Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations is devoted to inverse scattering problems (ISPs) for differential equations and their application to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, partial differential equations, equations of mathematical physics, and functions of a complex variable. This book is intended for a wide community working with inverse scattering problems and their applications; in particular, there is a traditional community in mathematical physics.
In this monograph, the problems are solved step-by-step, and detailed proofs are given for the problems to make the topics more accessible for students who are approaching them for the first time.
Features
The unique solvability of ISPs are proved. The scattering data of the considered inverse scattering problems (ISPs) are described completely.
Solving the associated initial value problem or initial-boundary value problem for the nonlinear evolution equations (NLEEs) is carried out step-by-step. Namely, the NLEE can be written as the compatibility condition of two linear equations. The unknown boundary values are calculated with the help of the Lax (generalized) equation, and then the time-dependent scattering data (SD) are constructed from the initial and boundary conditions.
The potentials are recovered uniquely in terms of time-dependent SD, and the solution of the NLEEs is expressed uniquely in terms of the found solutions of the ISP.
Since the considered ISPs are solved well, then the SPs generated by two linear equations constitute the inverse scattering method (ISM). The application of the ISM to solving the NLEEs is consistent and is effectively embedded in the schema of the ISM.
1 Inverse scattering problems for systems of rst-order ODEs on a
half-line. 1.1 The inverse scattering problem on a half-line with a potential
non-self-adjoint matrix. 1.2 The inverse scattering problem on a half-line
with a potential self-adjoint matrix. 2 Some problems for a system of
nonlinear evolution equations.on a half-line. 2.1 The IBVP for the system of
NLEEs. 2.2 Exact solutions of the system of NLEEs. 2.3 The Cauchy IVP problem
for the repulsive NLS equation. 3 Some problems for cubic nonlinear evolution
equations on a half-line . 3.1 The direct and inverse scattering problem. 3.2
The IBVPs for the mKdV equations. 3.3 Non-scattering potentials and exact
solutions. 3.4 The Cauchy problem for cubic nonlinear equation (3.3). 4 The
Dirichlet IBVPs for sine and sinh-Gordon equations. 4.1 The IBVP for the sG
equation. 4.2 The IBVP for the shG equation. 4.3 Exact soliton-solutions of
the sG and shG equations. 5 Inverse scattering for integration of the
continual system of nonlinear interaction waves. 5.1 The direct and ISP for a
system of rst-order ODEs. 5.2 The direct and ISP for the transport equation.
5.3 Integration of the continual system of nonlinear interaction waves. 6
Some problems for the KdV equation and associated inverse scattering. 6.1 The
direct and ISP. 6.2 The IBVP for the KdV equation. 6.3 Exact
soliton-solutions of the Cauchy problem for the KdV equation. 7 Inverse
scattering and its application to the KdV equation with dominant surface
tension. 7.1 The direct and inverse SP. 7.2 The system of evolution equations
for the scattering matrix. 7.3 The self-adjoint problem. 7.4 The
time-evolution of s(k; t) and solution of the IBVP. 8 The inverse scattering
problem for the perturbed string equation and its application to integration
of the two-dimensional generalization from Korteweg-de Vries equation. 8.1
The scattering problem. 8.2 Transform operators. 8.3 Properties of the
scattering operator. 8.4 Inverse scattering problem. 8.5 Integration of the
two-dimensional generalization from the KdV equation. 9 Connections between
the inverse scattering method and related methods. 9.1 Fokas's methodology
for the analysis of IBVPs, [ 29, 30, 31]. 9.2 A Riemann-Hilbert problem. 9.3
Hirota's method. 9.4 Backlund transformations.
Pham Loi Vu is a professor, doctor of mathematics, and a leading Vietnamese expert in inverse problems and their applications. He has authored 50 papers published in prominent journals, including Inverse Problems (Q1), Acta Applicandae Mathematicae (Q2), Journal of Nonlinear Mathematical Physics (Q2), and others.
His previous research focused on seismic waves in seismic prospecting for petroleum-gas complexes and on determining coordinates of epicenter in near earthquakes at the Institute of Geophysics of the Vietnam Academy. He is now a researcher at VAST's Institute of Mechanics and the National Foundation for Science and Technology Development (NAFOSTED).
