Solitary Waves In Nonlinear Physics

The theories of solions, chaos, fractals and dissipations are the foundation of nonlinear physics. Nowadays, the study of solitary waves is one of the hot topics in nonlinear physics.In this paper, by using reductive perturbation technique, we studied the solitary waves in nonlinear vibrating string, plasma, blood vessel and nonliear transimission lines respectively.In chapter three, According to variational principles, we obtained the equation of nonlinear vibrating string. For a nonlinear vibrating string, the governing equation is utt-2a2uxuxx - 2uuxxxx = 0, where uu, uxuxx and uxxxx are corresponding the kinetic energy, the tensile and flexural potential energy of the string respectively. With reductive perturbation technique, we derived a Korteweg de Vries equation (ut - 6uux + uxxx = 0) of the nonlinear vibrating string equation, which is easier to solve than the original form. By using hyperbola function method, a class of exact kink solitary wave solutions to the nonlinear vibrating string are obtaind.In chapter four, by using reductive perturbation technique, Ion acoustic solitary wave in weakly relativistic plasmas under transverse perturbation, the propagation of solitons in an inhomogeneous dusty plasma and the modulational instability of dust-acoustic waves in warm dusty plasmas are all discussed respectively. 1. For unmagnetized, relativistic and hot ion plasmas, the ion-acoustic waves can be described by the Kadomtsev-Petviashvili (KP) equation((ut + auux + buxxx)x + duyy = 0). It suggests that the nonlinear ion-acoustic solitary waves in a relativistic hot ion plasma are stable even there are some higher order transverse perturbations. There are only compressive solitary waves in the relativistic hot ion plasmas which has been vertified analytically. 2. In the lowest order, for the propagation of solitons in an inhomogeneous dusty plasma composed by two kinds of dust grains with different masses, if the interface of two kinds of dust grains in an inhomogenous dusty plasma is discontinuous, the transmision and reflection waves both can be described by the KdV equation. The numbers and amplitudes of both transimtted and reflected solitons from an incident soliton are given analytically for this case. If the interface of two kinds of dust grains is continuous, neglecting the reflection, the nonlinear dust-acoustic wave can be described by a KdV-type equation in the lowest order. The amplitudes, propagating velocities of these quasi-solitons for this case are also given analytically. 3. For a warm dusty plasma with variable dust charge, the modulational instability of the dust-acoustic waves can be described by the nonlinear Schrodinger equation(iut 4- auxx + b|u|2u = 0). Itshows that the dust-acoustic wave is modulational stable in this dusty plasma. Only the dark soliton exists in warm dusty plasma with variable dust charge.In chapter five, the reflection of solitons at more than two bifurcations are studied. The maximum reflections will take place when the radius of each branch is same. These results are consistent with experimental ones. The inhomogeneity of the artery will change the amplitude of the blood pulse wave which is in good agreement with the numerical results or the experiments.In chapter six, we studied the nonlinear solitary wave solution under the transverse perturbations for a system of coupled nonlinear electrical transmission lines. In the continuum limit and suitably scaled coordinates, the voltage on the system is described by a modified Zakharov-Kuznetsov equation. The cut-off frequency of the growth rate for the solitary waves under transverse perturbations has been analytically obtained. It is in agreement with the case p = 1/2 and p = 1 which have been studied previously.