Recursion operators and bi-Hamiltonian structures in multidimensions. II

We analyze further the algebraic properties of bi-Hamiltonian systems in two spatial and one temporal dimensions. By utilizing the Lie algebra of certain basic (starting) symmetry operators we show that these equations possess infinitely many time dependent symmetries and constants of motion. The master symmetries for these equations are simply derived within our formalism. Furthermore, certain new functionsT ₁₂ are introduced, which algorithmically imply recursion operators ₁₂. Finally the theory presented here and in a previous paper is both motivated and verified by regarding multidimensional equations as certain singular limits of equations in one spatial dimension.     

AUTHOR INDEX (Volume 16)

It is shown that the deformed Nonlinear Schrödinger (NLS), Hirota and AKNS equations with (1 + 1) dimension admit infinitely many generalized (nonpoint) symmetries and polynomial conserved quantities, master symmetries and recursion operator ensuring their complete integrability. Also shown that each of them admits infinitely many nonlocal symmetries. The nature of the deformed equation whether bi-Hamiltonian or not is briefly analyzed.     

On the theory of recursion operator

The general structure and properties of recursion operators for Hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered. Weak and strong recursion operators are introduced. The conditions which determine weak and strong recursion operators are found.In the theory of nonlinear waves a method for the calculation of the recursion operator, which is based on the use of expansion into a power series over the fields and the momentum representation, is proposed. Within the framework of this method a recursion operator is easily calculated via the Hamiltonian of a given equation. It is shown that only the one-dimensional nonlinear evolution equations can posses a regular recursion operator. In particular, the Kadomtsev-Petviashvili equation has no regular recursion operator.     

BI-HAMILTONIAN REPRESENTATION,SYMMETRIES AND INTEGRALS OF MIXED HEAVENLY AND HUSAIN SYSTEMS

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.     

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.     

The bi-Hamiltonian structure of the short pulse equation

We prove the integrability of the short pulse equation derived recently by Schäfer and Wayne from a Hamiltonian point of view. We give its bi-Hamiltonian structure and show how the recursion operator defined by the Hamiltonian operators is connected with the one obtained by Sakovich and Sakovich. An alternative zero-curvature formulation is also given.     

Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)$SO\left(3\right)$-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)$SO\left(3\right)$-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrödinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrödinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.     

Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schr?dinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.     

On bi-Hamiltonian structure of two-component Novikov equation

In this Letter, we present a bi-Hamiltonian structure for the two-component Novikov equation. We also show that proper reduction of this bi-Hamiltonian structure leads to the Hamiltonian operators found by Hone and Wang for the Novikov equation.     

Recursion Operators of Two Supersymmetric Equations

From Lax representations, recursion operators for the supersymmetric KdV and the supersymmetric Kaup-Kupershimdt (SKK) equations are proposed explicitly. Under some special conditions, the recursion operator of the supersymmetricSawada-Kotera equation can be recovered by the one of the SKK equation.     

Some super systems with local bi-Hamiltonian operators

For a class of super skew-symmetric operators with eight parameters, two kinds of super Hamiltonian operators are identified by the method of functional multi-vectors. Appropriate decompositions of these operators lead to compatible super Hamiltonian pairs which in turn produce nonlinear super bi-Hamiltonian systems from the trivial x-translation flow. As examples, besides the super Korteweg–de Vries equations and other known ones, new super generalizations are obtained for the Riemann equation, the Hunter–Saxton equation and the Camassa–Holm equation, all of them admit two compatible local super Hamiltonian operators.     

Integrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem

Based on a new discrete three-by-three matrix spectral problem, a hierarchy of integrable lattice equations with three potentials is proposed through discrete zero-curvature representation, and the resulting integrable lattice equation reduces to the classical Toda lattice equation. It is shown that thehierarchy possesses a Hamiltonian structure and a hereditary recursion operator. Finally, infinitely many conservation laws of corresponding lattice systems are obtained by a direct way.     

The Riccati and Ermakov-Pinney hierarchies

Abstract

The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315–337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201–225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.     

Two Hierarchies of New Differential-Difference Equations Related to the Darboux Transformations of the Kaup–Newell Hierarchy

Two hierarchies of new nonlinear differential-difference equations with one continuous variable and one discrete variable are constructed from the Darboux transformations of the Kaup–Newell hierarchy of equations. Their integrable properties such as recursion operator, zero-curvature representations, and bi-Hamiltonian structures are studied. In addition, the hierarchy of equations obtained by Wu and Geng is identified with the hierarchy of two-component modified Volterra lattice equations.     

A Super Extension of Heisenberg Hierarchy

A super Heisenberg equation associated with a 3 × 3 matrix spectral problem is proposed take advantage of the zero-curvature equation. Its super bi-Hamiltonian structures are obtained through super trace identity and infinitely many conserved quantities of the super Heisenberg equation are constructed by means of spectral parameter expansions.     

Modified Kadomtsev-Petviashvili Equation in (3+1) Dimensions:Multiple Front-Wave Solutions

A modified Kadomtsev-Petviashvili(mKP) equation in(3+1) dimensions is presented.We reveal multiple front-waves solutions for this higher-dimensional developed equation,and multiple singular front-wave solutions as well.The constraints on the coefficients of the spatial variables,that assure the existence of these multiple front-wave solutions are investigated.We also show that this equation fails the Pamlevi test,and we conclude that it is not integrable in the sense of possessing the Pain/eve property,although it gives multiple front-wave solutions.     

SYMMETRIES AND DROMION SOLUTION OF A (2+1)-DIMENSIONAL NONLINEAR SOHR?DINGER EQUATION

The Painlevé property, infinitely many symmetries and exact solutions of a (2+1)-dimensional nonlinear Schr?dinger equation, which are obtained from the constraints of the Kadomtsev-Petviashvili equation, are studied in this paper. The Painlevé property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method and the dromion-like solution which is localized exponentially in all directions is obtained by a variable separation method.     

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.