Integrable Soliton Equations Research Articles

The Darboux Transformation for a New Fractional Schrödinger Equation Hierarchy

It is well established that many integrable soliton equations can be constrained from the famous AKNS system. In this paper, aiming to generate new integrable soliton equations, the potentials of the AKNS system are constrained to a pair of fractional conjugate ones. The resulted equations include two fractional Scrödinger equations which are integrable for their own Lax pairs. Furthermore, odd‐soliton solutions are obtained through N‐fold Darboux transformations. Dynamic properties of the obtained solutions are analyzed in detail. These results suggest that the AKNS system allows a new type of potential constraints.PACS: 02.30.Ik, 05.45.Yv.

Asymptotic theory of not completely integrable soliton equations.

We develop the theory of transformation of intensive initial nonlinear wave pulses to trains of solitons emerging at asymptotically large time of evolution. Our approach is based on the theory of dispersive shock waves in which the number of nonlinear oscillations in the shock becomes the number of solitons at the asymptotic state. We show that this number of oscillations, which is proportional to the classical action of particles associated with the small-amplitude edges of shocks, is preserved by the dispersionless flow. Then, the Poincaré-Cartan integral invariant is also constant, and therefore, it reduces to the quantization rule similar to the Bohr-Sommerfeld quantization rule for a linear spectral problem associated with completely integrable equations. This rule yields a set of "eigenvalues," which are related to the asymptotic solitons' velocities and their characteristics. It is implied that the soliton equations under consideration give modulationally stable solutions; therefore, these "eigenvalues" are real. Our analytical results agree very well with the results of numerical solutions of the generalized defocusing nonlinear Schrödinger equation.

GAUGE EQUIVALENCE BETWEEN THE TWO-COMPONENT GENERALIZATION OF THE (2+1)-DIMENSIONAL DAVEY-STEWARTSON I EQUATION AND 𝜞 - SPIN SYSTEM

In recent years, multidimensional nonlinear evolutionary equations have been actively studied within the framework of the theory of solitons. Their relevance is confirmed by numerous scientific publications. In this work the gauge equivalence between the (2+1)-dimensional integrable two-component Davey-Stewartson I (DSI) equation and the Г - spin system is established. Multicomponent generalizations of nonlinear integrable equations are of current interest from both physical and mathematical points of view. In the theory of integrable (soliton) equations, one of the key models is integrable nonlinear Schrodinger-type (NLS) equations. A two-component integrable generalization of the (2+1)-dimensional DSI equation, obtained on the basis of its one-component representation, and its corresponding Lax representation were proposed. A geometric connection between the twolayer spin system and the integrable two-component Manakov system is found. The nonlinear equations are integrated using the inverse scattering problem method by means of a linear system. For each integrable nonlinear equation, as is known, there is a Lax pair of two linear equations, a compatibility condition, that is, a condition of zero curvature, which this equation serves. We have obtained Lax pair whose zero curvature condition gives the Г - spin system.

Nonlocal conservation laws, nonlocal symmetries and exact solutions of an integrable soliton equation

Abstract We compute nonlocal symmetries and obtain kink type soliton solutions to an integrable soliton equation. We construct a tree of nonlocally related partial differential equations (PDEs) of integrable soliton equation which consists of potential systems arising from conservation laws based method and inverse potential systems (IPS) from symmetry based method. We show that the integrable soliton equation admits two nonlocal symmetries among these one of which results from potential system and the other from locally related subsystem of IPS. We propose a systematic procedure to obtain exact solutions of a given PDE system using nonlocal symmetry which arises from IPS or its locally related subsystem. Finally, we obtain exact solutions to integrable soliton equation using nonlocal symmetries and discuss the physical behavior of the explicit solution graphically.

Time-dependent defects in integrable soliton equations.

We study (1 + 1)-dimensional integrable soliton equations with time-dependent defects located at x = c(t), where c(t) is a function of class C 1. We define the defect condition as a Bäcklund transformation evaluated at x = c(t) in space rather than over the full line. We show that such a defect condition does not spoil the integrability of the system. We also study soliton solutions that can meet the defect for the system. An interesting discovery is that the defect system admits peaked soliton solutions.

Resonant multiple wave solutions to some integrable soliton equations**Project supported by the Yue-Qi Scholar of the China University of Mining and Technology (Grant No. 102504180004) and the 333 Project of Jiangsu Province, China (Grant No. BRA2018320).

