Dimensional Integrable Systems Research Articles

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the `dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.

NONTRIVIAL SOLITON SCATTERING IN PLANAR INTEGRABLE SYSTEMS

The behavior of solitons in integrable theories is strongly constrained by the integrability of the theory, that is by the existence of an infinite number of conserved quantities that these theories are known to possess. As a result, the soliton scattering of such theories is expected to be trivial (with no change of direction, velocity or shape). In this paper we present an extended review on soliton scattering of two spatial dimensional integrable systems which have been derived as dimensional reductions of the self-dual Yang–Mills equations and whose scattering properties are highly nontrivial.

Multilinear Variable Separation Approach in (3+1)-Dimensions:the Burgers Equation

The multi-linear variable separation approach has been proved tobe very useful in solving many (2+1)-dimensional integrablesystems. Taking the (3+1)-dimensional Burgersequation as a simple example, here we extend the multi-linear variableseparation approach to (3+1)-dimensions. The form of the universalformula obtained from many (2+1)-dimensional system is stillvalid. However, a more general arbitrary function (with threeindependent variables) has been included in the formula. Startingfrom the universal formula, one may obtain abundant(3+1)-dimensional localized excitations. In particular, we display a specialparaboloid-type camber soliton solution and a dipole-type dromionsolution which is localized in all directions.

Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems

The multilinear variable separation approach and the related “universal” formula have been applied to many (2+1)-dimensional nonlinear systems. Starting from the universal formula, abundant (2+1)-dimensional localized excitations have been found. In this paper, the universal formula is extended in two different ways. One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the (2+1)-dimensional symmetric sine-Gordon system. The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved. Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons. The FSWs and foldons may be “folded” in quite complicated ways and possess quite rich structures and multiplicate interaction properties.

Algebro-geometric solutions to a hierarchy of (1 + 1)-dimensional and two new (2 + 1)-dimensional nonlinear evolution equations

Two new 2 + 1 dimensional nonlinear evolution equations are presented. The 2 + 1 dimensional equations closely relate with a hierarchy of 1 + 1 dimensional soliton equations. Through nonlinearizing of Lax pairs, the 1 + 1 dimensional evolution equations are decomposed to the finite dimensional integrable Hamiltonian systems. Finally by applying Riemann–Jacobi inversion technique, the algebro-geometric solutions of the 1 + 1 dimensional soliton equation hierarchy as well as two 2 + 1 dimensional equations are obtained.

Painlevé Analysis and Some Solutions of (2+1)-Dimensional Generalized Burgers Equations

Búrgers equation describes a lot of phenomena in physics fields, and it has attracted much attention. In this paper, the Burgers equation is generalized to (2+1) dimensions. By means of the Painlevé analysis, the most generalized Painlevé integrable (2+1)-dimensional integrable Burgers systems are obtained. Some exact solutions of the generalized Burgers system are obtained via variable separation approach.

NEW JACOBI ELLIPTIC FUNCTION SOLUTIONS OF (2 +1)-DIMENSIONAL COMPLEX NONLINEAR INTEGRABLE SYSTEM

Recently sixteen types of doubly-periodic solutions were obtained for the new (2 +1)-dimensional complex nonlinear integrable system. In this paper with the aid of computerized symbolic computation, our extended Jacobi elliptic function expansion method is extended to this system. As a result, another eight families of Jacobian elliptic functions solutions are also found.

Variable Separation Solutions for the (2+1)-DimensionalBurgers Equation

Considering that the multi-linear variable separation approach has beenproved to be very useful to solve many (2+1)-dimensional integrablesystems, we obtain the variable separation solutions of the Burgersequation with arbitrary number of variable separated functions. TheY-shaped soliton fusion phenomenon is revealed.

Classification of Integrable Equations on Quad-Graphs. The Consistency Approach

A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed.

Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

One of the more interesting solutions of the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in 2+1 dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in 2+1 dimensions. It is interesting that neither of the (2+1)-dimensional integrable systems considered admit Virasoro-type subalgebras.

