The Dynamic Behaviors Of The Nonlinear Schr(?)dinger Equation With The Structure-preserving Method

The nonlinear Schr(o|¨)dinger equation as a ubiquitous nonlinearevolution equation plays an important role in nonlinear physics;itarises as an asymptotic limit of a slowly varying dispersive waveenvelope in a nonlinear medium and as such has significantapplications, e. g. nonlinear optics, plasma physics, laser fusion andcondensed physics.The nonlinear Schr(o|¨)dinger equation is an infinite-dimensionalHamiltonian system. The fundamental theorem of Hamiltonianmechanics says that the time-evolution of Hamiltonian system is theevolution of symplectic transformation. In this sense, we say that theHamiltonian system has a symplectic structure. Therefore, Ruth andFeng Kang presented the symplectic algorithm for solving theHamiltonian system. After that, the study and application ofsymplectic algorithm has been developed;at present, the symplecticalgorithm has been widely used in celestial mechanics, atmospheremechanics, geologic, plasma physics, molecular dynamics andquantum mechanics and so on. It is a better method in the calculationof long-time many-step and preserving the structure of system.In the thesis we investigate the dynamic behavior of the cubicnonlinear Schr?dinger equation. In the calculation, we first discretizethe spatial derivative of the nonlinear Schr?dinger equation: substitutethe space difference approximation for the spatial derivative is onemethod, in the thesis we have show the coefficients up to 14 order,then find the discrete Hamiltonian, and transform the nonlinearSchr?dinger equation to Hamiltonian canonical equations;For anothermethod, we will expand the wave function with the B-spline, in thisthesis we find the discrete Hamiltonian successfully, and transform thenonlinear Schr?dinger equation to Hamiltonian canonical equations.Hamiltonian system has a symplectic structure. So we solve theequation with symplectic algorithm. The nonlinear Schr?dingerequation has infinite conservative quantities, the first three are thequasiparticle number, the momentum and the energy, in thecomputation we compare the conservative quantities mentioned above.Then we construct a phase space, discuss the dynamic behavior of thenonlinear Schr?dinger equation, and compare the phase trajectorieswith different discrete order, the results have lots of difference. Thereare two pattern structure in the time evolution of the dynamic behaviorof the nonlinear Schr?dinger equation, we note that when , thecritical value ofq =1θ is not exactly 45? as mentioned in the literaturesbefore, it is between 45.1428o and 45.1429o. For the different nonlinearcoefficient q , there are different critical values of θ . In the thesis wegive a critical curve of the pattern structure about the nonlinearparameter q and the parameter θ of the initial amplitude of themodulation...