Periodic parabola solitons for the nonautonomous KP equation

TL;DR

This paper derives and analyzes periodic parabola soliton solutions for the nonautonomous KP equation using Painleve analysis and Hirota bilinear method, revealing parameter constraints and convergence conditions.

Contribution

It introduces a novel class of solutions for the nonautonomous KP equation with detailed parameter constraints and convergence analysis.

Findings

Six undetermined parameters in the solutions.

Conditions for the convergence of solutions.

Features of typical solution cases.

Abstract

Kadomtsev-Petviashvili (KP) equation, who can describe different models in fluids and plasmas, has drawn investigation for its solitonic solutions with various methods. In this paper, we focus on the periodic parabola solitons for the (2+1) dimensional nonautonomous KP equations where the necessary constraints of the parameters are figured out. With Painleve analysis and Hirota bilinear method, we find that the solution has six undetermined parameters as well as analyze the features of some typical cases of the solutions. Based on the constructed solutions, the conditions of their convergence are also discussed.

Equations

$$(u_t+6u{u}_x+u_{xxx}{)}_x+3\sigma u_{yy}=0,\sigma =\pm 1,$$

$$[u_t+a(t)u{u}_x+b(t)u_{xxx}{]}_x+\sigma c(t)u_{yy}+d(t)u_{xy}+[e_1(y,t)+e_2(t)x]u_{xx}+f(t)u_x=0,$$

$$a(t)=6\rho b(t{)}^{\frac{3}{4}}c(t{)}^{\frac{1}{4}}{e}^{\int [f(t)-2e_2(t)]dt},$$

$$e_1(y,t)=-\frac{e_2(t{)}^2{y}^2}{2\sigma c(t)}-\frac{3b^{\prime }(t{)}^2{y}^2}{16b(t{)}^2\sigma c(t)}+\frac{3c^{\prime }(t{)}^2{y}^2}{16\sigma c(t{)}^3}+\frac{3e_2(t)b^{\prime }(t)y^2}{8b(t)\sigma c(t)}$$

$$+\frac{e_2(t)\sigma c^{\prime }(t)y^2}{8c(t{)}^2}-\frac{e_2^{\prime }(t)y^2}{2\sigma c(t)}+\frac{b^{\prime \prime }(t)y^2}{8b(t)\sigma c(t)}-\frac{\sigma c^{\prime \prime }(t)y^2}{8c(t{)}^2}+{\alpha }_2(t)y+{\alpha }_1(t),$$

$$u=\frac{2}{\rho }b(t{)}^{\frac{1}{4}}c(t{)}^{-\frac{1}{4}}{e}^{-\int [f(t)-2e_2(t)]dt}(\log \Phi {)}_{xx}+\Psi (x,y,t),$$

$$\Psi (x,y,t)={\Psi }_1(t)x+{\Psi }_2(y,t),$$

$$\hskip 28.45274pt\Psi_{1}(t)=\frac{1}{a(t)}\Big{[}\beta_{3}(t)-e_{2}(t)+\frac{b^{\prime}(t)}{4b(t)}-\frac{c^{\prime}(t)}{4c(t)}\Big{]}\,,$$

$${\Psi }_2(y,t)=-\frac{a(t){\Psi }_1(t{)}^2{y}^2}{2\sigma c(t)}-\frac{f(t){\Psi }_1(t)y^2}{2\sigma c(t)}-\frac{{\Psi }_1^{\prime }(t)y^2}{2\sigma c(t)}+{\beta }_2(t)y+{\beta }_1(t),$$

$$\hskip 28.45274pt\Big{[}D_{x}D_{t}+b(t)\,D_{x}^{4}+\sigma c(t)\,D_{y}^{2}+d(t)\,D_{x}\,D_{y}+\varphi_{1}(x,y,t)\,D_{x}^{2}+\varphi_{2}(t)\frac{\partial}{\partial x}\Big{]}\Phi\cdot\Phi=0\,,$$

$$\displaystyle\hskip 28.45274pt\varphi_{1}(x,y,t)=\alpha_{2}(t)y+\alpha_{1}(t)+6\,\rho\,b(t)^{\frac{3}{4}}\,c(t)^{\frac{1}{4}}\,e^{\int{\![f(t)-2e_{2}(t)]dt}}\Big{[}\beta_{2}(t)y+\beta_{1}(t)\Big{]}$$