Integrable Systems in n-Dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in on+1 with the vector modified Korteweg-De Vries equation (vmKDV) ut = uxxx + 3 ||u|| 2 ux as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the natural or parallel frame and we show how this can be gauged by a generalized Hasimoto ¯t = ¯

ClassicalR-matrix theory of dispersionless systems: I. (1 1)-dimension theory

A systematic way of construction of (1+1)-dimensional dispersionless integrable Hamiltonian systems is presented. The method is based on the classical R-matrix on Poisson algebras of formal Laurent series. Results are illustrated with the known and new (1+1)-dimensional dispersionless systems.

ClassicalR-matrix theory of dispersionless systems: II. (2 1) dimension theory

A systematic way of construction of (2+1)-dimensional dispersionless integrable Hamiltonian systems is presented. The method is based on the so-called central extension procedure and classical R-matrix applied to the Poisson algebras of formal Laurent series. Results are illustrated with the known and new (2+1)-dimensional dispersionless systems.

Relationships among three kinds of finite dimensional integrable Hamiltonian systems

A new type of finite dimensional integrable Hamiltonian systems containing many restricted soliton flows are considered. Their relations with the generalized classical Gaudin models and the generalized classical Gaudin models with boundary are established. As an application, the relationships among three different restricted mKdV flows are exhibited.

Lie algebraic structures of some (1+2)-dimensional Lax integrable systems

The paper proposes an approach to constructing the symmetries and their algebraic structures for isospectral and nonisospectral evolution equations of (1+2)-dimensional systems associated with the linear problem of Sato theory. To do that, we introduce the implicit representations of the isospectral flows { K m } and nonisospectral flows { σ n } in the high dimensional cases. Three examples, the Kodomstev–Petviashvili system, BKP system and new CKP system, are considered to demonstrate our method.

Liouville integrability of the finite dimensional Hamiltonian systems derived from principal chiral field

For finite dimensional Hamiltonian systems derived from 1+1 dimensional integrable systems, if they have Lax representations, then the Lax operator creates a set of conserved integrals. When these conserved integrals are in involution, it is believed quite popularly that there will be enough functionally independent ones among them to guarantee the Liouville integrability of the Hamiltonian systems, at least for those derived from physical problems. In this article, we give a counterexample based on the U(2) principal chiral field. It is proved that the finite dimensional Hamiltonian systems derived from the U(2) principal chiral field are Liouville integrable. Moreover, their Lax operator gives a set of involutive conserved integrals, but they are not enough to guarantee the integrability of the Hamiltonian systems.

The Exact Solutions of Some (2+1)-Dimensional Integrable Systems**The project supported by the Ministry of Education of China, and National Outstanding Youth Foundation of China

The equivalence of three (2+1)-dimensional soliton equations is proved, and the quite general solutions with some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many special types of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutions are obtained.

An example of (3 + 1)-dimensional integrable system

In this paper an example of the (3+1)-dimensional integrable system is considered and the infinite series of divergent forms are described. The classical symmetries for this system, the factor-equation for these symmetries, and exact solutions of this system are found.

Homoclinic orbits and chaos in three– and four–dimensional flows

We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations. We first consider perturbations of a three–dimensional integrable system possessing a line of degenerate saddle points connected by a two–dimensional manifold of homoclinic loops. We show that this manifold splits to create transverse homoclinic orbits, and then appeal to geometrical and symbolic dynamic arguments to show that homoclinic bifurcations occur in which ‘simple’ connecting orbits are replaced by a countable infinity of such orbits. We discover a rich variety of connections among equilibria and periodic orbits, as well as more exotic sets, including Smale horseshoes. The second problem is a four–dimensional Hamiltonian system. Using symmetries and classical estimates, we again find countable sets of connecting orbits. There is no small parameter in this case, and the methods are non–perturbative.

Super Brockett Equations:¶A Graded Gradient Integrable System

Rather recently equations of Lax type defined by a double commutator, the so-called Brockett equations, have received considerable attention. In this paper we prove that a supersymmetric version of a Brockett hierarchy is an infinite dimensional integrable gradient system. As far as we know, this is the only graded system of this type existing in the literature.