Nonlinear Equations for Soliton Eigenfunctions Are the IST Integrable Equations

  • B. G. Konopelchenko
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)


A starting and basic point of the inverse spectral transform (IST) method is the representation of the given nonlinear partial differential equation as the compatibility condition of some nontrivial linear system
$$\begin{array}{*{20}{c}} {{{L}_{1}}(U;\lambda )\Psi } \hfill & = \hfill & {0,} \hfill \\ {{{L}_{2}}(U;\lambda )\Psi } \hfill & = \hfill & 0 \hfill \\ \end{array}$$
where L 1 and L 2 are usually differential operators the coefficients of which depend on the field U(x,t), U x , U xx ,... and spectral parameter λ (see e.g. [1–4]). Integrable equation for U arises after the elimination of the eigenfunction Ψ from the system (1) [1–4].


Compatibility Condition Nonlinear Integrable Equation mKdV Equation Eigenfunction Equation Linear Singular Integral Equation 
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Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • B. G. Konopelchenko
    • 1
  1. 1.Institute of Nuclear PhysicsNovosibirskUSSR

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