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Four types of bounded wave solutions of CH-\(\gamma\) equation. (English) Zbl 1117.35310

Summary: Recently, many authors have studied the following CH-\(\gamma\) equation: \[ u_t + c_0 u_x + 3uu_x - \alpha ^2(u_{xxt} + uu_{xxx} + 2u_x u _{xx}) + \gamma u_{xxx} = 0, \] where \(\alpha ^2\), \(c_0\) and \(\gamma\) are parameters. Its bounded wave solutions have been investigated mainly for the case \(\alpha^2 > 0\). For the case \(\alpha^2 < 0\), the existence of three bounded waves (regular solitary waves, compactons, periodic peakons) was pointed out by H. R. Dullin et al. [Fluid Dyn. Res. 33, No. 1–2, 73–95 (2003; Zbl 1032.76518)]. But the proof has not been given. In this paper, not only the existence of four types of bounded waves: periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case \(\alpha^2 < 0\). Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.

MSC:

35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
76B25 Solitary waves for incompressible inviscid fluids
34C12 Monotone systems involving ordinary differential equations

Citations:

Zbl 1032.76518
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References:

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