Analytic Solutions For The Ultrashort Laser Pulse In Nonlinear Media

The ultrashort (few-cycle) intense laser pulses are applied to observe and con-trol many physical phenomena, such as electron motion in atoms, molecular vi-brations, and chemical reactions of complex molecules. The interactions between ultrashort laser pulses and matters generate the transient coherent optical effects, such as superradiation, Rabi oscillations, free induction decay, optical nutation, photon, self-induction transparency, and optical soliton. It gives rise to a lot of attention of physicists. In experiment, experimental physicists can realize to let the pulse width of few-cycle pulse(FCP) reach to attosecond order, it provides possibility for observing the electron motion. In theory, many theoretical models beyond the slowly varying the envelope approximation(SVEA) were put froward for, such as (1+1)-dimensional mKdV equation, sG equation, short-pulse equa-tion, mKdV-sG equation, and (2+1)-dimensional KP equation. Additionally, the breathe soliton solution of mKdV-sG equation can well describe the propa-gation characteristics of FCP soliton in dielectric media among the non-slowly varying envelope approximation models. In thesis, using the multi-scale expan-sion method and considering the short-wave approximation, a (2+1)-dimensional nonlinear coupled equation is deduced from Maxwell-Bloch equation with the two-level atomic system. For this model, people just got the numerical results, and did not give the analytic solution. At first, we investigate integrability of the coupled equations with the Painleve singularity analysis and find that it is not an integrable system. Meanwhile, we obtain general exact solutions expressed by one arbitrary function or two arbitrary functions by using consistent tanh expansion (CTE) method. By assuming the specific form of arbitrary function, we obtain some interaction solutions, such as the one kink-one periodic wave in-teraction solution, one kink-two periodic waves interaction solution, one kink-one dipole soliton interaction solution, one kink-two dipole solitons interaction solu-tion and one kink-bright dark soliton-one periodic wave interaction solution. We also get the kink-kink interaction solution and localized lump solution by using Painleve truncated expansion method. We also look for some analytic solutions of the (2+1)-dimensional Boussinesq equation which describes the propagation of short pulse in quadratic medium, we get many types of soliton solutions by using CTE method and Painleve truncated expansion method, such as bright soliton-periodic wave interaction solution, variational amplitude periodic wave interaction solution, bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution.