^{2}u_{n}+\sum_{k_{1}-k_{2}+k_{3}=n}u_{k_{1}}\overline{u}_{k_{2}}u_{k_{3}}$. Fix $m\in\mathbb{Z}$, for every $n\in\mathbb{Z}$, we define $v_{n}:=u_{n+m}e^{i(m^{2}+2nm)t}$. Then $\|v(t)\|_{L^{2}}=\|u(t)\|_{L^{2}}$ and we have

[TABLE]

If $u$ is localized in the $m$-th Fourier mode, then $v$ is localized on the zero mode. Thus the orbital stability of the traveling wave $\mathbf{e}_{m}$ can be reduced to the case $m=0$. In Faou–Gauckler–Lubich $[\mathbf{\ref{faou--Gauckler--lubich Sobolev Stability of Plane Wave}}]$, $H^{s}$-orbital stability of plane wave solutions is established by limiting the mass of the initial data $\|u_{0}\|_{H^{s}}$ to a certain full measure subset of $(0,+\infty)$ for the defocusing cubic Schrödinger equation with the time interval $[-\epsilon^{-N},\epsilon^{-N}]$, for all $N\geq 1$ and $s\gg 1$. However, this above transformation $u\mapsto v$ does not preserve the $L^{2}$ norm for the NLS-Szegő equation and formula $(\ref{v equation for comparation NLS NLSSzego})$ fails too.

On the other hand, the approach that we use to prove $\mathbf{Theorem}$ 1.8 does not work for $(\ref{NLS comparation})$. In fact, the negative high frequency Fourier modes in the term $\mathcal{H}^{m}_{1}(v)=\mathrm{Re}\int_{\mathbb{S}^{1}}e^{-imx}|v|^{2}v$ can not be cancelled by the homological equation. The energy functional of the equation of $v$ can not be reduced as $\mathcal{H}_{0}^{m}+\mathcal{O}(\epsilon^{2})$ by using the same method in this paper. The Szegő filtering to $(\ref{NLS comparation})$ makes it possible to cancel all the high frequency resonances in the term $\mathcal{R}_{m}=\{\mathcal{F}_{m},\mathcal{H}_{0}^{m}\}+\mathcal{H}^{m}_{1}$. Then we use a bootstrap argument to deal with the equation $\partial_{t}w(t)=X_{\mathcal{H}^{m,\epsilon}(t)\circ\chi_{1}^{m}}(w(t))$ after the Birkhoff normal form transformation.

6 Appendix

We give the details of the homological equation in $\mathbf{Proposition}$ 4.4 and prove $(\ref{Solution of linear system of all a _kln})$ and $\mathbf{Proposition}$ 4.3 in the case $\alpha=0$.

For all $v\in\bigcup_{n\in\mathbb{N}}\mathbb{P}_{n}(C^{\infty}_{+})$, $\mathcal{H}_{0}^{m}(v)=\frac{1}{2}\|\partial_{x}v\|_{L^{2}}^{2}+\frac{1-m^{2}}{2}\|v\|_{L^{2}}^{2}+\frac{1}{2}\int_{\mathbb{S}^{1}}\mathrm{Re}(e^{2imx}\overline{v}^{2})\mathrm{d}x$ and ${\mathcal{F}_{m}}(v)=\sum_{\begin{smallmatrix}j-l+k=m\\ j,k,l\in\mathbb{N}\end{smallmatrix}}\mathrm{Re}(a_{j,l,k}v_{j}\overline{v}_{l}v_{k})$. With the convention $v_{n}=0$, for all $n<0$, we have

[TABLE]

Combing $(\ref{Poisson bracket})$, we have the Poisson bracket of $\mathcal{F}_{m}$ and $\mathcal{H}^{m}_{0}$,

[TABLE]

The term $\mathcal{A}_{4}=\sum_{\begin{smallmatrix}k-n+l=m\\ k,l,n\in\mathbb{N},n\leq 2m\end{smallmatrix}}a_{k,n,l}v_{k}v_{l}v_{2m-n}$ has only finite term depending on low frequency Fourier modes $v_{1},\cdots,v_{3m}$. We divide $\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$ into two parts. The first part consists of only low frequency resonances, the second part consists of high frequency resonances.

