Complex Solutions Of Boussinesq Equation And Conversed Laws Of Two Fully Discrete Equations

The infinite number of conservation laws and exact solutions of soliton equations are important topics in the study of integrable systems.This thesis is devoted to the study of complex solutions of Boussinesq equation,especial the PT-symmetric so-lutions,and constructing infinitly many conserved laws of the fully discrete Liouville equation and the special Q1 equation in ABS classification.The paper consists of two parts.The second chapter focuses to construct complex solutions of Boussinesq equa-tion.First,a real periodic solution in terms of Jacobi elliptic functions of the Boussinesq equation is constructed.Its special case leads to 1-soliton solution of Boussinesq equa-tion.Based on the solution,a PT-symmetric solution in Jacobi elliptic functions of the real variable,which can reduce to a periodic PT symmetry solution and the com-plex 1-solution of Boussinesq equation is obtained.Second,by applying Hirota direct method,a complex 1-soliton solution of the Boussinesq equation is constructed.It is shown that the complex soliton solution with a pure image phase is PT-symmetrically invariant.Finally,the complex 1-soliton solution whether PT symmetry or not has real total mass and energy.A PT-symmetric 2-soliton solution of Boussinesq is also pointed out.The third chapter is to construct infinitely many conserved laws of the fully dis-crete Liouville equation and the special Q1 equation in ABS classification.It is shown that the Lax matrices which are normalized at in the vicinity of the singular point may be diagonalized to formal series with diagonal matrices coefficients.The Lax repre-sents of the fully discrete integrable systems are become the form of conserved laws.As an application,the conversed laws of the fully discrete Liouville equation and the special Q1 equation are obtained.