$$+\frac{x{b}^{\prime }(t)}{4b(t)}-\frac{x{c}^{\prime }(t)}{4c(t)}-\frac{{\beta }_3(t{)}^2{y}^2}{2\sigma c(t)}+\frac{{\beta }_3(t)b^{\prime }(t)y^2}{8b(t)\sigma c(t)}+\frac{3{\beta }_3(t)\sigma c^{\prime }(t)y^2}{8c(t{)}^2}-\frac{{\beta }_3^{\prime }(t)y^2}{2\sigma c(t)}+x{\beta }_3(t),$$

$${\phi }_2(t)=\frac{b^{\prime }(t)}{4b(t)}-\frac{c^{\prime }(t)}{4c(t)},$$

$$\frac{\partial }{\partial x}\Phi \cdot \Phi =2\Phi {\Phi }_x,$$

$$\displaystyle D_{x}^{m}D_{y}^{n}D_{t}^{p}a\cdot b\equiv\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial x^{{}^{\prime}}}\right)^{m}\,\left(\frac{\partial}{\partial y}-\frac{\partial}{\partial y^{{}^{\prime}}}\right)^{n}\,\left(\frac{\partial}{\partial t}-\frac{\partial}{\partial t^{{}^{\prime}}}\right)^{p}\,a(x,y,t)\,b(x^{{}^{\prime}},y^{{}^{\prime}},t^{{}^{\prime}})\bigg{|}_{x^{{}^{\prime}}=x,\,y^{{}^{\prime}}=y,\,t^{{}^{\prime}}=t}\,.\$$

$$\Phi =b_1{e}^{k_1(t)x+l_{11}(t)y+l_{12}(t)y^2+w_1(t)}+b_2\cos \left[k_2(t)x+l_{21}(t)y+l_{22}(t)y^2+w_2(t)\right]$$

$$+b_3{e}^{-[k_1(t)x+l_{11}(t)y+l_{12}(t)y^2+w_1(t)]}.$$

$$k_i(t)=C_{ki}b(t{)}^{-\frac{1}{4}}c(t{)}^{\frac{1}{4}}{e}^{-\int {\beta }_3(t)dt},$$

$$l_{i2}(t)=\frac{1}{2}\sigma C_{ki}b(t{)}^{-\frac{1}{4}}c(t{)}^{-\frac{3}{4}}{\beta }_3(t)e^{-\int {\beta }_3(t)dt}.$$

$$\displaystyle\hskip 28.45274ptm(t)=\int b(t)^{-\frac{1}{4}}\,c(t)^{-\frac{3}{4}}\,e^{\int\beta_{3}(t)\,dt}\Big{[}6\,\rho\,b(t)^{\frac{3}{4}}\,c(t)^{\frac{5}{4}}\,\Lambda_{2}(t)\,e^{\int\left(f(t)-2e_{2}(t)\right)\,dt}$$

$$\displaystyle\hskip 65.44133pt+c(t)\alpha_{10}(t)+\sigma d(t)\beta_{3}(t)\Big{]}\,dt\,,$$

$$\displaystyle\hskip 28.45274ptn_{1}(t)=b(t)^{\frac{1}{4}}c(t)^{\frac{3}{4}}\bigg{[}-C_{{k1}}\left(-3C_{{k2}}^{4}+\sigma C_{{l1}}^{2}-\sigma C_{{l2}}^{2}\right)-2\sigma C_{{l1}}m(t)\left(C_{{k1}}^{2}+C_{{k2}}^{2}\right)$$

$$\displaystyle\hskip 65.44133pt-\sigma C_{{k1}}m(t)^{2}\left(C_{{k1}}^{2}+C_{{k2}}^{2}\right)+2C_{{k1}}^{3}C_{{k2}}^{2}-C_{{k1}}^{5}-2\sigma C_{{k2}}C_{{l1}}C_{{l2}}\bigg{]}\,,$$

$$\displaystyle\hskip 28.45274ptn_{2}(t)=b(t)^{\frac{1}{4}}c(t)^{\frac{3}{4}}\bigg{[}-C_{{k2}}\left(-3C_{{k1}}^{4}+\sigma C_{{l1}}^{2}-\sigma C_{{l2}}^{2}\right)-2\sigma C_{{l2}}m(t)\left(C_{{k1}}^{2}+C_{{k2}}^{2}\right)$$

