Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations

ebook ∣ Chapman & Hall/CRC Monographs and Research Notes in Mathematics

By Pham Loi Vu

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Format

ebook

ISBN

9781032429212

Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

Author

Pham Loi Vu

Publisher

CRC Press

Release

15 May 2023

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Subjects

Mathematics Nonfiction

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Inverse Scattering Problems and Their Applications to Nonlinear Integrable Equations, Second Edition is devoted to inverse scattering problems (ISPs) for differential equations and their applications to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, differential equations, and equations of mathematical physics. This book is intended for a wide community working with ISPs and their applications. There is an especially strong traditional community in mathematical physics.

In this monograph, the problems are presented step-by-step, and detailed proofs are given for considered problems to make the topics more accessible for students who are approaching them for the first time.

New to the Second Edition

  • All new chapter dealing with the Bäcklund transformations between a common solution of both linear equations in the Lax pair and the solution of the associated IBVP for NLEEs on the half-line
  • Updated references and concluding remarks

    • Features

  • Solving the direct and ISP, then solving the associated initial value problem (IVP) or initial-boundary value problem (IBVP) for NLEEs are carried out step-by-step. The unknown boundary values are calculated with the help of the Lax (generalized) equations, then the time-dependent scattering data (SD) are expressed in terms of preassigned initial and boundary conditions. Thereby, the potential functions are recovered uniquely in terms of the given initial and calculated boundary conditions. The unique solvability of the ISP is proved and the SD of the scattering problem is described completely. The considered ISPs are well-solved.
  • The ISPs are set up appropriately for constructing the Bӓckhund transformations (BTs) for solutions of associated IBVPs or IVPs for NLEEs. The procedure for finding a BT for the IBVP for NLEEs on the half-line differs from the one used for obtaining a BT for non-linear differential equations defined in the whole space.
  • The interrelations between the ISPs and the constructed BTs are established to become new powerful unified transformations (UTs) for solving IBVPs or IVPs for NLEEs, that can be used in different areas of physics and mechanics. The application of the UTs is consistent and efficiently embedded in the scheme of the associated ISP.

    • Format

    ebook

    ISBN

    9781032429212

    Series

    Chapman & Hall/CRC Monographs and Research Notes in Mathematics

    Author

    Pham Loi Vu

    Publisher

    CRC Press

    Release

    15 May 2023

    Share

    Subjects

    Mathematics Nonfiction
    Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations