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Traveling wave solutions for a class of nonlinear dispersive equations. (English) Zbl 1011.35014

Consider the following nonlinear dispersion equations \[ K(m,n): u_t+a(u^m)_x+(u^n)_{xxx}=0,\quad m,n\geq 1,\tag{1} \] where \(m,n\) are integers, \(a\) is a real parameter, \(u(x,t)\) is the unknown functions of \(x\) and \(t\). The traveling solutions of (1): \(u=\varphi(x-ct)=\varphi(\xi)\) satisfy the system \[ \frac{d\varphi}{d\xi}=y,\quad \frac{dy}{d\xi}=\frac{-n(n-1)\varphi^{n-2}y^2-a\varphi^m+c\varphi+g}{n\varphi^{n-1}},\tag{2} \] where \(g\) is the integral constant. (2) has a singular straight line \(\varphi=0\) and a first integral \[ H(\varphi, y)=\varphi^n \biggl(n\varphi^{n-2}y^2+\frac{2a}{m+n}\varphi^m-\frac{2c}{n+1}\varphi-\frac{2g}{n}\biggr)=h. \] By using the method of phase plane analysis to (2), the authors investigate the solitary and periodic traveling wave solutions of (1). The bifurcation theory of dynamical systems is employed to do qualitative analysis. All possible phase portraits, as \(n=2\), \(n=2l\) \((l\geq 2)\), \(n=2l+1\), in the parametric space for the traveling system (2) are obtained and drew out. It is shown that the existence of the singular straight line in the traveling system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. While the parameter pair \((g,c)\) lies on the bifurcation curve, the dynamics of (2) is also analyzed. At last, the authors consider the existence and the relationship for the different solitary and periodic wave solutions, and give a complete description of them. This is a wonderful work for the investigation of solitary and periodic wave solutions for the nonlinear dispersion system.

MSC:

35B32 Bifurcations in context of PDEs
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References:

[1] doi:10.1006/jdeq.1995.1043 · Zbl 0820.35118
[2] doi:10.1103/PhysRevLett.70.564 · Zbl 0952.35502
[3] doi:10.1103/PhysRevLett.73.1737 · Zbl 0953.35501
[4] doi:10.1016/S0375-9601(97)00241-7 · Zbl 1052.35511
[5] doi:10.1016/S0167-2789(98)00148-1 · Zbl 0938.35172
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