[TABLE]

where $\mathcal{A}_{1}^{low}$ consists of all the resonances $v_{j}\overline{v}_{l}v_{k}$ such that $j,k\leq 2m$,

[TABLE]

and $\mathcal{A}_{1}^{\geq 2m+1}$ contain other resonances $v_{j}\overline{v}_{l}v_{k}$ such that at least one of $j,k$ is strictly larger than $2m$.

[TABLE]

Then, we calculate $\mathcal{A}_{2}=\sum_{\begin{smallmatrix}k-n+l=m\\ k,l,n\in\mathbb{N}\end{smallmatrix}}(1+n^{2}-m^{2})a_{k,n,l}v_{k}\overline{v}_{n}v_{l}=\mathcal{A}_{2}^{low}+\mathcal{A}_{2}^{\geq 2m+1}$. $\mathcal{A}_{2}^{low}$ consists of all the resonances $v_{k}\overline{v}_{l}v_{n}$ such that $k,n\leq 2m$ or $l\leq 2m$.

[TABLE]

$\mathcal{A}_{2}^{\geq 2m+1}$ plays the same role as $\mathcal{A}_{1}^{\geq 2m+1}$.

[TABLE]

At last, $\mathcal{A}_{3}=\sum_{\begin{smallmatrix}k-l+n=m\\ k,l,n\in\mathbb{N},n\leq 2m\end{smallmatrix}}2\overline{a}_{k,l,n}v_{l}\overline{v}_{k}v_{2m-n}=\sum_{\begin{smallmatrix}j-l+k=m\\ j,k,l,n\in\mathbb{N},j\leq 2m\end{smallmatrix}}2\overline{a}_{l,k,2m-j}v_{k}\overline{v}_{l}v_{j}$. Using the same idea, we have $\mathcal{A}_{3}=\mathcal{A}_{3}^{low}+\mathcal{A}_{3}^{\geq 2m+1}$, with

[TABLE]

and

[TABLE]

Recall that $\mathcal{H}_{1}^{m}(v)=\mathrm{Re}(\int_{\mathbb{S}^{1}}e^{-imx}|v|^{2}v)=\sum_{k-l+n=m}\mathrm{Re}(v_{k}\overline{v}_{l}v_{n})$. A similar calculus as in the case of $\mathcal{A}_{1}$ shows that $\mathcal{H}_{1}^{m}(v)=\mathrm{Re}(B^{low}+B^{\geq 2m+1})$ with

[TABLE]

[TABLE]

At last we define \mathrm{Reson}^{low}(v)\Big{|}_{\alpha=0}=-i(\mathcal{A}_{1}^{low}+\mathcal{A}_{2}^{low}+\mathcal{A}_{3}^{low}+\mathcal{A}_{4})+B^{low} and

[TABLE]

Then we have \{\mathcal{F}_{m},\mathcal{H}^{m}_{0}\}(v)+\mathcal{H}_{1}^{m}(v)=\mathrm{Re}(\mathrm{Reson}^{low}(v)\Big{|}_{\alpha=0}+\mathrm{Reson}^{\geq 2m+1}(v)\Big{|}_{\alpha=0}). Since \mathrm{Reson}^{low}(v)\Big{|}_{\alpha=0} contains only finite terms and depends only on $v_{1},\cdots,v_{3m}$, so is \mathcal{R}_{m}=\mathrm{Re}(\mathrm{Reson}^{low}(v)\Big{|}_{\alpha=0}). For high frequency resonances, we compute

[TABLE]

After replacing $j$ by $2m-j$ in the sum $m+1\leq j\leq 2m$, we have the equivalence between \mathrm{Reson}_{\geq 2m+1}\Big{|}_{\alpha=0}=0 and $(\ref{Solution of linear system of all a _kln})$ in the case $\alpha=0$.