mKdV Hierarchy Research Articles

The Modified Korteweg-de Vries Hierarchy: Lax Pair Representation and Bi-Hamiltonian Structure

Abstract We consider equations in the modified Korteweg-de Vries (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV equations. We also show that the complex modified KdV (cmKdV) equation follows from the action principle to have a Lagrangian representation. This representation not only provides a basis to write the cmKdV equation in the canonical form endowed with an appropriate Poisson structure but also help to construct a semianalytical solution of it. The solution obtained by us may serve as a useful guide for purely numerical routines which are currently being used to solve the cmKdV equation

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.

Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics

We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are ‘N-adapted’, metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in solitonic hierarchies described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrödinger map equation. All geometric constructions can be re-defined for the Levi-Civita connection but with “noholonomic mixing” of solitonic interactions. Finally, we speculate why certain methods and results from the geometry of nonholonmic manifolds and solitonic equations have general importance in various directions of modern mathematics, geometric mechanics, fundamental theories in physics and applications, and briefly analyze possible nonlinear wave configurations for modeling gravitational interactions by effective continuous media effects.

Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

Methods in Riemann–Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections ( N -connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces G / S O ( n ) provided with an N -connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle.

Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources

We propose a systematic method for generalizing the integrable Rosochatius deformations for finite-dimensional integrable Hamiltonian systems to integrable Rosochatius deformations for infinite-dimensional integrable equations. An infinite number of the integrable Rosochatius deformed higher-order constrained flows of some soliton hierarchies, which includes the generalized integrable Hénon–Heiles system, and the integrable Rosochatius deformations of the KdV hierarchy with self-consistent sources, of the AKNS hierarchy with self-consistent sources and of the mKdV hierarchy with self-consistent sources as well as their Lax representations are presented.

A new loop algebra [formula omitted] and its applications to integrable system

We construct a new loop algebra A ∼ 5 which is used to set up a linear isospectral problem. By making use of Tu scheme, a new integrable couplings of the generalized MKdV hierarchy is obtained. Then its tri-Hamiltonian structure is constructed by the quadratic-form identity. Furthermore, we point out that the integrable couplings is integrable in the sense of Liouville.

The quadratic-form identity for constructing Hamiltonian structures of the NLS–MKdV hierarchy and multi-component Levi hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the NLS–MKdV hierarchy, and integrable coupling of multi-component Levi hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type

The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of non- stretching curves in Riemannian symmetric spaces G/SO(N). These spaces are exhausted by the Lie groups G = SO(N + 1),SU(N). The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curve, tied to a zero curvature Maurer-Cartan form on G, and this yields the mKdV recursion operators in a geometric vectorial form. The kernel of these recursion operators is shown to yield two hyperbolic vector generalizations of the sine-Gordon equation. The corresponding geometric curve flows in the hierarchies are described in an explicit form, given by wave map equations and mKdV analogs of Schrodinger map equations.

Integrable deformations of the mKdV and SG hierarchies and quasigraded Lie algebras

We construct a new family of quasigraded Lie algebras that admit the Kostant–Adler scheme. They coincide with special quasigraded deformations of twisted subalgebras of the loop algebras. Using them we obtain new hierarchies of integrable equations in partial derivatives. They coincide with the deformations of integrable hierarchies associated with the loop algebras. We consider the case g = g l ( 2 ) in detail and obtain integrable hierarchies that could be viewed as deformations of mKdV, sine-Gordon and derivative non-linear Shrödinger hierarchies and some other integrable hierarchies, such as the (w3) non-linear Shrödinger hierarchy and its doubled form.

