Modified Korteweg-de Vries Hierarchy Research Articles

The algebraic structure of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy

We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine–Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product—the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’ representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.

Integrability of linear structure of components of modified Korteweg–De Vries hierarchy: multiple soliton solutions and breathers solutions

PurposeThe purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.Design/methodology/approachThe authors investigate Painlevé integrability of the constructed linear structure.FindingsThe Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.Research limitations/implicationsThe research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.Practical implicationsThe authors also furnish a variety of solitonic solutions and complex solutions as well.Social implicationsThe work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.Originality/valueThe paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.

Higher Order Airy and Painlevé Asymptotics for the mKdV Hierarchy

In this paper, we consider Cauchy problem for the modified Korteweg-de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of $t^{-1/(2n+1)}$ with smooth coefficients of the variable $(-1)^{n+1}x((2n+1) t)^{-1/(2n+1)}$ in the self-similarity region for the solution of $n$-th member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlev\'e II hierarchy, which generalize the classical Ablowitz-Segur solution for the Painelv\'{e} II equation and appear in a variety of random matrix and statistical physics models. We establish the connection formulas for this family of solutions. In the special case that the reflection coefficient vanishes at the origin, the solutions of Painlev\'e II hierarchy in the leading coefficient vanishes as well, the leading and subleading terms in the asymptotic expansion are instead given explicitly in terms of derivatives of the generalized Airy function.

Duality of positive and negative integrable hierarchies via relativistically invariant fields

It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

Traveling wave reduction of the modified KdV hierarchy: The Lax pair and the first integrals

The traveling wave reduction of the modified Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found in the general form. The Lax pair is used to obtain the first integrals for the traveling wave reduction of this hierarchy. The first three members of the mKdV hierarchy are considered in more detail. Exact solutions for the traveling wave reduction for the mKdV hierarchy and its first integrals are given. The obtained first integrals can be considered as the expansion of the list for the nonlinear integrable differential equations.

From Euler’s elastica to the mKdV hierarchy, through the Faber polynomials

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler’s elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.

The full positive flows of Manakov hierarchy, Hamiltonian structures and conservation laws

Based on four sets of Lenard recursion sequences and zero-curvature equation, we derive the full positive flows of the Manakov hierarchy associated with a 3×3 matrix spectral problem, from which some new nonlinear evolution equations are proposed. With the help of the Darboux transformation, soliton solutions of two new nonlinear evolution equations in the Manakov hierarchy are constructed. As two special reductions, the full positive flows of the coupled modified Korteweg–de Vries hierarchy and the Sasa–Satsuma hierarchy are deduced, in which some new nonlinear evolution equations are included. And then, we construct the Hamiltonian structures of the Manakov hierarchy and infinite conservation laws of several nonlinear evolution equations.

Nonlinear integrable couplings of a nonlinear Schrödinger—modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B2, a new extended Lie algebra F is constructed. Based on the Lie algebras B2 and F, the nonlinear Schrödinger—modified Korteweg de Vries (NLS—mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived. With the help of the variational identity, their Hamiltonian structures are generated.

Mixed hierarchy of soliton equations

The mixed hierarchy of soliton equations in (1+1) dimensions is introduced. It contains nonholonomic deformations of soliton equations such the KdV6 equation and the Kupershmidt deformations of soliton equations as special members. Based on the commutator representation method, a recipe for constructing zero curvature representations of mixed hierarchy is proposed. As applications, we obtain the mixed hierarchies and their zero curvature representations for the Korteweg–de Vries hierarchy, the Ablowitz–Kaup–Newell–Segur hierarchy, the modified Korteweg–de Vries hierarchy, the Toda lattice hierarchy, and the Volterra lattice hierarchy.

On the kinematics of 2+1-dimensional motions of a fibre-reinforced fluid. Integrable connections

Summary Evolution of foliations of the plane is shown, under a condition of constant divergence, to be linked to the scattering problem for the integrable modified Korteweg–de Vries hierarchy. This result is applied to a set of kinematic relations which arise in the theory of ideal fibre-reinforced fluids. In particular, it is established that the fibres, which are convected with the fluid, constitute generalized tractrices.

Vanishing residue characterization of the sine-Gordon hierarchy

The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.

