Dimensional Integrable Systems Research Articles

Some new fractals behaviors in (2 + 1)-dimensional integrable system

By introducing the extended homogeneous balance approach into the (2 + 1)-dimensional integrable system, a linearized form of this physical model is established in this paper. Subsequently, after applied the Bäcklund transformation in the system, a variable separation solution with the entrance of different arbitrary functions is obtained. Furthermore, by using the Weierstrass, Bessel and Jacobian elliptic functions, some interesting fractal structures are produced.

On the Space of KdV Fields

The space of functions A over the phase space of KdV-hierarchy is studied as a module over the ring D generated by commuting derivations. A D-free resolution of A is constructed by Babelon, Bernard and Smirnov by taking the classical limit of the construction in quantum integrable models assuming a certain conjecture. We propose another D-free resolution of A by extending the construction in the classical finite dimensional integrable system associated with a certain family of hyperelliptic curves to infinite dimension assuming a similar conjecture. The relation of two constructions is given.

Darboux transformation and explicit solutions to a (2+1)-dimensional integrable system

Darboux transformation and explicit solutions to a (2+1)-dimensional integrable system

Higher dimensional bright solitons and their collisions in a multicomponent long wave–short wave system

Bright plane soliton solutions of an integrable (2+1)-dimensional (n + 1)-wave system are obtained by applying Hirota's bilinearization method. First, the soliton solutions of a three-wave system consisting of two short-wave components and one long-wave component are found and then the results are generalized to the corresponding integrable (n + 1)-wave system with n short waves and a single long wave. It is shown that the solitons in the short-wave components (say S(1) and S(2)) can be amplified by merely reducing the pulse width of the long-wave component (say L). Study of the collision dynamics reveals some interesting behaviour: the solitons which split up in the short-wave components undergo shape changing collisions with intensity redistribution and amplitude-dependent phase shifts. Even though a similar type of collision is possible in (1+1)-dimensional multicomponent integrable systems, to our knowledge we report this kind of collision in (2+1) dimensions for the first time. However, solitons which appear in the long-wave component exhibit only elastic collision though they undergo amplitude-dependent phase shifts.

Infinite-dimensional Lie algebras in 4D conformal quantum field theory**Lecture at the workshops ‘Lie Theory and Its Applications in Physics’, 18–24 June 2007, Varna, Bulgaria; ‘Infinite- Dimensional Algebras and Quantum Integrable Systems’, 23–27 July, 2007, Faro, Portugal; and ‘Supersymmetries and Quantum Symmetries’, 30 July–4 August, 2007, Dubna, Russia.

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra closed under transposition. The associative algebra is irreducible iff its commutant coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of corresponding to the field of reals, of u(∞, ∞) associated with the field of complex numbers, and of so*(4∞) related to the algebra of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and , respectively.

On (2+1)-Dimensional Non-isospectral Toda Lattice Hierarchy and Integrable Coupling System

By considering (2+1)-dimensional non-isospectral discrete zero curvature equation, the (2+1)-dimensional non-isospectral Toda lattice hierarchy is constructed in this article. It follows that some reductions of the (2+1)-dimensional Toda lattice hierarchy are given. Finally, the (2+1)-dimensional integrable coupling system of the Toda lattice hierarchy is obtained through enlarging spectral problem.

Gauge Transformation and Reciprocal Link for (2+1)-Dimensional Integrable Field Systems

Appropriate restrictions of Lax operators which allows to construction of (2+1)-dimensional integrable field systems, coming from centrally extended algebra of pseudo-differential operators, are reviewed. The gauge transformation and the reciprocal link between three classes of Lax hierarchies are established.

Multi-linear Variable Separation Approach to Solve a(1+1)-Dimensional Coupled Integrable Dispersionless System**The projectsupported by the Science Foundation of Zhejiang University of Technology

The multi-linear variable separation approach (MLVSA) is veryuseful to solve (2+1)-dimensional integrable systems. In thisletter, we extend this method to solve a (1+1)-dimensional coupledintegrable dispersion-less system. Namely, by using a Bäcklundtransformation and the MLVSA, we find a new general solution anddefine a new ``universal formula". Then, some new (1+1)-dimensionalcoherent structures of this universal formula can be found byselecting corresponding functions appropriately. Specially, insome conditions, bell soliton and kink soliton can transform eachother, which are illustrated graphically.