$$\displaystyle\hskip 65.44133pt-\sigma C_{{k2}}m(t)^{2}\left(C_{{k1}}^{2}+C_{{k2}}^{2}\right)+2C_{{k2}}^{3}C_{{k1}}^{2}-C_{{k2}}^{5}-2\sigma C_{{k1}}C_{{l1}}C_{{l2}}\bigg{]}\,.$$

$$l_{1i}(t)=e^{-2\int {\beta }_3(t)dt}\left(C_{li}-C_{ki}m(t)\right),$$

$$\displaystyle\hskip 14.22636ptw_{i}(t)=\int e^{-3\int\beta_{3}(t)\,dt}\bigg{[}\frac{1}{C_{{k1}}^{2}+C_{{k2}}^{2}}n_{i}(t)-6\rho b(t)^{\frac{1}{2}}c(t)^{\frac{1}{2}}C_{ki}\Lambda_{1}(t)e^{\int(f(t)-2e_{2}(t)+2\beta_{3}(t))\,dt}$$

$$\displaystyle\hskip 51.21495pt-b(t)^{-\frac{1}{4}}c(t)^{\frac{1}{4}}C_{{ki}}\alpha_{9}(t)e^{\int 2\beta_{3}(t)\,dt}+d(t)e^{\int\beta_{3}(t)\,dt}\left(C_{{ki}}m(t)-C_{{li}}\right)\bigg{]}\,dt\,.$$

$$3\left(C_{k1}^2+C_{k2}^2\right){}^2\left(4b_1{b}_3{C}_{k1}^2+b_2^2{C}_{k2}^2\right)-\sigma \left(b_2^2-4b_1{b}_3\right)\left(C_{k2}{C}_{l1}-C_{k1}{C}_{l2}\right){}^2=0,$$

$$g_1=k_1(t)x+l_{11}(t)y+l_{12}(t)y^2+w_1(t),$$

$$g_2=k_2(t)x+l_{21}(t)y+l_{22}(t)y^2+w_2(t),$$

$$\Phi =b_1{e}^{g_1}+b_2\cos g_2+b_3{e}^{-g_1}.$$

$$u$$

$$\displaystyle=A(t)\Big{[}\frac{b_{1}k_{1}(t){}^{2}e^{g_{1}}+b_{3}k_{1}(t){}^{2}e^{-g_{1}}-b_{2}k_{2}(t){}^{2}\cos g_{2}}{b_{1}e^{g_{1}}+b_{3}e^{-g_{1}}+b_{2}\cos g_{2}}-\frac{\left(b_{1}k_{1}(t)e^{g_{1}}-b_{3}k_{1}(t)e^{-g_{1}}-b_{2}k_{2}(t)\sin g_{2}\right){}^{2}}{\left(b_{1}e^{g_{1}}+b_{3}e^{-g_{1}}+b_{2}\cos g_{2}\right){}^{2}}\Big{]}\,.$$

$$u=A(t)\Big{[}\frac{b_{1}k_{1}(t){}^{2}e^{g_{1}}+b_{3}k_{1}(t){}^{2}e^{-g_{1}}}{b_{1}e^{g_{1}}+b_{3}e^{-g_{1}}+b_{2}\cos g_{2}}-\frac{\left(b_{1}k_{1}(t)e^{g_{1}}-b_{3}k_{1}(t)e^{-g_{1}}\right){}^{2}}{\left(b_{1}e^{g_{1}}+b_{3}e^{-g_{1}}+b_{2}\cos g_{2}\right){}^{2}}\Big{]}\,.$$

$$\Phi (x,y,t)=\frac{\epsilon b_2{C}_{l2}}{\sqrt{C_{l2}^2+3{\text{σC}}_{k1}^4}}\cosh \left[k_1(t)x+l_{12}(t)y^2+l_{11}(t)y+w_1(t)\right]$$

Full text

PACS numbers: 05.45.Yv, 02.30.Ik, 47.35.Fg

Keywords: Periodic Parabola solitons; Nonautonomous Kadomtsev-Petviashvili equation; Bilinear method

