The Continuous and Discrete Extended Korteweg-de Vries Equations and Their Applications in Hydrodynamics and Lattice Dynamics

This dissertation, "The Continuous and Discrete Extended Korteweg-de Vries Equations and Their Applications in Hydrodynamics and Lattice Dynamics" by Cheuk-man, Edmond, Shek, 石焯文, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of the thesis entitled THE CONTINUOUS AND DISCRETE EXTENDED KORTEWEG-DE VRIES EQUATIONS AND THEIR APPLICATIONS TO HYDRODYNAMICS AND LATTICE DYNAMICS submitted by Edmond Cheuk-Man Shek For the degree of Master of Philosophy At the University of Hong Kong in July 2006 Solitons are waves of permanent shapes and constant speeds, which are achieved as a consequence of balance between dispersive and nonlinear effects. These nonlinear waves have been studied with interest in recent decades and have been shown to have various physical applications. The extended Korteweg-de Vries equation with negative cubic nonlinearity the negative eKdV is one of the Korteweg-de Vries (KdV) family equations which exhibit different solitons. Its continuous and discrete versions could have applications in modeling the propagation of internal oceanic solitons (hydrodynamics) and the cubic nonlinear case of intermolecular interactions under thermal processes (lattice dynamics). The objective of this research project was therefore to study the dynamical behavior of both continuous and discrete solitons and make a comparison between them. This evolution is integrable such that Hirota's bilinear method can be adopted to obtain the exact soliton solutions. The negative eKdV has one family of solitons whose polarity depends on the sign of the quadratic nonlinearity. Its expressions of 1-soliton and 2-soliton are re-derived by setting up new bilinear differential equations and transformation variables such that all negative eKdV solitons including solitary waves of elevation or depression, pedestals or plateau solitons and kink-type solitons (either kink or antikink solitons) are found. The profiles of the solitary waves depend on their wave numbers. When the numbers increase Ito a certain extent, the solitary waves become plateaus with limited amplitudes. Further increasing the wave numbers would lead merely to broader widths but not larger amplitudes and thus would result in infinitely thick solitons, called kink-type solitons. Followed by the establishment of 2-soliton expressions, the interactions between any two kinds of solitons with the same polarity are analyzed. The combined value of the amplitudes determines the result of an interaction. With the total amplitude less than the limiting value, the interaction is either a merger (distinct solitary waves) or an exchanging of identities with combination (similar solitary waves). On the contrary, if the quantity value exceeds the critical one, the smaller soliton undergoes a reversal of polarity during the superposition process. The collision between a pedestal and a smaller soliton refers to such kind of interaction. The smaller soliton can regain its original form after the interaction. The interaction between a kink soliton and an ordinary solitary wave is also studied. The reversed soliton, interpreted as the dig, appears on the infinite summit of the kink soliton. Because of their different velocities, they tend to spread apart and the smaller soliton then interacts with the back slope of the kink one. Finally, the solitary wave restores its original sign and both solitons become separated and move independently. The discrete modified Korteweg-de Vries equation with negative quadratic nonlinearity (negative DmKdV) under non-vanishing boundary conditions at infinity, one of the discrete neg