Dressing Method And Discrete Integrable Systems

The dressing method was developed by Zakharov and Shabat in 1970s.The starting point of the procedure involves the factorization of an integral operator F on the line as the product of two Volterra type integral operators K_±, from which the Gel'fand-Levitan equation is obtained. These Volterra operators are then used to construct dressed differential operators M_j(j =1,2) starting with a pair of initial operators My which are commuting, constant-coefficient differential operators. This dressing method allows one to construct pairs of commuting differential operators M_j(j =1,2) with [M₁.M₂] = 0, which gives the nonlinear evolution equations. In order to give the solution of the obtained equations, one need to explicitly construct the kernel F of integral operator F from the commutative relation between integral operator F and initial differential operators My, the kernel K of Volterra operators is obtained from the Gel' fand-Levitan equation which is the equation about the F and K, then the solution of the nonlinear evolution equation can be obtained.The generalized dressing method is that the constant-coefficient initial differential operators are generalized to be variable-coefficient differential operators, which satisfy the generalized commuting relation. By using the Theorem 2.3 and the above similar method, one can obtain a set of variable-coefficient nonlinear evolution equations and their solutions. This generalized version of dressing method allows one to construct a set of equations instead of only one equation.In the following, two concrete problems are considered by using the generalized version of dressing method. The AKNS spectral problem is considered , firstly. Two set of variable-coefficient initial differential operators are used to construct two variable-coefficient evolution equations, namely variable-coefficient coupled mKdV equation and variable-coefficient coupled NLS equation. Their solutions are also obtained. A (2+1)-dimensional variable-coefficient KP equation are discomposed into the two (1+1)-dimensional variable-coefficient coupled equations, if the two (l+l)-dimensional equations have the compatible solutions, then the solution of the variable-coefficient KP equation can be constructed by using the compatible solution of the two (1+1)-dimensional equations. The second concrete problem is DS equation, the explicit solution of which is obtained.The next important steps in the development of the dressing method are mainly...