A new loop algebra and its application to the multi-component S-mKdV hierarchy

Abstract An efficient, straightforward method for obtaining multi-component integrable hierarchy and the multi-component integrable couplings system is proposed in this paper. A new 4M dimensional loop algebra X ∼ is constructed firstly, whose commutation operations defined by us are as simple and straightforward as that in the loop algebra A 1 ∼ . As an application example, a new isospectral problem is established, then the well-known multi-component Schrodinger hierarchy and the multi-component mKdV hierarchy are obtained. So we call it the multi-component S-mKdV hierarchy. Finally, an expanding loop algebra F ∼ M of the loop algebra X ∼ is presented. Based on the F ∼ M , the multi-component integrable couplings system of the multi-component S-mKdV hierarchy has been worked out. The method in this paper can be applied to other nonlinear evolution equations hierarchies. It is easy to find that we can construct any finite-dimensional Lie algebra by this approach in this paper.

Asymptotic solitons on a non-zero mean level

Abstract The collision of solitary waves for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sech-type and algebraic (which only exist on a non-zero mean level) profiles is considered. An asymptotic transformation, valid if the higher-order coefficients satisfy a certain algebraic relationship, is used to transform the higher-order mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higher-order collision is asymptotically elastic and to derive the higher-order phase shifts. Numerical simulations of both elastic and inelastic collisions are performed. For the example covered by the asymptotic theory the numerical results confirm that the collision is elastic and the theoretical predictions for the higher-order phase shifts. For the example not covered by the asymptotic theory the numerical results show that the collision is inelastic; an oscillatory wavetrain is produced by the colliding solitary waves. The higher-order phase shift for the faster (sech-type) solitary wave is found. For the slower (algebraic) wave, however, the shed radiation never completely separates from the solitary wave. Interaction with the shed radiation causes the phase shift to evolve long after collision is over, with no final higher-order phase shift able to be determined. An asymptotic mass-conservation law is to test the accuracy of the finite-difference scheme for the numerical solutions. It is shown that, in general, mass is not conserved by the higher-order mKdV equation, but varies during the interaction of the solitary waves.

A generalized MKdV hierarchy, tri-Hamiltonian structure, higher-order binary constrained flows and its integrable couplings system

A subalgebra of higher-dimension loop algebra A 2 ˜ is constructed, from which a new 3 × 3 isospectral problem is designed. By making use of Tu’s scheme, an integrable Hamiltonian hierarchy of equations in the sense of Liouville is obtained, which possesses tri-Hamiltonian structure. As reduction cases of the hierarchy presented in this paper, the generalized MKdV equation is engendered. By establishing binary symmetric constraints, three constrained flows of the hierarchy are presented, which are then reduced to Hamiltonian systems. Finally, an integrable coupling system is obtained by constructing a high-dimension loop algebra.

Two kinds of new integrable decompositions of the mKdV equation

Two kinds of integrable decompositions of the mKdV equation are presented. We apply a new kind of binary nonlinearization approach of the spectral problem to the well-known 2 × 2 matrix spectral problems of the mKdV equation and obtain a pair of integrable Hamiltonian systems in 4 N dimensions. To get the integrabilities of the resulting 4 N-dimensional systems, we introduce a group of new 3 × 3 matrix spectral problems for the mKdV hierarchy and apply the traditional binary nonlinearization approach of the spectral problem to them. As a result, we obtain another pair of integrable Hamiltonian systems in 6 N dimensions. We show that the 4 N-dimensional integrable systems are just the restrictions of the 6 N-dimensional systems to a special symplectic submanifold.

Quantum supersymmetric Toda–mKdV hierarchies

In this paper we generalize the quantization procedure of Toda–mKdV hierarchies to the case of arbitrary affine (super)algebras. The quantum analogue of the monodromy matrix, related to the universal R-matrix with the lower Borel subalgebra represented by the corresponding vertex operators is introduced. The auxiliary L-operators satisfying RTT-relation are constructed and the quantum integrability condition is obtained. General approach is illustrated by means of two important examples.

Poles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy

The first Painlevé hierarchy, which is a sequence of higher order analogues of the first Painlevé equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the first Painlevé hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painlevé equation, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic solutions, and the other is a lower estimate for the frequency of α-points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution.

