New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems.(English)Zbl 0873.65111

The purpose is to analyze the behaviour of the solution of the Korteweg-de Vries (KdV) equation $$u_t- 6uu_x+ \varepsilon^2 u_{xxx} =0$$, $$u(x,0) =u_0(x)$$, $$\varepsilon>0$$ when $$\varepsilon\downarrow 0$$. The approach to this small dispersion problem is based on the Riemann-Hilbert (RH) formulation of the inverse scattering transform, involving a steepest descent method for oscillatory RH problems.
A systematic procedure for determining the contour of the main contribution to the solution of the RH problem is introduced. The initial value problem of the modulation equations is reduced to the solution of a set of algebraic equations constrained by algebraic inequalities.

MSC:

 65Z05 Applications to the sciences 35Q53 KdV equations (Korteweg-de Vries equations) 35Q15 Riemann-Hilbert problems in context of PDEs 81U40 Inverse scattering problems in quantum theory
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