The Boussinesq equation is the first model of surface waves in shallow water that considers the nonlinearity and the dispersion and their interaction as a reason for wave stability known as the Boussinesq paradigm. This balance bears solitary waves that behave like quasi-particles. At present, there are some Boussinesq-like equations. The prevalent part of the known analytical and numerical solutions, however, relates to the 1d case while for multidimensional cases, almost nothing is known so far. An exclusion is the solutions of the Kadomtsev-Petviashvili equation. The difficulties originate from the lack of known analytic initial conditions and the nonintegrability in the multidimensional case. Another problem is which kind of nonlinearity will keep the temporal stability of localized solutions.
The system of coupled nonlinear Schroedinger equations known as well as the vector Schroedinger equation is a soliton supporting dynamical system. It is considered as a model of light propagation in Kerr isotropic media. Along with that, the phenomenology of the equation opens a prospect of investigating the quasi-particle behavior of the interacting solitons. The initial polarization of the vector Schroedinger equation and its evolution evolves from the vector nature of the model. The existence of exact (analytical) solutions usually is rendered to simpler models, while for the vector Schroedinger equation such solutions are not known. This determines the role of the numerical schemes and approaches. The vector Schroedinger equation is a spring-board for combining the reduced integrability and conservation laws in a discrete level.
The experimental observation and measurement of ultrashort pulses in waveguides is a hard job and this is the reason and stimulus to create mathematical models for computer simulations, as well as reliable algorithms for treating the governing equations. Along with the nonintegrability, one more problem appears here - the multidimensionality and necessity to split and linearize the operators in the appropriate way.
About the Author: Michail Todorov graduated in 1984 and received Ph.D. degree in 1989 from the St. Kliment Ohridski University of Sofia, Bulgaria. Since 1990 he has been Associate Professor and Full Professor (2012) with the Department of Applied Mathematics and Computer Science by the Technical University of Sofia, Bulgaria. He has worked as a Senior Research Fellow in the Joint Institute for Nuclear Research at Dubna, Russia (2004) and as a Visiting Professor, a Visiting Scholar, and a Visiting Consultant in the University of Texas at Arlington, USA (2008, 2009 and 2011) and Texas A&M University at Commerce, USA (2011), Sabbatical Professor at Southeastern Louisiana University at Hammond, LA (2013) and Embry-Riddle Aeronautical University, Daytona Beach, FL (2017). Since 2000 he has been also part-time employed instructor on Computer Science and Technology in the St. Kliment Ohridski University of Sofia, Bulgaria. In 2004-2008 he was part-time employed instructor on Theoretical Electrodynamics in the Paisii Khilendarski University of Plovdiv, Bulgaria. For the last few years his primary research areas have been mathematical modeling, computational studies, and scientific computing of nonlinear phenomena including soliton interactions, nonlinear electrodynamics, nonlinear optics, mathematical biology and bioengineering, and astrophysics.
Dr Todorov is an editor of eleven peer reviewed books in the Conference Proceedings Series by the American Institute of Physics, Melville, NY, Guest-Editor in Wave Motion (Elsevier) and Springer Proceedings. He is a coordinator, chair and/or special session organizer, member of Program and Scientific Committees of about 40 conferences on Applied Mathematics, Dynamical Systems and Differential Equations in Bulgaria, Russia, USA, and Taiwan.