A subalgebra of loop algebra 2 and its applications

A subalgebra of loop algebra A2 and its expanding loop algebra are constructed. It follows that both resulting integrable Hamiltonian hierarchies are obtained. As a reduction case of the first hierarchy, a generalized nonlinear coupled Schrodinger equation, the standard heat-conduction and a formalism of the well known Ablowitz, Kaup, Newell and Segur hierarchy are given, respectively. As a reduction case of the second hierarchy, the nonlinear Schrodinger and modified Korteweg de Vries hierarchy and a new integrable system are presented. Especially, a coupled generalized Burgers equation is generated.

Integration of the modified Korteweg-de Vries hierarchy with an integral type of source

The modified Korteweg-de Vries hierarchy with an integral type of source (mKdVHWS), which consists of the reduced AKNS eigenvalue problem with r = q and the mKdV hierarchy with an extra term of the integration of a square eigenfunction, is investigated. We propose a method to find the explicit evolution equation for the eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations of scattering data corresponding to the mKdVHWS, and solve the equation in the mKdVHWS by inverse scattering transformation.

Recursion formulae for IKK system with two hierarchies

In a previous paper an inverse system with two hierarchies for a given symmetry was proposed. The scheme was found to have so constructive structure that relation between members of the hierarchies was systematically determined. In this Letter, with the proposed system we find simply and directly recursion formulae for two hierarchies for the su(2) algebra, i.e., the localized induction hierarchy and the hierarchy with the dispersionless equation. We show that, as a special case, the recursion formulae reduce to those of the modified Korteweg–de Vries hierarchy and the sine–Gordon hierarchy.

On Huygens' Principle for Dirac Operators and Nonlinear Evolution Equations

We exhibit a class of Dirac operators that possess Huygens’ property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering the rational solutions of the modified Korteweg-de Vries hierarchy.

On Huygens’ Principle for Dirac Operators and Nonlinear Evolution Equations

We exhibit a class of Dirac operators that possess Huygens’ property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering the rational solutions of the modified Korteweg-de Vries hierarchy.

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painlevé equation, . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of equations, and a little known Bäcklund transformation for itself. We then use our results on Bäcklund transformations to obtain, for each member of the hierarchy, a sequence of special integrals.

GEOMETRICAL CONSTRUCTION OF THE HIROTA BILINEAR FORM OF THE MODIFIED KORTEWEG-DE VRIES EQUATION ON A THIN ELASTIC ROD: BOSONIC CLASSICAL THEORY

Recently there have been several studies of a nonrelativistic elastic rod in R2 whose dynamics is governed by the modified Korteweg-de Vries (MKdV) equation. Goldstein and Petrich found the MKdV hierarchy through its dynamics [Phys. Rev. Lett. 69, 555 (1992).] In this article, we will show the physical meaning of the Hirota bilinear form along the lines of the elastica problem after we formally complexify its arc length.

An involutive system and integrable C. Neumann system associated with the modified Korteweg–de Vries hierarchy

In this article, a system of finite-dimensional involutive functions is presented and proven to be integrable in the Liouville sense. By using the nonlinearization method, the C. Neumann system associated with the modified Korteweg–de Vries (mKdV) hierarchy is obtained. Thus, the C. Neumann system is shown to be completely integrable via a gauge transformation between it and an integrable Hamiltonian system. Finally, the solution of a stationary mKdV equation and the involutive solutions of the mKdV hierarchy are secured. As two examples, the involutive solutions are given for the mKdV equation: vt+1/4vxxx−3/2v2vx=0 and the 5th mKdV equation vt−1/16vxxxxx+5/8v2vxxx +5/2vvxvxx +5/8v3x−3/40v4vx=0.

An elementary geometric characterization of the integrable motions of a curve

We show that the following elementary geometric properties of the motion of a curve select hierarchies of integrable dynamics: (i) the curve moves in an N-dimensional sphere of radius R; (ii) the motion is nonstretching; (iii) the dynamics does not depend explicitly on the radius of the sphere. For N = 2 we obtain the modified Korteweg-de Vries hierarchy, for N = 3 the nonlinear Schrödinger hierarchy and for N > 3 we obtain integrable multicomponent generalizations of the above hierarchies.