# Analysis of Stability of Traveling Wave for Kadomtsev-Petviashvili Equation

Authors: Jun Liu; Xi Liu; Gui Mu; Chunyan Zhu; Jie Fu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper presents the boundedness and uniform boundedness of traveling wave solutions for the Kadomtsev-Petviashvili (KP) equation. They are discussed by means of a traveling wave transformation and Lyapunov function.


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## Body

## 1. Introduction

We consider the Kadomtsev-Petviashvili (KP) equation:(1)utx+6uxuxx+uxxxx+uyy+cu=0.
It is well known that Kadomtsev-Petviashvili equation arises in a number of remarkable nonlinear problems both in physics and mathematics. By using various methods and techniques, exact traveling wave solutions, solitary wave solutions, doubly periodic solutions, and some numerical solutions have been obtained in [1–6].In this paper, (1) can be changed into an ordinary differential equation by using traveling wave transformation; the boundedness and uniform boundedness of solution for the resulting ordinary differential equation are discussed using the method of Lyapunov function.


## 2. The Boundedness

Taking a traveling wave transformationξ=αx+βy+γt in (1), then (1) can be transformed into the following form:
(2)u(4)+(γα3+β2α4+6α2u)u′′+6α2u′2+cα4u=0.In general, we use the following system, which is equivalent to (2):
(3)u(4)+au′′′+f(t,u,u′′)+g(u′)+du=p(t,u,u′,u′′,u′′′),
where
(4)f(t,u,u′)=(γα3+β2α4+6α2u)u′′,g(u′)=6α2u′2,p(t,u,u′,u′′,u′′′)=-au′′′,d=cα4.We consider the following system, which is equivalent to (3):
(5)x1′=x2,x2′=x3,x3′=x4,x4′=-ax4-f(t,x1,x2,x3)-g(x2)-dx1+p(t,x1,x2,x3,x4).Theorem 1.
If the following conditions hold for the system (5):(i)
there are positive constantsa, b, d, δ, k, and λ such that
(6)k≤b2λ,abg(x2)x2-[g(x2)x2]2-a2d≥δ,(x2≠0).(ii)
f(t,x1,x2,0)=0, 0≤f(t,x1,x2,x3)/x3-b≤2δλ/k  (x2≠0).(iii)
x3ft′(t,x1,x2,x3)+x2x3fx1′(t,x1,x2,x3)+x32fx2′(t,x1,x2,x3)≤0.(iv)
|p(t,x1,x2,x3,x4)|≤q(t)(x12+x22+x32+x42)1/2, where q(t) is a nonnegative continuous function and ∫0∞q(t)dt<∞.
Then, all the solutions of system (5) are bounded.Proof.
We first construct the Lyapunov functionV=V(t,x1,x2,x3,x4) defined by
(7)V=b2(2x4+ax3+bx2)2+2bd(2x3+ax2+bx1)2+(b2-4d)(ax4+bx2)2+4ab2×∫0x2[g(x2)x2-adb]x2dx2+[2b(b2-4d)+4a2d]x32+8b2∫0x3[f(t,x1,x2,x3)x3-b]x3dx3.
It follows from conditions (i) and (ii) that(8)b2-4d≥0,0≤∫0x2[g(x2)x2-adb]x2dx2≤a(b2-d)2bx22,0≤∫0x3[f(t,x1,x2,x3)x3-b]x3dx3≤δλkx32.
Summing up the above discussions, we get
(9)V≥2b(b2-4d)x32+4a2dx32.
Thus, we deduce that the functionV(t,x1,x2,x3,x4) defined in (7) is a positive definite function which has infinite inferior limit and infinitesimal upper limit. Hence, there exsits a positive constant ε(>0) such that
(10)V(t,x1,x2,x3,x4)≥ε(x12+x22+x32+x42).
Taking the total derivative of (7) with respect to t along the trajectory of (5), we obtain
(11)dVdt=-2ab2[x4+1ag(x2)]2-2b3x2x3[f(t,x1,x2,x3)x3-b]-2ab2[f(t,x1,x2,x3)x3-b]x32-2b2a×[abg(x2)x2-g2(x2)x22-a2d]x22+4b2∫0x3ft′(t,x1,x2,x3)dx3+4b2x2∫0x3fx1′(t,x1,x2,x3)dx3+4b2x3∫0x3fx2′(t,x1,x2,x3)dx3+2b2(bx2+ax3+2x4)p(t,x,x2,x3,x4).
By using conditions (i) and (iii), it follows that
(12)dVdt≤-2b2δax22-2b3x2x3[f(t,x1,x2,x3)x3-b]-2ab2[f(t,x1,x2,x3)x3-b]x32+2b2(bx2+ax3+2x4)p(t,x,x2,x3,x4).
According to (ii), we have
(13)2b3x2x3[f(t,x1,x2,x3)x3-b]+2ab2[f(t,x1,x2,x3)x3-b]x32=-b42a[f(t,x1,x2,x3)x3-b]x22+2b2a[f(t,x1,x2,x3)x3-b]·(ax3+b2x2)2≥-b42a[f(t,x1,x2,x3)x3-b]x22=-b42a·2δλkx22=-b4δλakx22.
Hence,
(14)dVdt≤-2b2δax22+b4δλakx22+2b2(bx2+ax3+2x4)p(t,x,x2,x3,x4)=-b2δax22+b2δak(b2λ-k)x22+2b2(4+a2+b2)1/2(x22+x32+x42)1/2×(x22+x32+x42)1/2q(t)≤-b2δax22+2b2(4+a2+b2)1/2(x22+x32+x42)q(t)≤-b2δax22+2b2(4+a2+b2)1/2·q(t)·Vε≤2b2(4+a2+b2)1/2·q(t)ε·V≡φ(V,t).
Thus, all the solutions of system (5) are bounded.Theorem 2.
Let conditions (i)–(iv) of Theorem1 be satisfied for the system (5), and let the following condition hold:
(15)(4+a2+b2)1/2·q(t)ε·V-δax22≤0.
Then, all the solutions of system (5) are uniformly bounded.Proof.
It is clear that the functionV(t,x1,x2,x3,x4) defined in (7) satisfies the conditions (15), therefore, all the solutions of system (5); are uniformly bounded [7].



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*Source: 230871-2013-03-06.xml*