Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations
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
- 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.
1. Inverse scattering problems for systems of first-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 n first-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 . 9.2. A Riemann–Hilbert problem. 9.3. Hirota’s method. 10. The B¨acklund 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. 10.1. The BTs for NLEEs defined in the whole space. 10.2. The BT between a constructed common solution of both equations in the Lax pair and the solution of the associated IBVP for NLEEs on a half line. 10.3. The BT between a common solution of both systems (7.70), (7.71) and the solution of the IBVP for KdV equation (7.1) with a negative dispersive coefficient.