Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

Abstract

In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N . We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmanians Gr R (k, N ) and Gr C (k, N ).

Direct linearisation of the discrete-time two-dimensional Toda lattices

“…The direct linearisation scheme was established for the discrete-time 2DTL equations of A ∞ -type and A (1) r−1type. For each algebra, a class of nonlinear (including bilinear) equations arise and their integrability is guaranteed in the sense of having Lax pairs and direct linearising solutions.…”

“…Periodic reductions. Performing the r-periodic reduction (for integer r ≥ 2) of the discrete-time 2DTL of A ∞ -type is equivalent to considering sub-algebra A (1) r−1 , see e.g. [16].…”

“…We can now construct the bilinear and nonlinear equations in the class of the discrete-time 2DTL of A (1) r−1 -type, with the help of the constraints generated by the periodic reduction.…”

“…Exact solutions to the 2DTLs of different types are also obtained by using various methods, see e.g. [15,19,25] for soliton solutions, and also [1] for higher-rank solutions. For the algebraic structure of the 2DTL, we refer to Ueno and Takasaki [29] (see also [27]).…”

“…In section 2, we introduce the general theory of the direct linearisation, in the language of infinite matrix. Sections 3 and 4 are contributed to the direct linearisation schemes of the discrete-time 2DTL equations of A ∞ -type and A (1) r−1 -type, respectively, and their integrability properties.…”