I. Introduction

In several aspects of physics, some dynamical systems can be described by nonlinear partial differential equations (PDEs) [1]. While investigating them, we pay attention to some soliton solutions for their significance both in theoretical and practical values . The Kadomtsev-Petviashvili (KP) equation, as follows:

[TABLE]

is such a nonlinear PDE which can describe surface wave with low amplitude [2]. Several researches focusing on its solutions have emerged including algebraically decaying solutions [3], lump solutions [4], rogue waves [5] and periodic solitons [6]. For more complicated models, such as the ones considered the variation of depth and density, nonautonomous KP equation with variable coefficients should be investigated[7, 8] and some researches have been finished [9, 10, 11, 12, 13, 14, 15, 16, 17]. In this paper, we set the nonautonomous KP equation in this form:

[TABLE]

where $x$ and $y$ are scaled space coordinates, $t$ is scaled time coordinate, $a(t)$, $b(t)$, $c(t)$, $d(t)$, $e_{1}(y,t)$, $e_{2}(t)$ and $f(t)$ are inhomogeneous coefficients while $b(t)$ and $c(t)$ are both positive and $\sigma=\pm 1$. In the following, Eq. (2) with $\sigma=+1$ and $\sigma=-1$ will be named KP-I and KP-II equation, respectively. The convergence and interactions of parabola exponent solitons have been already investigated for KP-I equation [14], but it remains unknown when trigonometric function is considered. In this case, the features and convergence of the solutions will become more complicated where the difference of KP-I and KP-II equation can be distinct. Such characteristics are drawing more and more attention in fluid and plasma physics, especially the singularities, which may appeal to some novel physics phenomena [11].

II. Periodic parabola solitons

The Painlevé analysis [18] for KP-I equation has been finished in Ref. [14], whose $c(t)$ should be replaced by $\sigma c(t)$ somewhere when KP-II equation is taken into consideration. The integrable conditions are:

[TABLE]

[TABLE]

where $\rho$ is a nonzero constant, $\alpha_{1}(t)$ and $\alpha_{2}(t)$ are introduced arbitrary functions of $t$, and ′ denotes the derivative with respect to $t$.

In this paper, we propose the similar generalized dependent variable transformation with Ref. [14],

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\Phi$ is a function of $x$, $y$ and $t$, $\beta_{1}(t)$, $\beta_{2}(t)$ and $\beta_{3}(t)$ are arbitrary functions of $t$. Under the conditions (3)-(8), Eq. (2) can be transformed into its bilinear form as below,

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and $D_{x}^{m}D_{t}^{n}$ is the Hirota bilinear derivative operator [19, 20] defined by

[TABLE]

Similar to the periodic linear soliton solutions in Ref. [20], the solution of Eq. (9) can be set periodic parabola solitonic as below (without loss of generality, we assume $b_{2}>0$):

[TABLE]

Substituting Eq. Periodic parabola solitons for the nonautonomous KP equation into Eq. 9, we can derive an equation consisting of different terms, whose coefficient should be equaled to 0 due to the arbitrary coordinates. Such equations yield eight explicit constraints and one implicit constraint. The explicit ones are ($i=1,2$):

[TABLE]

[TABLE]

To simplify our process, we introduce three new functions $m(t)$, $n_{1}(t)$ and $n_{2}(t)$. They are defined as

[TABLE]

Then we have

[TABLE]

[TABLE]

Besides, the implicit constraint is:

[TABLE]

which is the key to determine the convergence.

In one word, the independence of $k_{1}(t),k_{2}(t),l_{11}(t),l_{12}(t),l_{21}(t),l_{22}(t),w_{1}(t),w_{2}(t),b_{1},b_{2},b—_{3}$ can be changed to be any six ones of seven parameters $C_{k1},C_{k2},C_{l1},C_{l2},b_{1},b_{2},b_{3}$.

III. Solutions’ convergence

Here we set the moving characteristic line as $g_{1}$ ans $g_{2}$ for simplification:

[TABLE]

Since there is a term of $(\text{log}\Phi)_{xx}$ in the solutions, $u$ will be infinite in some area when $\Phi=0$. Thus, we want to find the maximum or the minimum value of $\Phi$ in different situations. In the following discussion, $h_{1}$ and $h_{2}$ represent the extreme line for $g_{1}$ and $g_{2}$, respectively.

