The Exact Travelling Wave Solutions For Three Types Of Generalized KdV Equations

The mathematical techniques of the reduction of order for solvingdifferential equations and the auxiliary differential equation approaches are usedto study three types of nonlinear generalized KdV equations. The analytical ex-pressions of the exact solutions for the equations are obtained. The main factorsleading to the change in the physical structures of solutions are highlighted.In chapter 2, using a mathematical technique based on the reduction oforder for solving differential equations, we study two types of generalized KdVequations, which are ut + (aun ff bu2n)ux + [uk(um)xx]x = 0 and ut + (aun ffbu2n)ux + uk(um)xxx = 0, and obtain exact travelling solutions. The solutionspresented in this chapter possess various forms including bell type or kink typesolitary wave solutions, solitons, compactons, periodic solutions and algebraictravelling wave solutions. It is shown that the exponents of the nonlinear terms,the wave speed of the solutions and the coeffcients of the derivatives of theequations play a major role in the qualitative change of the physical structuresof the solutions.Chapter 3 studies a prototypical and nonlinear K(n,n) equation withvariable coeffcients by using a mathematical technique based on auxiliarydifferential equations and the symbolic computation system Maple. The exact solutions to the equation are constructed analytically under variouscircumstances. It is shown that the variable coe?cients and the exponentappearing in the equation determine the quantitative change in the physicalstructures of the solutions.