Traveling wave solutions of degenerate coupled Korteweg-de Vries equation
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.
scientific article; zbMATH DE number 6358464