Travelling Wave Solutions on a Non-zero Background for the Generalized Korteweg-de Vries Equation

Abstract

In presenting this thesis, we try to find all non-periodic travelling waves of the generalized Korteweg-de Vries (gKdV) equation u_t +\alpha u^p u_x +\beta u_{xxx}=0 using an energy analysis method. Since the power p in the gKdV equation is arbitrary, we consider positive integer values for $p$. We first check the method for two cases where p=1 and p=2 which are known as the KdV and the mKdV equations, respectively. Then, we look at the general case where p greater than or equal 3 is arbitrary.

By applying the energy analysis method on the KdV and the mKdV equations, we will find an explicit form of solitary waves on a non-zero background. Afterwards, we reparametrize the derived solutions in terms of speed and the background size to interpret these solutions physically. We also look at some limiting cases in which heavy-tailed and kink waves arise in the mKdV equation.

At last, we split up the gKdV equation into two cases of odd and even $p$ powers and apply a similar derivation. In each case, the implicit solutions are introduced and characterized by their features.