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.