To transform the exponential traveling wave solutions to bilinear differential equations, a sufficient and necessary condition is proposed. Motivated by the condition, we extend the results to the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation, the (3+1)-dimensional generalized Kadomtsev–Petviashvili (g-KP) equation, and the B-type Kadomtsev–Petviashvili (BKP) equation. Aa a result, we obtain some new resonant multiple wave solutions through the parameterization for wave numbers and frequencies via some linear combinations of exponential traveling waves. Finally, these new resonant type solutions can be displayed in graphs to illustrate the resonant behaviors of multiple wave solutions.

On periodic wave solutions of the KdV6 equation via bilinear Bäcklund transformation

The KdV6 equation is an integrable soliton equation. By using the bilinear Bäcklund transformation, we derive the one and two Riemann theta function periodic wave solutions of the KdV6 equation. The dynamical behaviors of the periodic wave solutions are analyzed by numerical simulations. Finally, we establish the relations between the periodic wave solutions and the soliton solutions.

Smooth soliton and kink solutions for a new integrable soliton equation

We establish a relationship between a new integrable soliton equation and Gardner’s equation by a transformation. Then, we use this transformation and solutions of Gardner’s equation to derive some exact solutions of the new integrable soliton equation. The obtained solutions include smooth soliton solution, kink solution and periodic wave solution. In particular, the hyperbolic type kink and the rational type kink waves are given.

On Generating Integrable Dynamical Systems in 1+1 and 2+1 Dimensions by Using Semisimple Lie Algebras

Abstract We deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are one-order polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λ t of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.

Some properties for a new integrable soliton equation

In this paper, we introduce an integrable soliton equation and present its Lax pair and bi-Hamiltonian structure. We demonstrate that this integrable equation possesses special kink waves. The relationship between the integrable soliton equation and Gardner's equation is established by a reciprocal transformation. Our study extends previous research through a comparison between the soliton equation and its generalized version.

Classical integrable systems and soliton equations related to eleven-vertex R-matrix

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest case – the 11-vertex R-matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n-particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau–Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is described.

TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES

Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview

The Landau-Lifshitz-Gilbert (LLG) equation is a fascinating nonlinear evolution equation both from mathematical and physical points of view. It is related to the dynamics of several important physical systems such as ferromagnets, vortex filaments, moving space curves, etc. and has intimate connections with many of the well-known integrable soliton equations, including nonlinear Schrödinger and sine-Gordon equations. It can admit very many dynamical structures including spin waves, elliptic function waves, solitons, dromions, vortices, spatio-temporal patterns, chaos, etc. depending on the physical and spin dimensions and the nature of interactions. An exciting recent development is that the spin torque effect in nanoferromagnets is described by a generalization of the LLG equation that forms a basic dynamical equation in the field of spintronics. This article will briefly review these developments as a tribute to Robin Bullough who was a great admirer of the LLG equation.

Two 3 × 3 discrete matrix spectral problems and associated Liouville integrable lattice soliton equations

Two 3 × 3 discrete matrix spectral problems are introduced and the corresponding lattice soliton equations are derived. By means of the discrete trace identity the Hamiltonian structures of the resulting equations are constructed. Liouville integrability of the discrete Hamiltonian systems is proved.

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchywith complex-valued initial data and prove unique solvability globally intime for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inversealgebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

Quantum deformations of associative algebras and integrable systems

Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which have a geometrical meaning of a vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the Witten–Dijkgraaf–Verlinde–Verlinde equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.

The Darboux transformation associated with two-parameter lattice soliton equation

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding two-parameter integrable lattice soliton equation is obtained through the discrete zero curvature representation, and the resulting integrable lattice equation reduce to the Toda lattice in rational form for a special choice of the parameters. A Darboux transformation (DT) for the lattice soliton equation is constructed. As an application, an explicit solution of the two-parameter lattice soliton equation is presented.

A NEW INTEGRABLE LATTICE HIERARCHY AND ITS TWO DISCRETE INTEGRABLE COUPLINGS

Starting from a discrete isospectral problem, a hierarchy of nonlinear Liouville integrable lattice soliton equations are derived. Through enlarging associated spectral problems, two kinds of discrete integrable couplings are constructed for the resulting lattice hierarchy.

The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data

We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Toda hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for generally non-self-adjoint Jacobi operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin equations), this yields the construction of global algebro-geometric solutions of the time-dependent Toda hierarchy. The inherent non-self-adjointness of the underlying Lax (i.e., Jacobi) operator associated with complex-valued coefficients for the Toda hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Toda hierarchy but applies generally to 1+1 -dimensional completely integrable discrete soliton equations.