bi-Hamiltonian Integrability Research Articles

Integrable systems from inelastic curve flows in 2– and 3– dimensional Minkowski space

Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2– and 3– dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrödinger (NLS) equation in 2– and 3– dimensional Euclidean space, respectively. In 2–dimensional Minkowski space, timelike/spacelike inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers’ equation and its symmetry integrability structure. In 3–dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.

Integrability of and differential–algebraic structures for spatially 1D hydrodynamical systems of Riemann type

A differential–algebraic approach to studying the Lax integrability of a generalized Riemann type hydrodynamic hierarchy is revisited and a new Lax representation is constructed. The related bi-Hamiltonian integrability and compatible Poissonian structures of this hierarchy are also investigated using gradient-holonomic and geometric methods.The complete integrability of a new generalized Riemann hydrodynamic system is studied via a novel combination of symplectic and differential–algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are obtained.In addition, the differential–algebraic approach is used to prove the complete Lax integrability of the generalized Ostrovsky–Vakhnenko and a new Burgers type system, and special cases are studied using symplectic and gradient-holonomic tools. Compatible pairs of polynomial Poissonian structures, matrix Lax representations and infinite hierarchies of conservation laws are derived.

The Differential-Algebraic Analysis of Symplectic and Lax Structures Related with New Riemann-Type Hydrodynamic Systems

A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic hierarchy, proposed recently by O. D. Artemovych, M. V. Pavlov, Z. Popowicz and A. K. Prykarpatski, is developed. In addition to the Lax-type representation, found before by Z. Popowicz, a closely related representation is constructed in exact form by means of a new differential-functional technique. The bi-Hamiltonian integrability and compatible Poisson structures of the generalized Riemann type hierarchy are analyzed by means of the symplectic and gradient-holonomic methods. An application of the devised differential-algebraic approach to other Riemann and Vakhnenko type hydrodynamic systems is presented.

Hidden Symmetries of Lax Integrable Nonlinear Systems

Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are reviewed. A new symplectic approach to constructing nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and based upon the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is described. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix method [1,2] is analyzed in detail. A new modified differential-algebraic approach to analyzing the Lax integrability of generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is suggested and the corresponding Lax representations are constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of these generalized Riemann type hierarchies are discussed by means of the symplectic, gradientholonomic and geometric methods.

Symplectically invariant soliton equations from non-stretching geometric curve flows

Bi-Hamiltonian hierarchies of symplectically invariant soliton equations are derived from geometric non-stretching flows of curves in the Riemannian symmetric spaces Sp(n + 1)/Sp(1) × Sp(n) and SU(2n)/Sp(n). The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, two new multi-component versions of the sine–Gordon equation and the modified Korteweg–de Vries (mKdV) equation exhibiting Sp(1) × Sp(n − 1) invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in both Sp(n + 1)/Sp(1) × Sp(n) and SU(2n)/Sp(n) are shown to be described by a non-stretching wave map and a mKdV analogue of a non-stretching Schrödinger map.

Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited

A differential-algebraic approach to studying the Lax integrability of the generalized Riemann type hydrodynamic hierarchy is revisited and its new Lax representation is constructed in exact form. The bi-Hamiltonian integrability of the generalized Riemann type hierarchy is discussed by means of the gradient-holonomic and symplectic methods and the related compatible Poissonian structures for N = 3 and N = 4 are constructed.

Quaternionic soliton equations from Hamiltonian curve flows in

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space . The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces M = G/H by viewing as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar–vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.

Integrability of a nonlinear evolution equation derived from isoperimetric plane curve motion

We provide a geometrical interpretation for the series of transformations used by Sakovich to map the third-order nonlinear evolution equation obtained by Chou and Qu to the mKdV equation. We also discuss its bi-Hamiltonian integrability as well as integrable equations associated with this system.

Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows γ ( t , x ) in Riemannian symmetric spaces M = G / H , including compact semisimple Lie groups M = K for G = K × K , H = diag G . The derivation of these soliton hierarchies utilizes a moving parallel frame and connection 1-form along the curve flows, related to the Klein geometry of the Lie group G ⊃ H where H is the local frame structure group. The soliton equations arise in explicit form from the induced flow on the frame components of the principal normal vector N = ∇ x γ x along each curve, and display invariance under the equivalence subgroup in H that preserves the unit tangent vector T = γ x in the framing at any point x on a curve. Their bi-Hamiltonian integrability structure is shown to be geometrically encoded in the Cartan structure equations for torsion and curvature of the parallel frame and its connection 1-form in the tangent space T γ M of the curve flow. The hierarchies include group-invariant versions of sine–Gordon (SG) and modified Korteweg–de Vries (mKdV) soliton equations that are found to be universally given by curve flows describing non-stretching wave maps and mKdV analogs of non-stretching Schrödinger maps on G / H . These results provide a geometric interpretation and explicit bi-Hamiltonian formulation for many known multicomponent soliton equations. Moreover, all examples of group-invariant (multicomponent) soliton equations given by the present geometric framework can be constructed in an explicit fashion based on Cartan’s classification of symmetric spaces.