Nonlinear Partial Differential Equations, Integral Method,

This paper mainly derived the symmetry and conserved quantities of the discrete wave equation, compound KdV-Burgers equation and Jimbo--Miwa equation. First we put partial differential equation into difference equation. Based on the invariance of discrete equations under infinitesimal transformation of independent and dependent variables, Lie symmetry is derived. Furthermore, we receive a linear determing equation, which not only avoids the Killing equation, but also obtains a series of conservation quantities.The first part, with new lattice equation and Ablowitz - Ladik - Lattice, we obtain two discrete wave equations, respectively. And then the difficult continuous wave equation has been turned into a simple linear discrete equation. We introduce the translational invariables and scale invariables to reduce the number of independent variables. Next, according to the invariance of discrete equations under infinitesimal transformation of independent and dependent variables, invariants and algebras are obtained. At last, the ratio invariant of the discrete wave equation has been concluded.The invariant of compound KdV--Burgers equation has been obtained in the second part.This chapter is divided into two parts---the reductions of dependent and independent variables. Burgers equation is a multi-dimensional partial differential equation, therefore, we use translation variable method and grid equation method for its simplify. Finally obtained one-dimensional equation. It is easy to seek exact solutions.The second part, we mainly adapt matrix spectrum method to turn continuous wave equation into a simple linear discrete wave equation .Based on the invariance of differential equation of infinitesimal transformation, the conserved quantities have been obtained.The third part, we use the direct substitution method to reduce the dimensions. Then we apply time and space steps to obtain discrete Jimbo - Miwa equation, eventually conserved quantities are concluded.