Generalized Cauchy Matrix Approach

This thesis consists of two main works. On the one hand, we generalize the Cauchymatrix approach and obtain the exact solutions of several discrete integrable systems byusing the generalized Cauchy matrix method, including lattice KdV-type equations, ABSlattice equations, lattice GD-type hierarchies, lattice BSQ-type equations and lattice KP-type equations. On the other hand, the extended lattice BSQ systems and the thirdmember of the extended lattice GD systems are constructed through the direct lineariza-tion method. The main achievements are as follows.In Chapter2and Chapter3, by using the generalized Cauchy matrix approach westudy the exact solutions for the lattice KdV-type equations, ABS lattice equations andlattice GD-type hierarchies,3-component lattice BSQ-type equations, respectively. Withthe condition equation set, the shift relation for some defined scalar functions are builded,which lead to the lattice equations. The all expressions for r and M are explicitly deter-mined by investigation of the condition equation set. With the formulae of r and M, thesoliton solutions, Jordan-block solutions and mixed solutions for the lattice equations areobtained in an explicit form.In Chapter4, the exact solutions for the lattice KP-type equations are discussed byusing the generalized Cauchy matrix approach. With the condition equation set, we setup the shift relation for some defined scalar functions, as well as the lattice equations.The condition equation set is used to determine the all explicit formulae of r,ts and M.The soliton solutions, Jordan-block solutions and mixed solutions for the lattice KP-typeequations are derived explicitly.In Chapter5and Chapter6, the direct linearization structure is presented for theconstructions of the extended lattice BSQ system and the third member of the extendedlattice GD type hierarchies. Meanwhile, the soliton solutions and Lax pairs for thesesystems are worked out.In Chapter7, we generalize the generalized Cauchy matrix approach for lattice BSQ-type equations, which given in Chapter3.