Reference Work Entry

Encyclopedia of Complexity and Systems Science

pp 4960-4971

Inverse Scattering Transform and the Theory of Solitons

  • Tuncay AktosunAffiliated withDepartment of Mathematics, University of Texas at Arlington

Definition of the Subject

A general theory to solve NPDEs does not seem to exist. However, there are certain NPDEs, usually first order in time, for which the corresponding IVPs can be solved by the IST method. Such NPDEs are sometimes referred to as integrable evolution equations. Some exact solutions to such equations may be available in terms of elementary functions, and such solutions are important to understand nonlinearity better and they may also be useful in testing accuracy of numerical methods to solve such NPDEs.

Certain special solutions to some of such NPDEs exhibit particle-like behaviors. A single-soliton solution is usually a localized disturbance that retains its shape but only changes its location in time. A multi-soliton solution consists of several solitons that interact nonlinearly when they are close to each other but come out of such interactions unchanged in shape except for a phase shift.

Integrable NPDEs have important physical app ...

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