The non-topological, stationary and propagating, soliton solutions of the classical continuous Heisenberg ferromagnet equation are investigated. A general, rigorous formulation of the Inverse Scattering Transform for this equation is presented, under less restrictive conditions than the Schwartz class hypotheses and naturally incorporating the non-topological character of the solutions. Such formulation is based on a new triangular representation for the Jost solutions, which in turn allows an immediate computation of the asymptotic behaviour of the scattering data for large values of the spectral parameter, consistently improving on the existing theory. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions (including breather-like and multipoles), and allowing their classification and description.
We study the low temperature thermodynamics of the quantum Heisenberg ferromagnet in three dimension for any value of the spin $ S \geq 1/2 $. We report on a rigorous proof of the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy as the inverse temperature $ β\to \infty $.
A generalized inhomogeneous higher-order nonlinear Schrodinger (GIHNLS) equation for the Heisenberg ferromagnetic spin chain system in (1+1)-dimensions under zero boundary condition at infinity is taken into account. The spectral analysis is first performed to generate a related matrix Riemann-Hilbert problem on the real axis. Then, through solving the resulting matrix Riemann-Hilbert problem by taking the jump matrix to be the identity matrix, the general bright multi-soliton solutions to the GIHNLS equation are attained. Furthermore, the one- and two-soliton solutions are written out and ana…
This paper concentrates on the Heisenberg ferromagnetic spin chain (HFSC) equation in (2+1)-dimensions modelling nonlinear wave propagation in ferromagnetic spin chain. A variable transformation is first employed to reduce the studied equation. And then an associated matrix Riemann-Hilbert problem is built on the real line through analyzing spectral problem of the reduced equation. As a consequence, solving the obtained matrix Riemann-Hilbert problem with the identity jump matrix, corresponding to the reflectionless, the general multi-soliton solutions to the HFSC equation in (2+1)-dimensions …
We present in this report 1+1 dimensional nonlinear partial differential equation integrable through inverse scattering transform. The integrable system under consideration is a pseudo-Hermitian reduction of a matrix generalization of classical 1+1 dimensional Heisenberg ferromagnet equation. We derive recursion operators and describe the integrable hierarchy related to that matrix equation.
The paper is dedicated to a system of matrix nonlinear evolution equations related to a Hermitian symmetric space of the type $\mathbf{A.III}$. The system under consideration extends the $1+1$ dimensional Heisenberg ferromagnet equation in the sense that its Lax pair has a form rather similar to the pair of the original Heisenberg ferromagnet model. We shall present here certain local and nonlocal reductions. A local integrable deformation and some of its reductions will be discussed too.
We study the negative flows of the hierarchy of the integrable Heisenberg Ferromagnet model and their soliton solutions. The first negative flow is related to the so-called short pulse equation. We provide a framework which generates Lax pairs for the other members of the hierarchy. The Inverse scattering, based on the dressing method is illustrated with the derivation of the one-soliton solution.
We investigate the soliton dynamics of the electromagnetic wave propagating in an inhomogeneous or deformed ferromagnet. The dynamics of magnetization and the propagation of electromagnetic waves are governed by the Landau-Lifshitz-Maxwell (LLM) equation, a certain coupling between the Landau-Lifshitz and Maxwell's equations. In the framework of multiscale analysis, we obtain the perturbed integral modified KdV (PIMKdV) equation. Since the dynamic is governed by the nonlinear integro-differential equation, we rely on numerical simulations to study the interaction of its mKdV solitons with vari…
The Quantum Heisenberg Ferromagnet can be naturally reformulated in terms of interacting bosons (called spin waves or magnons) as an expansion in the inverse spin size. We calculate the first order interaction correction to the free energy, as an upper bound in the limit where the spin size $S \to \infty$ and $βS$ is fixed ($β$ being the inverse temperature). Our result is valid in two and three spatial dimensions. We extrapolate our result to compare with Dyson's low-temperature expansion. While our first-order correction has the expected temperature dependance, in higher orders of the pertur…
We consider the Landau-Lifshitz equation for the spin torque oscillator - a uniaxial ferromagnet in an external magnetic field with polarised spin current driven through it. In the absence of the Gilbert damping, the equation turns out to be PT-symmetric. We interpret the PT-symmetry as a balance between gain and loss - and identify the gaining and losing modes. In the vicinity of the bifurcation point of a uniform static state of magnetisation, the PT-symmetric Landau-Lifshitz equation with a small dissipative perturbation reduces to a nonlinear Schrödinger equation with a quadratic nonlinear…
Using the algebro-geometric approach, we study the structure of semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic Landau-Lifshitz equation arise as coherent macroscopic low-energy fluctuations of the ferromagnetic ground state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. The internal magnon structure of classical spin waves is resolved by performing the semi-classical quantisation using the Riemann-Hilbert…
We give a polynomial-time algorithm for computing upper bounds on some of the smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with any graph $G$ for the underlying interactions. An important ingredient is the connection between Heisenberg models and the symmetric products of $G$. Our algorithms for computing upper bounds are based on generalized diameters of graphs. Computing the upper bounds amounts to solving the minimum assignment problem on $G$, which has well-known polynomial-time algorithms from the field of combinatorial optimization. We also study the possibilit…
