Soliton resolution for the short-pulse equation.

In this paper, we apply ∂ ‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equation u x t = u + 1 6 (u 3) x x , u (x , 0) = u 0 (x) ∈ H (R) , where H (R) = W 3 , 1 (R) ∩ H 2 , 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 cone C (y 1 , y 2 , v 1 , v 2) = { (y , t) ∈ R 2 : y = y 0 + v t , y 0 ∈ [ y 1 , y 2 ] , v ∈ [ v 1 , v 2 ] } , 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. [ABSTRACT FROM AUTHOR]