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.
We study a class of high dispersive cubic-quintic nonlinear Schrödinger equations, which describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. Applying bifurcation theory of dynamical systems and the Fan sub-equations method, more types of exact solutions, particularly solitary wave solutions, are obtained for the first time.
Communicated by P. M. MarianoBy using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution.
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