(a)

$b_{1}>0,b_{3}>0$ and $k_{1}(t)k_{2}(t)\neq 0$. In this case, the characteristic line will be parabolic and $\Phi$ must be positive in some area. The minimum of $b_{2}\cos g_{2}$ will be $-b_{2}$ when $h_{2}=(2n+1)\pi$, $n\in N$. Therefore, $b_{2}\cos g_{2}$ will take the minimum value $-b_{2}$ in a set consisted of an infinite number of uniformly spaced parabolas $\{h_{2}\}$. Similarly, $b_{1}e^{g_{1}}+b_{3}e^{-g_{1}}$ will take the minimum value $2\sqrt{b_{1}b_{3}}$ in a parabola: $h_{1}=\frac{1}{2}\ln\frac{b_{3}}{b_{1}}$. We can modify the ratio between $b_{1}$ and $b_{3}$ to shift $h_{1}$. However, unless $h_{1}$ and $h_{2}$ is just the same characteristic line, it’s obvious that the parabola $h_{1}=\frac{1}{2}\ln\frac{b_{3}}{b_{1}}$ will intersect with the set $\{h_{2}\}$ of $h_{2}=(2n+1)\pi$ in some area, where $\Phi$ take the minimum value $2\sqrt{b_{1}b_{3}}-b_{2}$.

With the restrict of Eq. (19), it’s easy to prove $\sigma\left(b_{2}^{2}-4b_{1}b_{3}\right)>0$. Thus, if $\sigma=1$, $2\sqrt{b_{1}b_{3}}-b_{2}<0$. In one word, for continuous function $\Phi$, there must be some place where $\Phi=0$, leading to the conclusion that $u$ will not always limited in this condition for KP-I unless $g_{1}$ and $g_{2}$ is the same characteristic line. To the opposite, when $\sigma=-1$, which means $2\sqrt{b_{1}b_{3}}-b_{2}>0$, $\Phi$ is positive everywhere so $u$ obtains convergence for KP-II.

For the case when $g_{1}$ and $g_{2}$ is the same characteristic line (which means $C_{k1}/C_{k2}=C_{l1}/C_{l2}$), due to Eq. (19), we can derive $b_{1}b_{3}<0$, which leads to (d).

(b)

$b_{1}<0,b_{3}<0$ and $k_{1}(t)k_{2}(t)\neq 0$. This is similar to Case 1. Here $b_{2}-2\sqrt{b_{1}b_{3}}$ will be the maximum value $\Phi$ takes and $\Phi$ must be negative some place. Like Case 1, for KP-I $b_{2}-2\sqrt{b_{1}b_{3}}>0$, then $\Phi$ will have zero point making $u$ unlimited. On the contrary, for KP-II the maximum value is still below zero so $u$ has a bound.

(c)

$b_{1}b_{3}>0$ while $k_{1}(t)k_{2}(t)=0$. In this case, one of the characteristic line will turn to be a straight line but the conclusion from Case 1 and 2 will not change because a parabola will always intersect with a set of infinite uniformly-spaced straight lines and a straight line will always intersect with a set of infinite uniformly-spaced parabolas or straight lines.

(d)

$b_{1}b_{3}\leq 0$. Here the high slope of exponent function will definitely bring $\Phi$ zero points, for both KP-I and KP-II.

In one word, if $b_{1}b_{3}\leq 0$, both KP-I and KP-II will be unlimited in some places. If $b_{1}b_{3}>0$, only KP-II can keep the convergence when the shapes of the two characteristic lines are different.

IV. Different solutions

When we solve Eq. (19) with symbolic computation, the solutions can be classified by 5 cases depending on the characteristic line and whether $b_{3}$ equals to 0. In the discussion and figures, we set $\beta_{3}(t)=e_{2}(t)-\frac{b^{{}^{\prime}}(t)}{4b(t)}+\frac{c^{{}^{\prime}}(t)}{4c(t)}$ and $\beta_{1}(t)=\beta_{2}(t)=0$, to assure the solutions’ decaying when $(x^{2}+y^{2})^{\frac{1}{2}}\to\infty$ [14]. Thus, the solution will become as (For simplification, we set $A(t)=\frac{2}{\rho}\,b(t)^{\frac{1}{4}}\,c(t)^{-\frac{1}{4}}\,\,e^{-\int{\![f(t)-2e_{2}(t)]dt}}$)

[TABLE]

In the following discussion, we set $b(t)=c(t)=d(t)=e_{2}(t)=\rho=\alpha_{1}(t)=1$, $f(t)=2$, $\alpha_{2}(t)=0$ to draw figures.

