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Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE. (English) Zbl 1010.35066

Summary: We study exchange of stability in the dynamics of solitary wave solutions under changes in the nonlinear balance in a 1+1 evolutionary partial differential equation related both to shallow water waves and to turbulence. We find that solutions of the equation \(m_t+um_x+bu_xm=\nu m_{xx}\) with \(m=u-\alpha^2u_{xx}\) for fluid velocity \(u(x,t)\) change their behavior at the special values \(b=0,\pm 1,\pm 2,\pm 3\).

MSC:

35L05 Wave equation
74J35 Solitary waves in solid mechanics
35L70 Second-order nonlinear hyperbolic equations
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References:

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