An integrable generalized Korteweg–de Vries equation with pseudo-peakons

Abstract

In this letter, we propose a hierarchy of new generalized Korteweg–de Vries equations and find that the generalized Korteweg–de Vries equation has a class of pseudo-peakons. The so-called pseudo-peakon looks like a peakon, but it is continuously differentiable everywhere and its second-order derivative goes to infinity at some point. The infinite sequence of conserved quantities of the generalized Korteweg–de Vries equation is constructed by using Riccati equations of the Lax pair. Based on the study of traveling-wave solutions, we obtain various exact solutions of the generalized Korteweg–de Vries equation and its modified equation, including pseudo-peakon, periodic pseudo-peakon, cuspon, periodic cuspon, loop and periodic loop solutions.