Pham Loi Vu

Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations (eBook, ePUB)

• Format: ePub

Produktbeschreibung

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 B¿ckhund 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.

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Produktdetails

• Verlag: Taylor & Francis eBooks
• Seitenzahl: 452
• Erscheinungstermin: 15. Mai 2023
• Englisch
• ISBN-13: 9781000872125
• Artikelnr.: 67577152

Autorenporträt

Pham Loi Vu, PhD is a professor, 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).

Inhaltsangabe

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 [38]. 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.