Solitons In Complex Nonlinear Systems

It is now commonly acknowledged that nonlinear science is another scientific revolution after the Quantum mechanics and the theory of relativity. One has now gradually recognized that linear system is only approximate to the complex real physical world, but the nonlinear one can be better close to the nature of the world. At present, there are three main nonlinear science research fields:chaos, soliton and fractal. We will focus on the soliton in this study.Here, we will mainly study the propagation of solitons in some complex nonlinear systems, including the ideal homogeneous systems and the inhomogeneous ones with discrete impurities. As for the homogeneous case, we will focus our interest on the research on developing method for solving exact solutions to nonlinear evolution equations (NEEs), and specifically, we will study the auxiliary equation method; and for the inhomogeneous case, we will study the impurity-soliton interactions (ISIs), and will investigate two types of nonlinear systems:the continuous hydrodynamic system and the discrete nonlinear electrical transmission line (NETL).In Chapter 1, we will briefly introduce the soliton, including its history and research methods, the developments of ISIs, and so on. Then, we will further present the main research in this study.At present, the propagation of hydrodynamic solitons influenced by discrete impurities, compared with that of non-propagating ones, still lacks the understanding. In Chapter 2, we will study the interactions between discrete impurity and propagating soliton in the hydrodynamics. Specifically, we will pay our attention to the interactions between depth defects of a trough and surface hydrodynamic propagating solitons (Korteweg-de Vries (KdV) solitons). First, we will use the perturbation method to derive the corresponding theoretical approximate ISI model from the hydrodynamic dynamic equations, and then do the numerical study based on this model. The results show that, the defect can decelerate or accelerate the propagating hydrodynamic solitons depending on its polarity, which is consistent with the case for the non-propagating ones. However, from the present study, we find that a dipole defect-induced effect is involved on propagating hydrodynamic solitons, which is not found in the present non-propagating cases influenced by single impurity. In addition, we also investigate different impurity intensity cases, and find that the ISIs in the hydrodynamics for propagating solitons are related to the impurity intensity.In Chapter 3, we will study the ISIs in the NETL. As we know, the present ISIs both in the Frenkel-Kontorova (FK) model and hydrodynamics for the non-propagating solitons have got fruitful results and reached a good agreement each other, namely, an impurity or defect can attract or repel non-propagating solitons depending on its polarity. And from Chapter 2, we can get that, a defect can change the propagating speeds of the propagating hydrodynamic solitons, accelerating or decelerating also depending on its polarity. The present ISIs in the NETL lack the systemic understanding though they have also got some results. This Chapter is aimed to systemically understand the ISIs in the NETL by simulating the influence of a new capacitor impurity on the NETL solitons. Here, a capacitor with a slight difference in linear capacitance will serve as the single impurity. We will study the influence of the capacitor impurity on the NETL solitons both by the simulation which is based on the Kirchhoffs laws and by the calculation of the corresponding theoretical ISI model. The results show that, compared with the present ISI results of the NETL, the impurity-induced influence in the NETL for different types of parameter impurities has the same nature and essentially shares the same physical mechanism. At the same time, they also show that, compared with the present ISI results of the FK model and hydrodynamics, the impurity-induced influence in the NETL, FK model, and hydrodynamics essentially shares the same physical mechanism and thus can be understood in a unified way.In addition, we also study the different intensity cases of the single linear capacitance impurity, and find that, the ISIs in the NETL are related to the impurity intensity. Besides, in view of the fact that single impurity is only a primary approximation to the real inhomogeneous systems, we tentatively consider the pair, dipole and quadrupole impurities in the homogeneous NETL, which are the different combination of the single linear capacitance impurity, to study the impurity-induced influence on the propagation of solitons.In Chapter 4, we will analytically study the method for solving exact solutions to ideal nonlinear systems. Solving exact solutions to nonlinear equations play an important role in nonlinear science. Up to now, a wealth of powerful methods have been proposed, such as the inverse scattering method, Backlund transformation, the Hirota's bilinear method, and so on. However, because the superposition principle is no longer valid in nonlinear cases, one can not give the general solutions to the different nonlinear equations, and usually, he only can seek the different solving method depending on the different nonlinear problems. In this Chapter, we will focus our interest on studying the auxiliary equation method for solving nonlinear evolution equations (NEEs). Specifically, we will further extend the Huang's method to the form which possesses an auxiliary equation of a first order nonlinear ordinary differential equation with at most an eighth-degree nonlinear term. Furthermore, by performing the auxiliary equation into a more general expansion form, we develop an algebraic method and apply it to the Sharma-Tasso-Olver (STO) equation and the generalized Camassa-Holm (GCH) equation. And some exact solutions to these two NEEs are obtained.Finally, in Chapter 5, we summarize the conclusions and also discuss the prospect of the future studies.