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A new integrable equation with cuspons and W/M-shape-peaks solitons. (English) Zbl 1112.37063
Summary: We propose a new completely integrable wave equation: $$m_{t}+m_{x}(u^{2} - u\_{x}^{2})+2m^{2}u_{x}=0, m=u - u_{xx}$$. The equation is derived from the two dimensional Euler equation and is proven to have Lax pair and bi-Hamiltonian structures. This equation possesses new cusp solitons–cuspons, instead of regular peakons $$ce^{ - |x - ct|}$$ with speed c. Through investigating the equation, we develop a new kind of soliton solutions–“W/M”-shape-peaks solitons. There exist no smooth solitons for this integrable water wave equation.

MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q51 Soliton equations 35Q58 Other completely integrable PDE (MSC2000) 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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References:
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