Solutions For High Dimensional Nonlinear Systems

The study of nonlinear partial differential equations(PDE)can be used in the fields of optical information transmission,plasma physics,Bose Einstein condensation and fluid mechanics.The nonlinear Schrodinger equation is one of the most important branches of partial differential equations.The main task we have to do is to study and analyze the structure and properties of its soliton analytical solutions.The main technical route of this paper is to use Hirota bilinear method as the core,complete the transformation from nonlinear to two or more linear forms by rational transformation and other forms combined with the related properties of D operator.The solution of the soliton is obtained,and further the transmission characteristics of solitons are analyzed based on the solution.Based on the(1+1)-dimensional dispersion decreasing fibers equation model,the analytical solutions of double soliton and triple soliton are obtained.Then we study the soliton propagation in the(2+1)-dimensional equation model,find the analytical solution of double solitons,and analyze the structure and properties of the solution by ignoring the anisotropic factor in the equation.The results show that the soliton transmission is more stable when the dispersion term is selected as Gaussian function above the two forms.By adjusting the value of some coefficients in the Gaussian term,the phase shift of the soliton can be better controlled.However,for the(2+1)-dimensional form,the soliton transmission amplitude near the origin will decrease sharply,showing a"circular concave Valley" shape.This situation can be obviously improved by adjusting the coefficients in the Gaussian term.The soliton transmission is gradually uniform and smooth from the original"truncated" mode,and the interaction between solitons can be significantly weakened.The above research results are helpful to the mechanism and application of optical fiber transmission.Taking a higher order nonlinear Schrodinger equation as the research model.Let ?=0,the single soliton,double soliton and triple soliton solutions of the model are obtained.Based on the above solutions,the propagation distance and time of solitons with odd order dispersion term are studied.The results show that with the change of the constraint value between the fifth order dispersion and the third order dispersion,the propagation direction of the solitons will also change.With the increase of the constraint value,the cohesion between the solitons will be enhanced.In addition,the influence of nonlinear term on soliton interaction is studied,and three different soliton dynamics characteristics are summarized for soliton binding and periodic variation.The above research results are not only helpful for the application in optical field,but also have guiding value for fuzzy adaptive control in nonlinear system.Based on the above high-order model,the ?=?=0 is further converted to the integrable model of the existing variable coefficient(2+1)-dimensional Heisenberg ferromagnetic spin chain.The analytical solution of the three soliton is obtained.The simultaneous interpreting of different solitons is studied and analyzed for different transmission modes due to the rejection and attraction between solitons.It is found that the change of dispersion term and nonlinear term affects the change of soliton interaction morphology.By adjusting the constraint value between dispersion term and nonlinear term,the cohesion of soliton is enhanced and the phase shift is changed.In addition,by adjusting the value of nonlinear term effectively,the interaction between solitons is weakened obviously,and the long-distance stable transmission can be realized.The above results further enrich and deepen the present model of Heisenberg ferromagnetic spin chain.