Ishimori Equation Research Articles

On the Study and Possible Applications of Minimally Complex Chaos

Abstract The nature of chaos is elusive and disputed, however it can be connected to sensitivity to initial conditions caused by nonlinearity of the equations describing chaotic phenomena. A nowhere-near comprehensive list of such equations can still be shown: the Boltzmann equation, Ginzburg-Landau equation, Ishimori equation, Korteweg-de Vries equation, Landau-Lifshitz-Gilbert equation, Navier-Stokes equation, and many more. This disproportionality between input and output creates an analytically-difficult situation, one that is complicated both algebraically and numerically – however, the study of equations that are both simple and chaotic may yield useful connections between algebraic complexity and chaos. Such connections can be used to determine the simplest possible chaotic function, which can be used as a “chaotic operator” for various non-chaotic or chaotic functions, thus reducing the problem of chaos to one based strictly on algebra.

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods.

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of to solution for arbitrary finite values of . Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.

Lie Symmetries of Ishimori Equation

The Ishimori equation is one of the most important (2+1)-dimensional integrable models, which is an integrable generalization of (1+1)-dimensional classical continuous Heisenberg ferromagnetic spin equations. Based on importance of Lie symmetries in analysis of differential equations, in this paper, we derive Lie symmetries for the Ishimori equation by Hirota's direct method.

Local well-posedness for hyperbolic–elliptic Ishimori equation

In this paper we consider the hyperbolic–elliptic Ishimori initial-value problem with the form:{∂ts=s×□xs+b(ϕx1sx2+ϕx2sx1)on R2×[−1,1];Δϕ=2s⋅(sx1×sx2);s(0)=s0 where s(⋅,t):R2→S2⊂R3, × denotes the wedge product in R3, □x=∂x12−∂x22, b∈R. We prove that such system is locally well-posed for small data s0∈HQσ0(R2;S2), σ0>3/2, Q∈S2. The new ingredient is that we develop the methods of Ionescu and Kenig (2006) [6] and (2007) [7] to approach the problem in a perturbative way.

Localized coherent structures of Ishimori equation I through Hirota’s bilinearization method: Time dependent/stationary boundaries

Abstract Ishimori equation is a (2 + 1) dimensional generalization of the (1 + 1) dimensional integrable classical continuous Heisenberg ferromagnetic spin equation. The richness of the coherent structures admitted by Ishimori equation I such as dromion, lump and rationally–exponentially localized solutions, have been demonstrated in the literature through ∂ ¯ technique and binary Darboux transformation method. To our knowledge Hirota’s method had been adopted to construct only the vortex solutions of Ishimori equation II. For the first time, the various types of localized coherent structures mentioned above have been constructed in this paper for the Ishimori equation I using the Hirota’s direct method. In particular we have brought out the significance of boundaries and arbitrary functions in generating all these types of localized structures and proved that the absence of such boundaries leads only to line soliton solutions.

Motion of curves and surfaces and nonlinear evolution equations in (2+1) dimensions

It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov-Strachan (2+1) dimensional nonlinear Schrödinger equations, respectively.

The Davey - Stewartson equation and the Ablowitz - Ladik hierarchy

A relation between the Davey - Stewartson (DS) system and the Ablowitz - Ladik hierarchy (ALH) is established. It is shown that the former can be `embedded' into the system of flows described by the ALH. This relation is used to work out a procedure for obtaining a wide range of solutions of the (2 + 1)-dimensional DS equation using the traditional inverse scattering transform that has been developed for (1 + 1)-dimensional systems, without invoking the multidimensional one. The `embedding into the ALH' method is used to derive the dark soliton, quasiperiodic and some other solutions for the DS-II as well for the Ishimori equation, which is gauge equivalent to the DS system.

The Cauchy Problem for the Ishimori Equation

The Cauchy problem for the Ishimori equation is solved when initial spins are almost parallel by relating the Ishimori equation to the Davey–Stewartson II equation through a gauge transformation.

Global existence of small solutions to the Davey-Stewartson and the Ishimori systems

We study the initial-value problems for the Davey-Stewartson systems and the Ishimori equations. Elliptic-hyperbolic and hyperbolic-elliptic cases were treated by the inverse scattering techniques ([2--4, 10, 13--15, 32] for the Davey-Stewartson systems and [28, 29, 33] for the Ishimori equations). Elliptic-elliptic and hyperbolic-elliptic cases were studied (in [16, 17] for the Davey-Stewartson systems and [31] for the Ishimori equations) without the use of the inverse scattering techniques. Existence of a weak solution to the Davey-Stewartson systems for the elliptic-hyperbolic case is also obtained in [16] with a smallness condition on the data in $L^2$ and a blow-up result was also obtained for the elliptic-elliptic case. By using the sharp smoothing property of solutions to the linear Schrödinger equations the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases was established in [30] in the usual Sobolev spaces with a smallness condition on the data. We prove the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces without a smallness condition on the data. Furthermore we prove existence of global small solutions of these equations for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces.

