Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations

Abstract

New Landau–Lifshitz (LL) and higher-order nonlinear systems gauge generated from nonlinear Schrödinger (NS) type equations are presented. The consequences of gauge equivalence between different dynamical systems are discussed. The gauge connections among various LL and NS equations are found and depicted through a schematic representation.

The Darboux transformation of the Kundu–Eckhaus equation

“…the KE equation as a modelling equation for ultra-short pulse propagation. Although the KE equation [16,17] was introduced 30 years ago as an integrable model from the mathematical point of view, its DT has not been constructed completely as mentioned in the introduction because of the occurrence of the complicated integrals in overall factor A N of the reported result in [20]. We overcome that problem by finding an explicit analytical form of the overall factor n−1 i=0 H [i] in equation (2.3) for the n-fold DT T n , and thus got the explicit determinant representation of the T n in theorem 2.3 and new solutions u [n] in theorem 2.4 for the KE equation.…”

“…ax + (−a 2 + 4β 2 c 4 + 2c 2 )t = 0. Moreover, a solution of the KE equation u and a solution of the NLS solution q can be connected through a nonlinear transformation q = u exp(2iβ |u| 2 dx) [16], and thus a phase difference between above two solutions is θ = 2β |u| 2 dx. The effect of β on the phase difference is plotted in figure 4, i.e.…”

“…the Kundu-Eckhaus (KE) equation iu t + u xx + 2|u| 2 u + 4β 2 |u| 4 u − 4iβ(|u| 2 ) x u = 0, β ∈ R, (1. 2) which is an integrable equation and introduced independently during 1984 by Kundu [16] and later by Calogero & Eckhaus [17] from different perspective. Perhaps, the KE equation is the simplest integrable extension of the NLS with a quintic term, because there is just one additional term-one cubic term except the quintic term.…”

New integrable systems of derivative nonlinear Schrödinger equations with multiple components

“…We call this system the Chen-Lee-Liu equation. There is a simple transformation of dependent variables which cast one into the other [3,4]. In this sense, these two types of the DNLS equation are gauge equivalent.…”

Higher-Order Nonlinear Dispersive Equations

“…In this case some modifications are needed since the Qk 's introduced by the change of variable involve nonlocal operators. The use of a change of dependent variable was suggested by recent works of Hayashi and Ozawa [4,5] on nonlinear Schrödinger equations (see also [13]). In these works this change of variable was called a gauge transformation.…”