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.
Author(s): Pham Loi Vu
Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics
Edition: 2
Publisher: CRC Press/Chapman & Hall
Year: 2023
Language: English
Pages: 452
City: Boca Raton
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Acronyms
Preface
Preface for the second edition
Author
Introduction
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
1.1.1. The representation of the solution of system (1.7)
1.1.2. The Jost solutions of system (1.7)
1.1.3. The scattering function S(λ) and non-real eigenvalues
1.1.4. Connection between the analytic solution and Jost solutions
1.1.5. The scattering data
1.1.6. Derivation of systems of fundamental equations
1.1.7. The estimates for the functions f(–x) and g(x)
1.1.8. The unique solvability of systems of fundamental equations
1.1.9. The description of the scattering data
1.2. The inverse scattering problem on a half-line with a potential self-adjoint matrix
1.2.1. The unique solvability of the self-adjoint problem
1.2.2. The Jost solutions of system (1.84)
1.2.3. The scattering function and its properties
1.2.4. The relation between the functions f(x, ξ), g(x, ξ) and f(ξ), g(ξ)
1.2.5. The inverse scattering problem
1.2.6. The complete description of the scattering function
2. Some Problems for A System of Non-Linear Evolution Equations on A Half-Line
2.1. The IBVP for the system of NLEEs
2.1.1. The Lax compatibility condition
2.1.2. The time-dependence of the scattering function
2.1.3. Evaluation of unknown BVs
2.1.4. The time-dependence of the scattering data
2.1.5. The solution of the IBVP for the system of NLEEs (2.5)
2.1.6. The IBVP for the attractive NLS equation
2.2. Exact solutions of the system of NLEEs
2.2.1. Exact solutions of fundamental equations
2.2.2. The time-dependence of standardized multipliers and an exact solution of system (2.5)
2.2.3. An exact solution of the attractive NLS equation
2.3. The Cauchy IVP problem for the repulsive NLS equation
3. Some Problems for Cubic Non-Linear Evolution Equations on A Half-Line
3.1. The direct and inverse scattering problem
3.1.1. The representation of the solution of system (3.4)
3.1.2. The Jost solutions of system (3.4)
3.1.3. The scattering function S(λ) and non-real eigenvalues
3.1.4. Connection between the analytic solution and Jost solutions
3.1.5. The scattering data
3.1.6. The systems of fundamental equations
3.1.7. The complete description of the scattering data
3.2. The IBVPs for the mKdV equations
3.2.1. The Lax compatibility condition
3.2.2. The time-dependence of the scattering function
3.2.3. Evaluation of unknown BVs
3.2.4. The time-dependence of the scattering data
3.2.5. The solution of the IBVPs for mKdV equations
3.2.6. Relation between solutions of the mKdV and KdV equations
3.3. Non-scattering potentials and exact solutions
3.3.1. Exact solutions of systems of fundamental equations
3.3.2. The time-dependence of standardized multipliers and an exact solution of system (3.41)
3.3.3. Exact solutions of equations mKdV and KdV
3.4. The Cauchy problem for cubic non-linear equation (3.3)
4. The Dirichlet IBVPs for sine and sinh-Gordon Equations
4.1. The IBVP for the sG equation
4.1.1. The Jost solutions
4.1.2. The Lax compatibility condition
4.1.3. Evaluation of unknown BVs
4.1.4. The time-dependence of the scattering data
4.1.5. The IBVP (4.14)–(4.16)
4.2. The IBVP for the shG equation
4.2.1. The self-adjoint problem associated with the shG equation
4.2.2. The Lax compatibility condition
4.2.3. Evaluation of unknown BVs
4.2.4. The time-dependence of the scattering function
4.2.5. 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 Non-Linear Interaction Waves
5.1. The direct and ISP for a system of n first-order ODEs
5.1.1. The transition matrix S(λ)
5.1.2. Representations of solutions of system (5.5)
5.1.3. The intermediate matrix S(λ)
5.1.4. The bilateral factorization of the transition matrix S(λ)
5.1.5. The analytic and bilateral factorizations of S(λ)
