On the existence of soliton solutions for systems with a polynomial potential and their numerical realization
The problem of existence of soliton solutions (solutions of the traveling wave type) for the Korteweg-de Vries equation with a polynomial potential is considered on the basis of the approach within which the presence of a one-to-one correspondence of such solutions with solutions of the induced functional differential equation of pointwise type is demonstrated. On this path, conditions for the existence and uniqueness of solutions of the traveling wave type, with the growth restrictions both in time and in space, arise. It is very important that the conditions for the existence of a traveling wave solution are formed in terms of the right-hand side of the equation and the characteristics of the traveling wave, without using either the linearization and spectral properties of the corresponding equation in variations. Conditions for the existence of periodic soliton solutions are considered separately, and the possibility of transition from systems with a quasilinear potential to systems with a polynomial potential with conservation of corresponding existence theorems is demonstrated. Numerical implementation of such solutions is given.
For equations of mathematical physics, which are the Euler-Lagrange equation of the corresponding variational problems, an important class of solutions are soliton solutions. The study of soliton solutions is based on the existence of a one-to-one correspondence between soliton solutions for initial systems and solutions of induced functional- differential equations of pointwise type (FDEPT). The existence and uniqueness theorem for an induced FDEPT guarantees the existence and uniqueness of a soliton solution with given initial values for systems with a quasilinear potential. For systems with a quasilinear potential, one can also formulate the conditions for the existence of a periodic solution. A system with a polynomial potential can be redefined so that the resulting potential turns out to be quasilinear. If a guaranteed periodic soliton solution for such an overdetermined system lies in a sphere, outside which the potential is redefined, then we obtain the conditions for the existence of a periodic soliton solution for the initial system with a polynomial potential. An important task is the numerical realization of periodic soliton solutions for systems with a polynomial potential, which has been successfully solved.
The main results obtained during the project № 10-01-96054 “Mathematical and computer modeling of ecological and economic state for a region: problems of identification, forecasting, attainability and control” are described in this paper in a popular form. The aim of the project was to develop mathematical and software bases for construction of mathematical models complex of ecological and economic state of the region taking into account the specifications, diversity and relationship processes occurring in the region. The created complex is focused on studying the stability of dynamic models, forecasting ecological and economic state of the region and solving control problems with finding the control variables and corresponding trajectory of development.
We address a specific but possible situation in natural water bodies when the three-layer stratification has a symmetric nature, with equal depths of the uppermost and the lowermost layers. In such case, the coefficients at the leading nonlinear terms of the modified Korteweg-de Vries (mKdV) equation vanish simultaneously. It is shown that in such cases there exists a specific balance between the leading nonlinear and dispersive terms. An extension to the mKdV equation is derived by means of combination of a sequence of asymptotic methods. The resulting equation contains a cubic and a quintic nonlinearity of the same magnitude and possesses solitary wave solutions of different polarity. The properties of smaller solutions resemble those for the solutions of the mKdV equation whereas the height of the taller solutions is limited and they become table-like. It is demonstrated numerically that the collisions of solitary wave solutions to the resulting equation are weakly inelastic: the basic properties of the counterparts experience very limited changes but the interactions are certainly accompanied by a certain level of radiation of small-amplitude waves.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
The geographical and seasonal distributions of kinematic and nonlinear parametersof long internal waves obtained on a base of GDEM climatology in the Baltic Sea region are examined. The considered parameters (phase speed of long internal wave, dispersion, quadratic and cubicnonlinearity parameters) of the weakly-nonlinear Korteweg-de Vries-type models (in particular, Gardner model), can be used for evaluations of the possible polarities, shapes of solitary internal waves, their limiting amplitudes and propagation speeds. The key outcome is an express estimate of the expected internal wave parameters for different regions of the Baltic Sea. The central kinematic characteristic is the near-bottom velocity in internal waves in areas where the density jump layers are located in the vicinity of seabed. In such areas internal waves are the major driver of sediment resuspension and erosion processes and may be also responsible for destroying the laminated structure of sedimentation regime (that frequently occurs in certain areas of the Baltic Sea).
This paper deals with the implementation of numerical methods for searching for traveling waves for Korteweg-de Vries-type equations with time delay. Based upon the group approach, the existence of traveling wave solution and its boundedness are shown for some values of parameters. Meanwhile, solutions constructed with the help of the proposed constructive method essentially extend the class of systems, possessing solutions of this type, guaranteed by theory. The proposed method for finding solutions is based on solving a multiparameter extremal problem. Several numerical solutions are demonstrated.
A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.