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The collisionless shock phenomenon for the focusing nonlinear Schrödinger equation with decaying initial data. (English. Abridged French version) Zbl 0844.35117
Summary: We investigate a phenomenon similar to the “collisionless shock” introduced by Sagdeev, Gurevich and Pitaevski, appearing in the long time asymptotics of the focusing NLS equation. We find that this is connected with the existence of a real spectral singularity for the associated Lax operator. We make use of the modern version of inverse scattering in terms of a Riemann-Hilbert problem.
35Q55 NLS equations (nonlinear Schrödinger equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
76L05 Shock waves and blast waves in fluid mechanics