5.1.6. The inverse scattering problem
5.2. The direct and ISP for the transport equation
5.2.1. The transition operator S(λ)
5.2.2. Volterra integral representations of solutions
5.2.3. Bilateral Volterra factorization of the S-operator
5.2.4. Analytic and bilateral Volterra factorizations of the intermediate operator S(λ)
5.2.5. The inverse scattering problem
5.3. Integration of the continual system of non-linear interaction waves
5.3.1. The generalized Lax equation
5.3.2. The time-evolution of the operators F(λ; t) and G(λ; t)
5.3.3. The Cauchy problem for the continual system (5.213)
6. Some Problems for The KdV Equation and Associated Inverse Scattering
6.1. The direct and ISP
6.1.1. The Jost solution and the analytic solution
6.1.2. The Parseval's equality and the fundamental equation
6.1.3. The necessary conditions of the scattering data
6.1.4. The necessary and sufficient conditions of a given data set
6.2. The IBVP for the KdV equation
6.2.1. The Lax compatibility condition
6.2.2. The time-dependent Jost solution
6.2.3. The normalization eigenfunction
6.2.4. The Sturm–Liouville scattering problem
6.2.5. Calculation of unknown BVs
6.2.6. The time-dependent scattering data
6.2.7. The IBVP (6.48)–(6.49)–(6.50)
6.3. Exact soliton-solutions of the Cauchy problem for the KdV equation
6.3.1. The direct and inverse problem (6.109)–(6.110)
6.3.2. Non-scattering potentials
6.3.3. The time-dependence of the reflection coefficient
6.3.4. Some examples
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
7.3.1. The linear change of dependent variables
7.3.2. The characters of the self-adjoint problem
7.3.3. The problem of finding the scattering function S(μ, t)
7.4. The time-evolution of s(k, t) and solution of the IBVP
7.4.1. The time-evolution of solution s(k, t) of system (7.86)
7.4.2. The solution of the IBVP (7.1)–(7.3)
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. Fokass methodology for the analysis of IBVPs [38]
9.2. A Riemann–Hilbert problem
9.3. Hirota's method
9.3.1. The scattering problem (SP) associated with the KdV equation
9.3.2. Bilinear equation for the KdV equation
9.3.3. The degree of standardized polynomials
9.3.4. The SP associated with the attractive NLS equation
9.3.5. The representations of F and G
9.3.6. The degree of standardized polynomials and solutions of the NLS equation
10. 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
10.1. The BTs for NLEEs defined in the whole space
10.1.1. The Cauchy–Riemann system
10.1.2. The Liouville equation
10.1.3. The BT for the solution of the two-dimensional generalization from the KdV equation (8.129) in C(E2; t)
10.1.4. The BT for the solution of the continual system of non-linear interaction waves (5.213) on the whole line
10.1.5. A note on the equivalence of BTs and the ISM for a class of non-linear differential equations [29]
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
10.2.1. The BT for the solution of the IBVP for the system of NLEEs (2.5)
10.2.2. The BT for the solution of the IBVP for the attractive NLS equation (2.60)
10.2.3. The BT for the solution of the Cauchy problem for the repulsive NLS equation (2.113)
10.2.4. The BTs for the solutions of the IBVP for the mKdV equations (3.59) and (3.60)
10.2.5. The Miura's transformations (3.103) and (3.107)
10.2.6. The BT for the solution of the Cauchy problem for the cubic NLEEs (3.144)
10.2.7. The BT for the solution of the IBVP for the sine-Gordon equation (4.14)
10.2.8. The BT for the solution of the IBVP for the sinh-Gordon equation (4.84)
10.2.9. The BT for the solution of the IBVP for the KdV equation (6.48)
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
10.3.1. The SP associated with the IBVP for KdV equation (10.5)
10.3.2. The time-dependence of eigenfunctions E(k, x, t) and W(k, x, t) of system (10.8)
10.3.3. The system of linear differential equations for s(k, t)
10.3.4. A common solution of both systems (10.8) and (10.10)
Bibliography
Index
