Asymptotics for the Camassa-Holm equation
Seminar Room 1, Newton Institute
I will present recent results on asymptotic behaviors for the Ca-massa–Holm (CH) equation ut - utxx + 2?ux + 3uux = 2uxuxx + uuxxx on the line, ? being a nonnegative parameter.
Firstly, I will describe the long-time asymptotic behavior of the solution u? (x, t), ? > 0 of the initial-value problem with fast decaying initial data u0(x). It appears that u? (x, t) behaves differently in different sectors of the (x, t)-half-plane. Then I will analyse the behavior of u? (x, t) as ? 0.
The methods are inverse scattering in a matrix Riemann-Hilbert approach and Deift and Zhou’s nonlinear steepest descent method.
Work in collaboration with Dmitry Shepelsky.
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