Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations

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.