Weak kink wave and peakon solutions with an arbitrary parameter

Abstract

In this paper, we reconsider the weak-kink and peakon solution of the integrable equation (Alber et al 2001 Commun. Math. Phys. 221, 197)mt=bux+12α1m(u2−ux2)x+12α2(2mux+mxu),m=u−uxx,where b, α1, α2 are three real numbers, through redefining the value of the conventional sign function sgn(x) at x = 0 as sgnθ (0) = θ, where θ is an arbitrary number. Then, our computation generates more generalized peakon and multi-peakon solutions involved in parameter θ, and all of those classical peakon solutions are special cases where we place θ = 0 here in the present paper. Furthermore, we also study a weak kink wave solution and the interactional dynamics of peakons with the weak kink wave if the parameter . All wave profiles are depicted through graphs for different values of parameter θ.