Traveling wave solutions of degenerate coupled Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: Traveling wave solutions of degenerate coupled Korteweg-de Vries equation
zbMATH Open document ID: 1303.35090
DOI: 10.1063/1.4893636
author: Metin Gürses
author: Aslı Pekcan
published in: Journal of Mathematical Physics
publication date: 2014-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1308.5649
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35C08
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35C07
zbMATH DE Number: 6358464
zbMATH Keywords: KdV equation
zbMATH Keywords: KB equations
zbMATH Keywords: Jacobi elliptic functions
MaRDI profile type: Publication
OpenAlex ID: W3124640855
arXiv ID: 1308.5649
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review text: Multi-component Kaup-Boussinesq (KB) equations can be obtained from the Lax operator NEWLINE\[NEWLINEL=D^2-\sum_{k=1}^l\lambda^{k-1}q^k(x,t),NEWLINE\]NEWLINE where \(q^k(x,t)\), \(k = 1, 2, ..., l\) are the multi-KB fields and \(l\geq2\) is a positive integer. The multi-system of KB equation is given as NEWLINE\[NEWLINE\begin{aligned} u_t &= \frac{3}{2} uu_x + q^2_x,\\ q^2_t &= q^2u_x + \frac{1}{2}uq^2_x + q^3_x,\\ \vdots \\ q^{l-1}_t& = q^{l-1}u_x + \frac{1}{2}uq^{l-1}_x + v_x, \\ v_t &= -\frac{1}{4}u_{xxx} + vu_x + \frac{1}{2}uv_x, \end{aligned}NEWLINE\]NEWLINE where \(q^1 = u\) and \(q^l = v\). The aim of this paper is to study symmetry reduced (traveling waves) equations of the Kaup-Boussinesq (KB) type of coupled degenerate KdV equations for \(l=2\), i.e., NEWLINE\[NEWLINE\begin{aligned} u_t &= \frac{3}{2} uu_x + v_x, \\ v_t &= -\frac{1}{4}u_{xxx} + vu_x + \frac{1}{2}uv_x.\end{aligned}NEWLINE\]NEWLINE Depending upon the zeros of a fourth degree polynomial, the authors have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutions. All such possible solutions are given explicitly in the form of Jacobi elliptic functions. Graphs of some exact solutions in solitary wave and periodic shapes are exhibited. The authors have also initiated the work on the cases for \(l=3\) and \(l=4\) and have given some results concerning these cases.
reviewed by: Ahmed Lesfari
description: scientific article; zbMATH DE number 6358464