Method for studing the multi-solitone solutions of the korteweg de-vries type equations based on the T-transformation
Journal Title: Вісник Тернопільського національного технічного університету - Year 2015, Vol 77, Issue 1
Abstract
In recent years the investigation of separated waves plays an important role in many applied scientific fields. Travelling wave solutions can describe various phenomena in fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology, meteorology and other fields. There are many models proposed to describe the physical phenomena of separated waves existence and variety of methods has been proposed to construct the exact and approximate solutions to nonlinear equations. In this paper, we propose a new technique of finding the PDE’s traveling wave solutions, which are based on the T- representation. This generalized representation of solutions has the advantages of classical δ-solitons in terms of their independence from the shape and profile, however, allowed to obtain limited smooth solution that simulates solitary waves.This technique guarantees isolation of a solution and allows to investigate the infinitesimal properties. Using T-representation method we found a new class of KdV solution, which simulate solitary wave and proved that well known KdV solution is a special case of our generalizations. The proposed method can be applied to finding solutions of a wide class of differential equations in partial derivatives in the form of solitary waves . T-representations can be useful for the investigation of multi-soliton solutions. Method for studying multi-soliton solutions is demonstrated on the example of KdV solutions. Multi-soliton solutions are represented as the sum of the T-representations with variable amplitudes. In this case, there are exact solutions for the amplitudes of the perturbations in special areas, which are determined by the maximum of perturbations. Problem of searching the perturbation amplitude is reduced to the Cauchy problem for some initial conditions. Changing conditions for maximum localization of perturbation we cover the area of solitones intersection by a system of curves, where exact solutions for the amplitude functions are found. For an approximate description of the waves interaction quite a few curves are needed. This approach allows to consider different laws of waves motion and simulate their amplitude in the region of intersection. An example of computer simulation is presented in this paper.
Authors and Affiliations
Andrii Bomba, Yuriy Turbal
Keywords
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