The exact solutions for the nonisospectral AKNS hierarchy through the inverse scattering transform

Exact solutions for the nonisospectral AKNS hierarchy are obtained through the inverse scattering transform. Some explicit forms of these solutions are further given. Reductions are also considered. Exact solutions of some nonisospectral hierarchies, such as the nonisospectral mKdV hierarchy, the nonisospectral KdV hierarchy, the nonisospectral nonlinear Schrödinger hierarchy and the nonisospectral sine-Gordon hierarchy, are obtained.

DISCRETE INTEGRABLE SYSTEMS ASSOCIATED WITH THE UNITARY MATRIX MODEL

The orthogonal polynomials on the unit circle associated with the unitary matrix model have various interesting properties, and have been studied in connection with different applications, such as double scaling, Riemann–Hilbert problems and integrable Fredholm operators. In this paper, we study the orthogonal polynomials on the unit circle with the weight function [Formula: see text]. We use the orthogonality of the polynomials to show that the orthogonal polynomials on the unit circle satisfy the linear problems associated with the discrete PainlevéII hierarchy, alternate discrete PainlevéII hierarchy and the discrete MKdV hierarchy. Thus the isomonodromy deformation method can be used to study the unitary matrix model. Typically, we focus on the orthogonal polynomials with the weight function exp {s(z+z-1)}. The recursion formula z(pn+vnpn-1)=pn+1+unpn (derived from Szegö's equation) for the orthogonal polynomials pn(z,s) is proved to be equivalent to the discrete AKNS-ZS system which is the base equation in the linear problem for the discrete equations. The variable xn=pn(0,s) satisfies the discrete PainlevéII equation and the discrete MKdV equation; un, vn satisfy the alternate discrete PainlevéII equation; 1/un-1=-xn-1/xn satisfies the PainlevéIII equation; and [Formula: see text] satisfies the PainlevéV equation. Also, we discuss the continuum limits of the discrete equations by using the double scaling method. Further, we present a procedure for finding the linear equations for the orthogonal polynomials and the consistency conditions of the linear equations by using the orthogonality of the polynomials.

Extension of noncommutative soliton hierarchies

A linear system, which generates a Moyal-deformed two-dimensional soliton equation as integrability condition, can be extended to a three-dimensional linear system, treating the deformation parameter as an additional coordinate. The supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter, the flow of which commutes with the flow of the deformed soliton equation. In this way, a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations. We prove the extended hierarchy properties for the deformed AKNS hierarchy, and specialize to the cases of deformed NLS, KdV and mKdV hierarchies. Corresponding results are also obtained for the deformed KP hierarchy. A deformation equation determines a kind of Seiberg-Witten map from classical solutions to solutions of the respective `noncommutative' deformed equation.

On gauge equivalent integrable systems andr-matrices for AKNS hierarchy and a coupled MKdV hierarchy

The new coupled MKdV hierarchy is obtained. By using gauge transformation, the constrained flow, the integrable system and Lax representation for the coupled MKdV hierarchy were first constructed from the AKNS hierarchy and then using the Lax representation, the r-matrix for the constrained flow of the coupled MKdV hierarchy was constructed. The second set of conserved integrals of this constrained flow and their involutivity were also given.

Fused Hierarchy Produced by mKdV Hierarchy and Sine-Gordon Hierarchy

By using the inverse problem, we have found a system with two hierarchies, that is, the localized induction hierarchy and the hierarchy including the dispersionless equation. Recently, with the same system, we have found a fused hierarchy which is produced by fusing two hierarchies with help of the recursion formulae. In this short note, we would like to show that the mKdV hierarchy and the sine-Gordon hierarchy produce another fused hierarchy. It is recognized that the recursion operators of the mKdV hierarchy and the sine-Gordon hierarchy is the inverse operator in each other. However, it has been misunderstood that two hierarchies are not connected since the boundary conditions have not been carefully considered. If each member of equations in the mKdV hierarchy is written by tðx; tÞ nðx; tÞ 1⁄4 0; ðn 1⁄4 1; 2; . . .Þ ð1Þ