SELF-DUAL CHERN-SIMONS SOLITONS IN NONLINEAR σ-MODELS

A self-dual Chern-Simons system and its Lax pair are derived from the tangent space representation of a two-dimensional nonlinear σ-model, endowed with a gauge field. The related “matter” field density obeys the Liouville equation, whose N-soliton solutions correspond to the magnetic vortices in the static self-dual planar Heisenberg model. It is shown that the topological charge and the total vorticity correspond to the electric charge and the magnetic flux for the Chern-Simons system, respectively. General holomorphic solutions of the system studied generate a large class of static solutions to the Davey-Stewartson and Ishimori equations.

Group theory of a nonlinear spin field model in 2 + 1 dimensions

We have studied both the compact and the noncompact version of an integrable nonlinear spin field system in 2 + 1 dimensions, the so-called Ishimori model, within a group theory framework. This model possesses an infinite-dimensional symmetry algebra with a loop algebra structure. Using the one-dimensional subalgebras of the symmetry algebra of the model under investigation, we find reduced equations which yield new classes of solutions of the original system from the known ones. The reduced equations obtained are of three types, i.e. the isotropic Heisenberg magnet model, the Landau-Lifshitz magnet model with uniaxial anisotropy, and a system in two independent variables resembling the static Ishimori equations. Our symmetry approach shows the physical meaning of the Ishimori model as being a well-defined phenomenological integrable nonlinear spin field model in 2 + 1 dimensions.

Boundary-value problems on the half-plane for the Ishimori equation that are compatible with the inverse scattering method

Examples of boundary conditions compatible with the integrability property of the Ishimori equation are found by adjoining a point symmetry of reflection type to Backlund and Miura transformations. Soliton solutions of the corresponding boundary-value problems on the half- and quarter-plane are constructed. The interaction of the boundary with the solitons is investigated.

The Cauchy problem for the Ishimori equations

In 1984, Y. Ishimori derived a nonlinear wave equation which is a two-dimensional analogue of the classical isotropic spin-chain in ferromagnetism. In this paper we show that the Cauchy problem associated to this equation is well-posed in Sobolev spaces when initial spins are almost parallel.

Coherent structures for the Ishimori equation II. Time-dependent boundaries

The localized solutions for the Ishimori equation are studied. Using the general formulae derived in paper I and the exact solutions of the inverse problem for the modified Kadomtsev-Petviashvili equation we construct the exponentially and rationally localized soliton type solutions of the Ishimori equation with time dependent boundaries. Explicit examples of such solutions are presented.

Coherent structures for the Ishimori equation: I. Localized solitons with stationary boundaries

The initial-boundary value problem for the Ishimori-II equation is studied. The time evolution of the inverse problem data in the case of nontrivial boundaries is found. A general formula for exact solutions of the Ishimori-II equation with nontrivial boundaries is derived. Localized soliton solutions of the Ishimori-II equation with time-independent boundaries are studied in detail. It is shown that there exist essentially three different types (ss, sb, bb) of localized solitons for the Ishimori-II equation. Some explicit examples of such solutions are presented.

Inverse spectral transform for the Ishimori equation: I. Initial value problem

The initial value problem for the (2+1)-dimensional integrable generalization of the (1+1)-dimensional continuous Heisenberg ferromagnet model (Ishimori-I,II equations) is solved by the ∂̄ and nonlocal Riemann–Hilbert problems (NRHP) method.

Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey‐Stewartson and Ishimori Equations

A 2 + 1‐dimensional nonlinear differential equation integrable by the inverse‐spectral‐transform method with the quartet operator representation is proposed. This GL(2, C)‐valued chiral‐field‐type equation is the generating (prototype) equation for the Davey‐Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey‐Stewartson eigenfunction ψDS. The initial‐value problem for this equation is solved by the techniques for the and the nonlocal Riemann‐Hilbert problem. The classes of exact solutions with the functional parameters and exponential‐rational solutions are constructed by the method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed.

On the inverse scattering transform for the Ishimori equation

The initial value problem for the 1 + 2 dimensional integrable generalisation of the 1 + 1 dimensional continuous Heisenberg ferromagnet model is solved by the ϖ -method.