Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy

Abstract

We construct the hierarchy of a multi-component generalisation of modified KdV equation and find exact solutions to its associated members. The construction of the hierarchy and its conservation laws is based on the Drinfel'd-Sokolov scheme, however, in our case the Lax operator contains a constant non-regular element of the underlying Lie algebra. We also derive the associated recursion operator of the hierarchy using the symmetry structure of the Lax operators. Finally, using the rational dressing method, we obtain the one soliton solution, and we find the one breather solution of general rank in terms of determinants.

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

“…Some of the earliest work on non-commutative integrable systems includes Fordy and Kulish [45], Nijhoff et al [79], Ablowitz et al [3], Ercolani and McKean [41] and Aden and Carl [6]. Again there has been more recent interest in such systems and their solutions, such as Treves [102,103], Hamanaka and Toda [59], Degasperis and Lombardo [24], Dimakis and Müller-Hoissen [28], Carillo and Schiebold [20,21,22] whose results are particularly relevant to those herein, Sooman [98], Pelinovsky and Stepanyants [83], Buryak and Rossi [17], Doikou et al [30], Stylianidis [99], Adamopoulou and Papamikos [5], Malham [68], Gürses and Pekcan [58] and Ma [67]. The role of Hankel operators in integrable systems first explored by Pöppe, has recently re-emerged as an active and fruitful research direction.…”

The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra

“…The coefficients are the χ-images of the signature entries shown. Each column shows the factor contributions to the real coefficients of the compositions of C (5) shown in the very left column, for each of the monomials in π 5 shown across the top row. The final column represents the right-hand side of the equation π 5 = 5.…”

“…See Reutenauer [60], Malham and Wiese [47], Ebrahimi-Fard et al [24] and also Ebrahimi-Fard et al [25] for more on shuffle algebras and references therein for Rota-Baxter algebras. For very recent work on the non-commutative Korteweg-de Vries equation, see Degasperis and Lombardo [18], Pelinovksy and Stepanyants [54] and Adamopoulou and Papamikos [5]. Hankel operators have also received a lot of recent attention, see for example Grudsky and Rybkin [36,37], Grellier and Ger-ard [35], and Blower and Newsham [12].…”

“…We observe that, given the leading term in π 5 , we should anticipate that c 5 = 1. The remaining combination of the quadratic terms 3 * 1, 2 * 2, 1 * 3 and the cubic term 1 * 1 * 1 are chosen because these are the only monomials of this form which when we substitute for the corresponding signature expansions for 1, 2 and 3 into them, and compute the Pöppe product of the resulting expansions, we will get compositions in C(5). And the compositions of C(5) represent the basis in R C * we should use, guided by the linear signature expansion for 5 which we can write in the form 5 = 5 − 'lower multi-part compositions'.…”