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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:
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q51Soliton-like equations
35Q58Other completely integrable PDE (MSC2000)
37K40Soliton theory, asymptotic behavior of solutions