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On a criterion of integrability by the inverse scattering transform. (English. Russian original) Zbl 0898.35094
Dokl. Math. 53, No. 1, 94-97 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 346, No. 1, 28-30 (1996).
Several nonlinear PDEs which are integrable by the method of the inverse scattering transform are considered in this note. The cases of (1) the Korteweg- de Vries equation, (2) the nonlinear Schrödinger equation and (3) the equations of self-induced transparency are under consideration. The author writes down these equations using the dependent variable \(E(x, t)\) and its inverse \(r=1/E\). The linear terms of the equations in this form are not varied from the usual form, and the nonlinear part in the right-hand side contains ordinary derivatives. The author constructs the solution of prescribed equations by seeking a simultaneous solution of the two equations. One of these equations is linear with respect to \(r\) and coincides with the linearized initial equation, the second equation is an ordinary differential equation for the function \(E\) or its module where the second independent variable serves as parameter. The solutions to equations (1)-(3) thus constructed coincide with the single-soliton solutions to these equations obtained by the inverse scattering transform.
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.)
35R30 Inverse problems for PDEs