Asymptotic Analysis of a Perturbed Periodic Solution for the KdV Equation

Jiang, Xinhua; Wong, R.

doi:10.1111/j.1467-9590.2005.00332.x

Cited by 2 publications

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“…In Ref. 19, a formal expansion was constructed for the solution of the KdV equation with periodic boundary conditions via the method of multiple scales. Small amplitude perturbation solutions were found in Ref.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of solutions to periodic problem for the Korteweg–de Vries–Burgers equation

Naumkin

Villela-Aguilar

2022

Stud Appl Math

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We consider the periodic problem for the Korteweg–de Vries–Burgers (KdVB) equation with pumping vtgoodbreak−vxxgoodbreak+α3vxxxgoodbreak=vgoodbreak+∂x()v2,x∈normalΩ,tgoodbreak>0,v(0,x)goodbreak=v0()x,x∈normalΩ,\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{c}v_{t}-v_{xx}+\frac{\alpha }{3}v_{xxx}=v+\partial _{x}{\left(v^{2}\right)} ,\text{ }x\in \Omega ,t>0,\\ v(0,x)=v_{0}{\left(x\right)} ,\text{ }x\in \Omega , \end{array} \right.} \end{equation*}where normalΩ=false[−π,πfalse]$\Omega =[ -\pi ,\pi ]$, α∈double-struckR$\alpha \in \mathbb {R}$. We study the solutions, which satisfy the periodic boundary conditions vfalse(t,xfalse)=vfalse(t,2π+xfalse)$v(t,x) =v(t,2\pi +x)$ for all x∈double-struckR$x\in \mathbb {R}$ and t>0$t>0$, with the 2π—periodic initial data v0(x)$v_{0}(x)$. Our aim is to find large time asymptotic profile of solutions. The large time asymptotic behavior of solutions to the Cauchy problem for the KdVB equations with different dissipative terms was extensively studied. In the present paper we find the asymptotic profile of solutions for large time. We develop the approach started in Refs. [1, 2, 3]. We prove the following asymptotics for the solutions: vt,x=met+ε1+2aε2tcosnormalΛ−b2alog()1+2aε2t+O1+ε2t−1\begin{align*} & v{\left(t,x\right)} =me^{t}\\ & +\frac{\varepsilon }{\sqrt {1+2a\varepsilon ^{2}t}}\cos {\left(\Lambda -\frac{b}{2a}\log {\left(1+2a\varepsilon ^{2}t\right)} \right)} \\ & +O{\left({\left(1+\varepsilon ^{2}t\right)} ^{-1}\right)} \end{align*}as t→∞$t\rightarrow \infty$ uniformly with respect to x∈normalΩ$x\in \Omega$, where m=∫Ωv0(x)dx$m=\int _{\Omega }v_{0}(x) dx$, ε=|∫Ωeixv0false(xfalse)dx|$\varepsilon =|\int _{\Omega }e^{ix}v_{0}(x) dx|$, a=12|μ|2$a=\frac{12}{|\mu |^{2}}$, b=8α|μ|2$b=\frac{8\alpha }{|\mu |^{2}}$, μ=3−2iα$\mu =3-2i\alpha$, Λ is a constant.

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Section: Introductionmentioning

confidence: 99%