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In this paper, we propose a new completely integrable wave equation: mt+mx(u2−ux2)+2m2ux=0, m=u−uxx. 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.

A new integrable equation with cuspons and W/M-shape-peaks solitons