Peaked and Smooth Solitons for K*(4,1) Equation

Abstract

This paper is contributed to explore all possible single peak solutions for the $K^{*}(\mathrm{4,1})$ equation $u_t=u_x{u}^2+2\alpha (u{u}_{xxx}+2u_x{u}_{xx})$. Our procedure shows that the $K^{*}(\mathrm{4,1})$ equation either has peakon, cuspon, and smooth soliton solutions when sitting on a nonzero constant pedestal ${\text{lim}}_{\xi \to \pm \infty }u=A\ne 0$ or possesses compacton solutions only when ${\text{lim}}_{\xi \to \pm \infty }u=A=0$. We present a new smooth soliton solution in an explicit form. Mathematical analysis and numeric graphs are provided for those soliton solutions of the $K^{*}(\mathrm{4,1})$ equation.