MAT 468E - Nonlinear Waves

Course Objectives

The course objective is to give the information of nonlinear waves, the equations governing the propagation of these waves, especially Korteweg-de Vries (KdV) equation, and their solutions which play an important part in many branches of mathematics, in engineering and in other areas of science.
In this connection,

1. To give knowledge and basic understanding of linear wave equations, dissipative waves and dispersion.
2. To introduce nonlinear wave equations and to give basic understanding of discontinuous solutions,
3. To teach the elementary solutions of KdV equation and the behavior of these solutions,
4. To introduce the scattering and inverse scattering problems to students.
5. To give an idea about the solitary wave and solitons by constructing the solutions of initial value problem for KdV equation,
6. To demonstrate the Lax formulation, Hirota’s method and Backlund transformation to students.
7. To introduce a brief knowledge about numerical methods to solve nonlinear evolution equations which admit solitary wave, soliton.

Course Description

Korteweg-de Vries i.e. KdV equation; linear wave equation, superposition of solutions. Linear dispersive wave equation, dispersion. The simplest nonlinear wave equation and discontinuous solutions. The balance between nonlinearity and dispersion, and the KdV equation.
Elementary solutions of the KdV equation; the qualitative behaviours of the traveling wave solutions of the KdV equation. Description of solutions in terms of the Jacobian elliptic functions. Limiting behaviours of the cnoidal wave and the solitary wave solutions.
The scattering and inverse scattering problems; the scattering problem, the inverse scattering problem, the solution of the Marchenko equation.
The initial-value problem for the KdV equation. Construction of the solution, Solitary wave and two-soliton solutions.
Further properties of the KdV equation;Conservation laws, Lax formulation and its KdV hierarchy, Hirota’s method, bilinear form of the KdV equation. Backlund transformations for the KdV equation. The Painleve property of the KdV equations and numerical methods for the soliton solutions