Exact Solutions Of Several Coupled Nonlinear Systems

As an important branch of nonlinear science, soliton theory has been attrac-tive and exciting since it came into being in 1960's. A large number of phenomenahave to be described by nonlinear models. Many classical soliton equations, such asKorteweg-de Vries (KdV) equation, modified Korteweg-de Vries (mKdV) equation,Kadomtsev-Petviashvili (KP) equation, nonlinear Schr¨odinger (NLS) equation and soon, have been widely applied in almost all of the physics fields such as the field the-ory, condensed matter, plasma physics, ?uid mechanics, optics, particle and nuclearphysics. Moreover, in other scientific and technological fields such as the communi-cations, chemistry, geography, atmospheric dynamics, biology etc, the theory is alsowielded widely. Owing to the complexity of the problem, various models are all approx-imate descriptions of science phenomena. Compared with single-component systems,coupled systems include more interactions of real factors, so they can better describephysical phenomena. It has been a hot topic to solve these coupled systems. Thisdissertation discusses exact solutions of several coupled nonlinear systems.In Chapter 2, the coupled displacement shallow water wave system and its exactsolutions are researched. Recently, Zhong and Yao have derived one (1+1)-dimensionaldisplacement shallow water wave system (1DDSWWS) to describe the shallow wa-ter waves by applying variational principle, and find the 1DDSWWS can better de-scribe real shallow water waves than the KdV equation. We extend 1DDSWWSfrom (1+1)-dimensions to (2+1)-dimensional displacement shallow water wave sys-tem (2DDSWWS). Compared with the traditional KP equation describing (2+1)-dimensional shallow water waves, the 2DDSWWS has the following advantages: (1) the line solitary wave solutions of the KP can move only in one direction (positive ornegative direction perpendicular to the line) and is nonsymmetric for space variables xand y, while those of the 2DDSWWS can move in both directions perpendicular to theline; (2) the vertical velocity of the ?uid has been neglected entirely for derivation ofthe KP equation in the ?uid mechanics, while it has not been neglected in the deriva-tion of the 2DDSWWS. Then we believe that the 2DDSWWS is more reasonable thanthe KP equation to describe (2+1)-dimensional shallow water waves and other physicalphenomena. The KP equation can also be derived from the 2DDSWWS by using theweak two dimensional long waves assumption. The general travelling wave solutionof the 2DDSWWS can be obtained by means of an elliptic integral. Especially, theelliptic integral solution can degenerate to Jacobi elliptic function and solitary wavesolutions under a special selection of one integral constant.With the development of computer science and the discrete property of microphenomena, the study of the integrable di?erential-di?erence equations (DDEs) hasbeen a new hot topic. Exact solutions of two integrable coupled discrete models areresearched in Chapter 3. The first model is the so-called"discrete mKdV equation".We find the discrete mKdV equation is also a discrete KdV equation. A coupled dis-crete system is derived from the discrete single-component equation by assuming itsfield complex. The Lax-integrability of the coupled system is proved and three types ofexact solutions are listed. The coupled discrete system is found to be discrete coupledKdV and discrete coupled mKdV systems. The second model researched in the disser-tation is the coupled Volterra system. The coupled Volterra system can converges tothe coupled KdV system and its Lax-integrability is proved. Meanwhile, The symme-tries of the discrete equations are discussed and the concept of"symmetry equationmethod"is introduce to research the Lie point symmetries of DDEs. On one hand,what we need to do is to solve the symmetry equations with the help of the researchedsystem, namely, we need not make use of prolongation of the vector field. On theother hand, discrete variables are equally important with the continuous variables inour method and we need not discuss the continuous and discrete transformations sep-arately. The coupled Volterra system is taken as an example to illustrate our methodconcretely and its periodic wave and solitary wave solutions are obtained by means of symmetry approach. The two coupled discrete models'continuous forms can describetwo-layer ?uids and Bose-Einstein condensation et al physical phenomena.Severe hazardous weather and climate have a great ruinous e?ect on entironmentand human society and the research on severe hazardous weather is hot and ?intytopic. Atmospheric gravity waves have a great e?ect on severe hazardous weather andclimate, so atmospheric gravity wave is one of the most significant and di?cult topicsof Atmospheric Sciences. Chapter 4 is devoted to a coupled nonlinear Schr¨odinger(CNLS) equation derived from the governing system for atmospheric gravity waves.The Painlev′e integrability, symmetries, exact solutions and application to atmosphericgravity waves are discussed. Calculation shows the CNLS equation is invariant undersome Galilean transformations, scaling transformations, phase shifts and space-timetranslations, and it will turn into Painlev′e integrable models if some restrictions areimposed on the parameters. Some ordinary di?erential equations are derived andseveral exact solutions for the CNLS equation are also obtained by making use ofsymmetries. Applying the rational expansions of fundamental Jacobi elliptic functions,we find 20 periodic cnoidal wave solutions for the CNLS equation. The generation andpropagation of atmospheric gravity waves on condition the background wind possessingperiodic form are discussed, and the evolutions of the perturbation stream function, thelatitudinal velocity perturbation and the vertical velocity perturbation are exhibitedby several pictures.The innovations of this dissertation are as follows:(1) 2DDSWWS is constructed to describe (2+1)-dimensional shallow water wavesystem through combining the analytic mechanics and the ?uid mechanics. The2DDSWWS is found to be more reasonable than the KP equation to describe (2+1)-dimensional shallow water waves and other physical phenomena. The exact solutionsof the 2DDSWWS are given.(2) A discrete mKdV equation is also found to be a discrete KdV equation. Twonew integrable discrete coupled systems are established, and their Lax-integrability isproved. The analytical solutions for the two discrete coupled systems are proposed.The concept of"symmetry equation method"is introduced to research the Lie pointsymmetries of DDEs. (3) Soliton theory is applied to the atmospheric dynamics. The Painlev′e prop-erty, symmetries and exact solutions of the CNLS equation for atmospheric gravitywaves are researched, and the transport and mixing of atmospheric gravity waves arediscussed.