The Methods For Exact Solutions To The Discrete Integrable Systems

In this thesis we study some methods for finding exact solutions to the discrete integrable systems. The methods we adopt here are Darboux transformations, the inverse scattering transform, and Hirota’s bilinear method.First, in Chapter3, we consider the Hirota-Miwa (dKP) equation and its potential form (dpKP), through introducing a differential-difference linear system. The differential form uses the first continuous flow x of the KP hierarchy which is compatible with the discrete flows in the Hirota-Miwa equation. So, their Darboux and binary Darboux trans-formations are given in differential, rather than the more usual difference form. These transformations are used to derive exact solutions in Wronskian and Grammian form, respectively. Particularly, in contrast with the classical theory, in discrete case, without complex conjugation we can still construct the binary Darboux transformations. Next we consider the reductions of the Hirota-Miwa equation. By taking appropriate reduction conditions, from the results of the Hirota-Miwa equation, we derive the lattice potential KdV, modified KdV and modified Boussinesq (mBSQ) equations, and their Lax pairs, and hence their Darboux and binary Darboux transformations and exact solutions.Then, in Chapter4, we present a discrete inverse scattering transform for the nonautonomous HI equation. In the direct scattering problem, we solve the initial value problem for a second-order difference equation with non-constant coefficients. Then we study the analytic properties of the wave functions. We obtain expressions of the scatter-ing data in Casoratians. In the inverse problem, through taking an appropriate transfor-mation with a kernel function, we derive an equation expressed by the kernel function, which was used to recover the potential. We also investigate a discrete Gel’fand-Levitan-Marchenko (GLM) integral equation which is used to solve the kernel function. So in the simple case of reflectionless potentials, through the discrete GLM equation, we can obtain the soltion solutions to the non-autonomous H1equation.Finally, in Chapter5, we investigate the non-autonomous H1, H2, H3δ and Q1δ equations in the ABS list, and obtain their bilinearization and Carsoratian solutions. The method we use is changing the autonomous bilinear equations into its non-autonomous form first, and then substituting them back into the variable transformations. So in this way we obtain both the integrable non-autonomous ABS lattice and their bilinearization. Their solutions are derived in Casoratian form. Besides, we see that these deforma-tions can well keep the correspondence of autonomous and non-autonomous ABS lattice equations.