The algebraic structure behind the derivative nonlinear Schrödinger equation

Abstract

The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schrödinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of aŝ 2 Kac-Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows, respectively. The equivalence between the latter and the massive Thirring model is explicitly demonstrated also. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.

Functional representation of the negative DNLS hierarchy

“…we reproduce results of [4,6,7,17] on the relationship between the DNLSH and the massive Thirring model [8,11,14,16].…”

“…The most straightforward continuation of this work is to use the advantages of the functional representation and to derive other classes of explicit solutions which complement the bright-soliton solutions derived in [4] and dark solitons presented above. This can be done starting from the bilinear equations of proposition 7.1 which can be associated with the Fay identities for the thetafunctions and used to derive the quasiperiodic solutions, or with various determinant identities that lead to Wronskian, Toeplitz and other solutions.…”

“…Consequently, we use the term DNLSH for the hierarchy whose simplest equations are (1.2) (see the recent paper [4] on the negative flows of the Kaup-Newell hierarchy).…”

A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup-Newell equation

“…Eq. (1.6) also can be derived from the generalized KN hierarchy [15] under n = 3 and proper parameter. Second, when β = 1 4 and x → ix, t → it, r = −q * the system (1.5) become…”

General bright and dark soliton solutions to the massive Thirring model via KP hierarchy reductions

“…The inverse scattering transform for the MT model was studied by Kuznetsov and Mikhailov [4] and many others [5,6,7,8,9]. The Darboux transformation, Bäcklund transformation of the MT model and its connection with other integrable systems have been investigated by Kaup and Newell [10], Lee [11,12], Prikarpatskii [13,14] , Franca et al [15] and Degasperis [16].…”