Application Of Bilinear Method In Solving Nonlinear Equations

Recently, the discrete integrable system has been applied to many fields,such as the optics, hydromechanics, magnetic fluid. Compared with the continuous integrable system, the discrete integrable system can describe the properties preferably and become a hotspot. In addition to, the rational solutions of the PDE equation play a role due to it can depict many other physical phenomena.In this thesis, we first construct a discrete spectral operator and get its integrable symplectic map as well as the Hamiltonian structure. Then we obtain some rational solution, lump solutions and rogue wave to two classes of PDE equations. The outline of this thesis is as follows:In section 1, we introduce the generating and the development of the soliton and explain the basic Hirota bilinear method.In section 2, based on the Tu scheme of discrete integrable system, we construct a new spectral operator and obtain its Hamiltonian structure with the help of zero curvature equation, especially, we list a coupling equation as example. Furthermore, with the Bargmann constraint, we present the integrable symplectic map. In the end, we solve this special coupling equation in virtue of the symmetry theory.In section 3, according to the Hirota bilinear method and the symbolic computation, we receive 5 classes of rational solutions to the (3+1)-dimensional Shallow Water equation and then depict its dynamical properties by choosing some appropriate parameters. Moreover, the lump solution of this equation are obtained via the quadratic function method, which can be guaranteed rationally localized in all directions in the (x, y)-plane under some constraints of the parameters.In the last section, we first give the lump solution by using the quadratic function method, which is similar to the lump solution in section 3, whereafter,the interaction between one stripe and lump solution is presented by extending the quadratic function method to the combination of the quadratic function method and exponential function, this phenomena can also be called fusion and fission because of the different moving directions of these two solitions. Finally,a new ansatz of combination of quadratic functions and hyperbolic functions is introduced, and thus an interesting phenomenon is generated. However, this obtained rogue wave is localized in the (x, y)-plane , which is different from the classic line rogue wave.It needs to be emphasized that the previously obtained solutions of two dimensional rogue waves are actually linear as they are only local in one dimension while the rogue wave referred in section 4 is localized in two dimensions independently.