High‐Order Soliton Solution of Landau–Lifshitz Equation

Abstract

The Landau–Lifshitz equation is analyzed via the inverse scattering method. First, we give the well‐posedness theory for Landau–Lifshitz equation with the frame of inverse scattering method. The generalized Darboux transformation is rigorous considered in the frame of inverse scattering transformation. Finally, we give the high‐order soliton solution formula of Landau–Lifshitz equation and vortex filament equation.

Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation

“…This formula -which differs from those obtained by means of the Darboux dressing method (e.g., see [59,54,8,14]), requiring one to treat the problem of inverting an N × N matrix featuring Jost solutions as its elements -contains and allows an immediate classification of all the reflectionless solutions, irrespective of the number and the nature of the discrete eigenvalues in the spectrum, providing their direct physical interpretation (e.g., explicit expressions for the speed and precession frequency, as well as the location and time of the interactions, period of the oscillations for the breather-like solutions, separation of the maxima for the creation of entangled states, etc, see Section 4). Indeed, by choosing in a proper way the parameters featured by this formula and naturally linked to the spectral data, we are able to generate explicit expressions for all the solutions already known in the literature [50,41,57,59,54,8,14], and, in particular, general, explicit expressions for the breather-like and multipole solutions (see Section 4). As for the latter, it is important to underline here that, in principle, the existence of multipole solutions for (1.2) might be inferred from the gauge equivalence to the nonlinear Schrödinger equation as derived in [61].…”

“…In order to derive these results we define a convenient set of Jost solutions (see Section 2.1) which enables the study of their asymptotic behaviour at large λ. Then, differently from [8] (where the conditions in Assumption 1.1 are used for developing the IST theory for (1.2) exploiting the gauge equivalence to the nonlinear Schrödinger equation and by solving the corresponding Riemann-Hilbert problem), in our treatment the inverse scattering problem is formulated directly in terms of the Marchenko integral equations. They are obtained by using a new triangular representation of the Jost solutions (see Propositions 2.3 and 2.4 in Section 2) which differs substantially from the triangular representations in [56] and [61] (e.g., see formulae (13) and (17) in [61]).…”

Fifth-order generalized Heisenberg supermagnetic models

“…The Heisenberg ferromagnet (HF) model [1,2] S t = S × S xx , S = (S 1 , S 2 , S 3 ), S · S = 1 (1) is the simplest integrable model of ferromagnetism, where S is a spin vector and × means vector product. HF model attracts a great interest mainly due to its widely applications roles in various fields in mathematics and physics, for example, the anti-de Sitter/conformal field theories [3]- [5], the two-dimensional gravity theory [6] and Eulerian vortex filament [7]. The HF model is shown to be gauge and geometrical equivalent to the nonlinear Schrödinger equation (NLSE) [2].…”

“…A variety of the work has focused on the investigation of the generalized HF models, such as higher-order deformations of HF models [9,10], the multi-component generalized HF models [11], the multidimensional extended HF models [12,13]. Later on the N-soliton solutions of the generalized HF models have also been analyzed [7,14].…”

General soliton solutions to a coupled Fokas–Lenells equation

“…Actually, the DT is a method related to the inverse scattering method which can be used to solve the initial value problem of integrable equations. Recently, there are some progresses to use the DT to construct some more general analytical solutions for NLS-type equations [30,[36][37][38][39][40]. However, in certain physical situations, two or more wave packets of different carrier frequencies appear simultaneously, and their interactions are governed by the coupled equations.…”