This newly updated volume of the Encyclopedia of Complexity and Systems Science (ECSS) presents several mathematical models that describe this physical phenomenon, including the famous non-linear equation Korteweg-de-Vries (KdV) that represents the canonical form of solitons. Also, there exists a class of nonlinear partial differential equations that led to solitons, e.g., Kadomtsev-Petviashvili (KP), Klein-Gordon (KG), Sine-Gordon (SG), Non-Linear Schrödinger (NLS), Korteweg-de-Vries Burger’s (KdVB), etc. Different linear mathematical methods can be used to solve these models analytically, such as the Inverse Scattering Transformation (IST), Adomian Decomposition Method, Variational Iteration Method (VIM), Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM). Other non-analytic methods use the computational techniques available in such popular mathematical packages as Mathematica, Maple, and MATLAB.
The main purpose of this volume is to provide physicists, engineers, and their students with the proper methods and tools to solve the soliton equations, and to discover the new possibilities of using solitons in multi-disciplinary areas ranging from telecommunications to biology, cosmology, and oceanographic studies.
Nonlinear Water Waves and Nonlinear Evolution Equations with
Applications.- Inverse Scattering Transform and the Theory of Solitons.-
Korteweg-de Vries Equation (KdV), Different Analytical Methods for Solving
the.- Korteweg-de Vries Equation (KdV), History, Exact N-Soliton Solutions
and Further Properties of the.- Semi-analytical Methods for Solving the KdV
and mKdV Equations.- Korteweg-de Vries Equation (KdV), Some Numerical Methods
for Solving the.- Nonlinear Internal Waves.- Partial Differential Equations
that Lead to Solitons.- Shallow Water Waves and Solitary Waves.- Soliton
Perturbation.- Solitons and Compactons.- Solitons: Historical and Physical
Introduction.- Solitons Interactions.- Solitons, Introduction to.- Tsunamis
and Oceanographical Applications of Solitons.
Mohamed Atef Helal received his BSc in Mathematics (with distinction 1st class honor in 1969) from the Faculty of Science, Cairo University, Egypt. He then received his MSc in Applied Mathematics from the Faculty of Science, Cairo University, in 1975. In 1976, he got his DEA from Institute de Mechanique (IMG) Grenoble, France. After that, he received his Doctorat 3ème Cycle in Fluid Mechanics from IMG, Grenoble, France, in 1979. Finally, he got his Ph.D. in Applied Mathematics from Cairo University in 1982. Mohamed is currently a Professor of Mathematics in the Faculty of Science, Cairo University. He is an active fellow in the Institute of Physics (England), Institute of Mathematics and its Applications (England), Royal Astronomical Society (England), and London Mathematical Society (England). Mohamed is also a member of several scientific societies in the USA and Europe. Mohamed also acted as a president of the Egyptian Mathematical and Physical Society and was a board member of the Egyptian Mathematical Society. Mohamed has various publications in different fields such as: non-linear partial differential equations and Soliton solutions, Tsunamis, computational fluid mechanics, physical oceanography, fluids in rotating circular basins, shallow water waves in stratified fluids, wavelets, induction in thin sheets, and its applications in oceans, dark energy and golden mean in cosmology and physics. Mohamed participated in writing and editing three books. He has guided many students who took up MSc and PhD in applied mathematics. He also participated in judging MSc and PhD theses from Egypt and other countries. He is an active reviewer in several well-known journals.
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