The Korteweg-de vries equation and solitons - Nurul Aliyah Hassim (2014)
This report presents a comprehensive study on a nonlinear partial differentiation equation (PDE) which is Korteweg-de Vries (KdV) equation. This includes other KdV variations which are Korteweg-de Vries (Cylindrical), Korteweg-de Vries (Generalized), Korteweg-de Vries (Modified) and Kortewg-de Vries-Burgers (KdVB). The report displays an extensive discussion of literature which covers solitary waves theory, explanation of dissipation and dispersion terms and historical background of KdV equation. Travelling wave solutions to the five equations will be handled using analytical methods such as variable transformation, tanh-coth and sine-cosine methods. Tanh-coth and sine-cosine methods were proved to be effective in handling nonlinear dispersive and dissipative equations especially in KdV and its other variation equations. Moreover, a new exact solution to the KdVB equation is obtained by using series of nonlinear transformations to reduce KdVB equation into Emden-Fowler equation and then, variable transformation is used to obtain the new exact solution which has not been found any previous literature. This new solution has proven that KdVB equation can also produce solitary wave. Different varying constant values, (c_0 ) ̃, in the reduced KdVB equation also do produce an effect on the exact solution. This new solution could be useful and applied in physical contexts such as liquid flow containing gas bubbles and flow of fluid in elastic tube. This could significantly contribute to the physics and mathematics research market in the future.