Nonlinear wave patterns in the complex KdV and nonlinear Schrodinger equations

Abstract

This thesis is on the theory of nonlinear waves in physics. To begin with, we develop from first principles the theory of the complex Korteweg-de Vries (KdV) equation as an equation for the complex velocity of a weakly nonlinear wave in a shallow, ideal fluid. We show that this is completely consistent with the well-known theory of the real KdV equation as a special case, but has the advantage of directly giving complete information about the motion of all particles within the fluid. We show that the complex KdV equation also has conserved quantities which are completely consistent with the physical interpretation of the real KdV equation.

When a periodic wave solution to the real KdV equation is expanded in the quasi-monochromatic approximation, it is known that the amplitude of the wave envelope is described by the nonlinear Schrodinger (NLS) equation. However, in the complex KdV equation, we show that the fundamental modes of the velocity are described by the split NLS equations, themselves a special case of the Ablowitz-Kaup-Newell-Segur system. This is a directly physical interpretation of the split NLS equations, which were primarily introduced as only a mathematical construct emerging from the Zakharov-Shabat equations.

We also discuss an empirically obtained symmetry of the rational solutions to the KdV equations, which seems to have been unnoticed until now. Solutions which can be written in terms of Wronskian determinants are well-known; however, we show that these are actually part of a more general family of rational solutions. We show that a linear combination of the Wronskians of orders $n$ and $n+2$ generates a new, multi-peak rational solution to the KdV equation.

We next move on to the integrable extensions of the NLS equation. These incorporate higher order nonlinear and dispersive terms in such a way that the system keeps the same conserved quantities, and is thus completely integrable. We obtain the general solution of the doubly-periodic solutions of the class I extension of the NLS equation, and discuss several special cases. These are the most general one-parameter first order solutions of the (class I) extended NLS equation.

Building on this, we also discuss second order solutions to the extended NLS equation. We obtain the general 2-breather solutions, and discuss several special cases; among them, semirational breathers, the degenerate breather solution, the second-order rogue wave, and the rogue wave triplet solution. We also discuss the breather to soliton conversion, which is a solution which does not exist in the basic NLS equation where only the lowest order dispersive and nonlinear terms are present.

Finally, we discuss a few possibilities for future research based on the work done in this thesis. Show more