**Case 1: Two different parabola characteristic lines

**In this case, $C_{k1}C_{k2}\neq 0$ and $C_{k1}/C_{k2}\neq C_{l1}/C_{l2}$, so the two characteristic lines will have the terms of $x$ and $y^{2}$, with different coefficients.

With expression ((d)), the solution can be regarded as an interaction between the periodic part and the exponent part. When we just set $b_{2}=0$, the solution will only have one peak along a parabola. Therefore, for expression ((d)), we can consider the trigonometric function offering some impact in some parallel parabolas. For example, when we set $C_{k1}=0.25$, $C_{k2}=0.7$, $C_{l1}=0$, $C_{l2}=4$, $b_{2}=4$, $b_{3}=1$, the solitons will be shown as in Fig. 1: (When dealing with divergence in contour plots, we analyze $\arctan u$ instead to show the features more clearly. Such method is utilized in other cases.)

Modifying the parameters can change the shape of solutions to some extent, including offering some symmetry. For KP equations with $m=C_{l1}/C_{k1}$, $l_{11}(t)$ will be 0 according to Eq. (17). The peak formed by the exponent part will be at a parabola symmetry upon $y=0$.

**Case 2: Straight characteristic line for trigonometric function part

**If $C_{k2}=0$, $g_{2}$ will turn to be a straight line perpendicular to the $x$-axis as $l_{21}(t)y+\omega_{2}(t)$. Meanwhile, the solution becomes as

[TABLE]

Here, the trigonometric term will affect the exponent one in equidistant vertical line. When we set $C_{k1}=1$, $C_{k2}=0$, $C_{l1}=1$, $C_{l2}=2$, $b_{1}=\frac{1}{3}$, $b_{2}=1$, $b_{3}=3$, such solution is demonstrated in Fig. 2

In this case, we get $b_{1}=\frac{b_{2}^{2}\sigma C_{l2}^{2}}{4b_{3}\left(3C_{k1}^{4}+\sigma C_{l2}^{2}\right)}$ due to Eq. (19) , where $b_{2},b_{3},C_{k1},C_{l1}$ and $C_{l2}$ are free parameters. When setting b_{1}=b_{3}=\frac{\varepsilon b_{2}C_{l2}}{2\sqrt{C_{l2}^{2}+3\text{\sigmaC}_{k1}^{4}}}\quad(\epsilon=\pm 1), we have

[TABLE]

When $\sigma=-1$ , the existence condition of solution for KP-II enquation is given by $C_{l2}^{2}-3C_{k1}^{4}>0$. However, it is only need that parameters $C_{l2},C_{k1}$,satisfy $C_{l2}C_{k1}\neq 0$ for KP-I equation ($\sigma=1$)

When $\sigma=-1$ and $C_{l2}^{2}-3C_{k1}^{4}<0$ ,taking $b_{1}=-b_{3}=\frac{\varepsilon b_{2}C_{l2}}{2\sqrt{3C_{k1}-C_{l2}^{2}}}$ ,we can obtain

[TABLE]

**Case 3: Straight characteristic line for exponent part

**Similar to case 2, if $C_{k1}=0$, $g_{1}$ will be perpendicular to the $x$-axis as $l_{11}(t)y+\omega_{1}(t)$, and the solution is

[TABLE]

Contrary to the above cases, now there only exist the trigonometric terms in the numerator, which dominate in the solution. Therefore, we will have peak in parallel parabolas, modulated by the exponent term periodically in $y$-direction. Fig. 3 shows one solution of this case with $C_{k1}=0$, $C_{k2}=1$, $C_{l1}=2$, $C_{l2}=1$, $b_{2}=1$, $b_{3}=0.5$:

