In this paper, we propose a new completely integrable wave equation: mt+mx u2 −ux 2 +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− −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.
Qiao, Zhijun, "A new integrable equation with cuspons and W/M-shape-peaks solitons" (2006). Mathematical and Statistical Sciences Faculty Publications and Presentations. 11.
Journal of Mathematical Physics