Inverse scattering problems and their application to nonlinear integrable equations

    • Vu, Pham Loi

著者

    • Vu, Pham Loi

書誌事項

Inverse scattering problems and their application to nonlinear integrable equations

Pham Loi Vu

(Monographs and research notes in mathematics)

CRC Press, c2023

2nd ed

大学図書館所蔵 件 / 6

この図書・雑誌をさがす

注記

Previous ed.: 2020

Includes bibliographical references (p. 409-420) and index

内容説明・目次

内容説明

The scattering data of the considered inverse scattering problems (ISPs) are described completely. Solving the associated IVP or IBVP 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 inverse scattering method (ISM) to solving the IVPs or IBVPs for NLEEs is consistent. It is effectively embedded in the schema of the ISM. Application of ISM to solving the NLEEs is effectively embedded in the scheme of the ISM.

目次

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 Backlund 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.

「Nielsen BookData」 より

関連文献: 1件中  1-1を表示

詳細情報

ページトップへ

AltStyle によって変換されたページ (->オリジナル) /