Soliton resolution for the short-pulse equation

Abstract

In this paper, we apply ∂‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equationuxt=u+16(u3)xx,u(x,0)=u0(x)∈H(R), where H(R)=W3,1(R)∩H2,2(R) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale (y,t). In any fixed space-time coneC(y1,y2,v1,v2)={(y,t)∈R2:y=y0+vt,y0∈[y1,y2], v∈[v1,v2]}, we compute the long time asymptotic expansion of the solution u(x,t), which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N(I)-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second |t|−1/2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O(|t|−1) from a ∂‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable.