Peaked and Smooth Solitons forK*(4,1)Equation

Abstract

This paper is contributed to explore all possible single peak solutions for theK*(4,1)equationut=uxu2+2α(uuxxx+2uxuxx). Our procedure shows that theK*(4,1)equation either has peakon, cuspon, and smooth soliton solutions when sitting on a nonzero constant pedestallimξ→±∞u=A≠0or possesses compacton solutions only whenlimξ→±∞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 theK*(4,1)equation.

Kink-like wave and compacton-like wave solutions for generalized KdV equation

“…Xie et al [15] studied the single peak solitary solutions of (4) under inhomogeneous boundary condition lim ξ →±∞ u =Ã. They obtained peakon, compacton, cuspon and smooth soliton solutions based on the method of phase portrait analysis technique.…”

Peakon, Cuspon, Compacton, and Loop Solutions of a Three-Dimensional 3DKP(3, 2) Equation with Nonlinear Dispersion

“…This case is completely similar to the case of < 0, ( , ) ∈ 3+ . (1) has a cuspon solution given by (23).…”

“…Remark 6. To the best of our knowledge, solutions (23) obtained for (1) have not been reported in the literature.…”

“…The straight line = in the --phase plane is called a singular straight line. There are some general theories and methods for investigating this type of singular traveling wave systems [21][22][23][24][25]. It is now well known that the existence of the singular straight lines implies the occurrence of some nonsmooth dynamical behaviors and curve breaking phenomena of the traveling wave solutions of such system, more precisely the so-called peakon and cuspon, and so forth.…”

Explicit kink-like and compacton-like wave solutions for a generalized KdV equation

“…For l = 3, p = 2, α > 0, Li [22] derived some exact explicit parametric representations of breaking loop solutions. Under inhomogeneous boundary condition lim ξ →±∞ u = A, Xie et al [23] and Fu et al [24] studied the single-peak solitary solutions for l = 4, p = 1 and l = 3, p = 1, respectively.…”