# Soliton thesis

The BSc Honours in Mathematics and Statistics is a flexible program designed to meet a wide range of interests. There are core courses in calculus, linear algebra, differential equations, probability and statistics, as well as the use of computers to explore and solve mathematical problems. Within this program, students have the option to select a specialized concentration in one of five areas: Students also have the opportunity to complete an honours project or thesis under the supervision of a faculty member. Soliton on a beach and related problems. It is shown that for a correct description, the full Boussinesq equations rather than a Korteweg-de Vries type approximation must be used. Using both energy conservation and two-timing expansions, the behaviour of the soliton is analysed.

The slowly varying soliton is found not to conserve mass and momentum and it has been suggested that to conserve these quantities, both forward and reflected waves must be added behind the soliton, these waves being solutions of the linear shallow water equations.

It is shown that to the order of approximation of the Boussinesq equations, only a forward wave or tail behind the soliton is necessary to fulfill mass and momentum conservation.

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A perturbed Korteweg-de Vries equation for which the perturbation adds energy to the soliton is considered. It is found that a tail is formed behind the soliton. The development of this tail into new solitons is analysed.

Using this solution, an exact averaged Lagrangian for slowly varying capillary waves is calculated.

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Modulation equations can be found from this averaged Lagrangian, but due to the algebraic complexity of the equations, the limit of waves on a thin fluid sheet is considered.

From the modulation equations, the stability of symmetric and antisymmetric capillary waves on a thin fluid sheet is found.

The modulation equations for antisymmetric waves form a hyperbolic system and the simple wave solutions for this system are calculated.

These simple wave solutions are interpreted physically.• Shailesh J. Mehta School of Management is equipped with the most modern state-of-the-art facilities. The SJMSOM building is a modern structure with its classrooms, seminar halls, computer labs, Wi-Fi enabled lobbies and wide open spaces for discussions and events. An Introduction to Wave Equations and Solitons Richard S.

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Palais TheMorningsideCenterofMathematics ChineseAcademyofSciences Beijing Summer Contents. The nonlinear Schrödinger equation is a classical field equation that describes weakly nonlinear wave-packets in one-dimensional physical systems. It is in a class of nonlinear partial differential equations that pertain to several physical and biological systems.

In this project we apply a pseudo-spectral solution-estimation method to a modified version of the nonlinear Schrödinger equation.

List of the new elected members to the European Academy of Sciences. In this thesis, chapter 1 is a general introduction to soliton theory and chapter 2 is an introduction to the main elements of this thesis. The contents of these chapters are taken from various references as indicated throughout the chapters.

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Introduction to Solitons Institute of Theoretical Physics and Astronomy Vilnius, University of Oldenburg and BSU Minsk Ya Shnir. Outline Historical remarks Linear waves. D'Alembert's solution.

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Dispersion relations. The Burger's equation. The Korteweg and de Vries equation. Soliton solution of the KdV equation.

"Soliton Solutions of the Nonlinear Schrödinger Equation." by Erin Middlemas