Here, we get $b_{1}=\frac{b_{2}^{2}\sigma C_{l1}^{2}-3b_{2}^{2}C_{k2}^{4}}{4b_{3}\sigma C_{l1}^{2}}$ , where $b_{2},b_{3},C_{k2},C_{l1}$ and $C_{l2}$ are free parameters. When $\sigma=1$ and $3C_{k2}^{2}-C_{l1}^{2}<0$ ,taking $b_{1}=-b_{3}=\frac{b_{2}\varepsilon\sqrt{3C_{k2}^{2}-C_{l1}^{2}}}{2C_{l1}}$ ,we can obtain

[TABLE]

If setting b_{1}=b_{3}=\frac{b_{2}\varepsilon\sqrt{C_{l1}^{2}-3\text{\sigmaC}_{k2}^{4}}}{2C_{l1}} ,we have

[TABLE]

When $\sigma=1$ , the existence condition of solution of solution for KP-I enquation is given by $C_{l1}^{2}-3C_{k2}^{2}>0$. However,it is only need that parameters $C_{l1},C_{k2}$,satisfy $C_{l1}C_{k2}\neq 0$ for KP-II equation ($\sigma=-1$).

**Case 4: Two parabola characteristic lines with the same shape.

**This means $C_{k1}/C_{k2}=C_{l1}/C_{l2}=K$, we get $b_{1}=-\frac{b_{2}^{2}k^{2}}{4b_{3}}$ by Eq. (19) , where $b_{2},b_{3},K,C_{k1}$ and $C_{l1}$ are free parameters. Since $b_{1}b_{2}<0$, the solution must be divergence along one characteristic line.

Similarly above, if we set $b_{1}=-b_{3}=\frac{\varepsilon b_{2}k}{2}$ , then we have

[TABLE]

**Case 5: $b_{3}=0$

**Here the solution changes into

[TABLE]

A little different with above, in this case the implicit restraint Eq. (19) will be:

[TABLE]

which is obvious $\sigma$ must be 1 here because both terms of this equation is positive. Therefore, periodic parabola solutions with $b_{3}=0$ only appears in KP-I equation. In Figs. 4(a) and  4(b), we set $C_{k1}=0.5$, $C_{k2}=0.5$, $C_{l1}=4\left(\frac{1}{2}-\frac{\sqrt{3}}{8}\right)$, $C_{l2}=2$, $b_{1}=3$, $b_{2}=1$.

According to Eq. (29), we will find $C_{k1}C_{l2}=0$ when $C_{k2}=0$. Furthermore,$C_{k1}=0$ will lead to straight characteristic line solution, and $C_{l2}=0$ will make $g_{2}$ as trivial $\omega_{2}(t)$. As a result, here the characteristic line of trigonometric function mush be parabola.

Considering $C_{k1}=0$ with Eq. (29) we can get $C_{k2}=\frac{\epsilon\sqrt{|C_{l1}|}}{\sqrt[4]{3}}$, $g_{1}$ will turn to be a straight line perpendicular to the $x$-axis as $l_{11}(t)y+\omega_{1}(t)$. Meanwhile, the solution becomes as

[TABLE]

Because $g_{1}$ turns to be linear as $l_{11}(t)y+\omega_{1}(t)$, the exponent term will adjust the amplitude of the periodic wave in $y$-direction. When $C_{k1}=0.5$, $C_{l1}=-\frac{\sqrt{3}}{4}$, $C_{l2}=2$, $b_{1}=3$, $b_{2}=1$, the solution is like Figs. 4(c) and  4(d):

V. Conclusions

Based on Painlevé analysis and Hirota bilinear method, periodic parabola solitons for nonautonomous (2+1) dimensional KP equation are obtained. The eleven undetermined parametric functions of periodic parabola solitons are limited to six independent coefficients with one implicit constraint and eight explicit ones. The condition of the solitons’ convergence is also found as KP-II equation of different characteristic lines while $b_{1}b_{3}>0$. Besides, five typical cases, classified upon the shape of characteristic lines of the solutions, are discussed and illustrated in this paper, which may appeal to various physics models. Here, all of the results base on the real coefficients. If complex coefficients were considered, we could discuss whether features of the solitons are more complicated and worthy classifying into more cases.

VI. Acknowledgements

This work has been supported by the National Natural Science Foundation of China under Grant No. 11302014, and by the Fundamental Research Funds for the Central Universities under Grant Nos. 50100002013105026 and 50100002015105032 (Beihang University).