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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.
MSC:
 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