ON A GENERALIZED TZITZEICA EQUATION

Extending the so called coupled KP hierarchy to negative time flows one obtains coupled Toda-type equations defined on a two-dimensional lattice. These equations allow for reductions to 1+1 dimensional integrable systems that are defined on a finite part of this lattice. A system of coupled Hirota bilinear equations, obtained from such a reduction and defined on only 5 points of the lattice, will be shown to correspond to a coupling of a Tzitzeica equation to two linear equations. The Lax representation of this system is also presented.

A higher dimensional loop algebra and integrable couplings system of evolution equations hierarchy

An extension of the lie algebra A n−1 has been proposed [Phys. Lett. A, 2003, 310:19–24]. In this paper, the new lie algebra was used to construct a new higher dimensional loop algebra \(\tilde G\). Based on the loop algebra \(\tilde G\) , the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.

Notes on Multi-linear Variable Separation Approach

The multi-linear variable separation approach method is very useful to solve (2+1)-dimensional integrable systems. In this letter,we extend this method to solve (1+1)-dimensional Boiti system, (2+1)-dimensional Burgerssystem, (2+1)-dimensional breaking soliton system, and (2+1)-dimensional Maccarisystem. Some new exact solutions are obtained and the universal formula obtained frommany (2+1)-dimensional systems is extended or modified.

STABILIZER ORBIT OF VIRASORO ACTION AND INTEGRABLE SYSTEMS

We study the stabilizer orbit of the coadjoint action of the Virasoro algebra on its dual. The vector field associated to the stabilizer orbit is called projective vector field. It was shown by Kirillov that the pair formed by the projective vector field and its square root comprises superalgebra. The Euler–Poincaré flow on the "Fermionic" part of the superalgebra yields a pseudospherical type equation. In this paper we express the solutions of the Ermakov–Pinney, Normalized Ermakov–Pinney, Painlevé II and C. Neumann system type (0+1) dimensional integrable systems in terms of the global and local projective vector field, a vector field leaves fixed a given projective connection.

Variable Separation Approach to Solve Nonlinear Systems

The variable separation approach method is very useful to solving (2+1)-dimensional integrable systems. But the (1+1)-dimensional and (3+1)-dimensional nonlinear systems are considered very little. In this letter, we extend this method to (1+1) dimensions by taking the Redekopp system as a simple example and (3+1)-dimensional Burgers system. The exact solutions are much general because they include some arbitrary functions and the form of the (3+1)-dimensional universal formula obtained from many (2+1)-dimensional systems is extended.

INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES IN MINKOWSKI 3-SPACE

In this paper, infinitesimal deformations of time-like surfaces are investigated in Minkowski 3-space R2,1. It is shown that some given deformations of the time-like surface can be described by 2+1 dimensional integrable systems. Moreover spectral parameters are introduced, and it is proved that deformation families are soliton surfaces’ families.

Lie-algebraic structure of (2 + 1)-dimensional lax-type integrable nonlinear dynamical systems

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Backlund transformation. The connection of this hierarchy with Lax-integrable two-metrizable systems is studied.

Lie-algebraic structure of (2 + 1)-dimensional Lax-type integrable nonlinear dynamical systems

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Båcklund transformation. The connection of this hierarchy with Lax-integrable two-metrizable systems is studied.

Efficient control of the energy exchange due to the Manakov vector-soliton collision.

By examining the concept of energy exchange among the orthogonally polarized components of each of two colliding (Manakov-like) vector solitons it is observed that a maximum or an efficient energy-exchange process is possible only for an appropriate choice of the initial physical parameters (namely, frequency separation, polarizations, time delay, and pulse-width separation between the colliding solitons) for which L(W) (walk-off length) >>L(NL) (nonlinear length). However, in this case only, the amount of energy-exchange can be considerably increased or decreased by appropriately changing the phases of colliding solitons without altering the walk-off length and the initial energy distributions between the soliton components. Moreover we observe that during the collision between two closely placed vector solitons of the practically interesting integrable Manakov model, nonuniform pulse broadening takes place in each of their components. Such an effect has not yet been reported in any (1+1) dimensional integrable soliton systems so far. In addition, the relation between walk-off length, polarization, and pulse width is briefly discussed.

Variable Separation Approach to Solve a (2+1)-Dimensional Integrable System**The project supported by the Natural Science Foundation of Zhejiang Province of China under Grant No. 102037

In this letter, starting from a Bäcklund transformation, a general solution of a (2+1)-dimensional integrable system is obtained by using the new variable separation approach.

New exact solutions of a new (2+1)-dimensional integrable system

In this Letter, a new (2+1)-dimensional integrable system is solved by using the modified Jacobi elliptic function expansion method. As a consequence, abundant families of Jacobi elliptic function solutions are obtained. When the modulus m→1 and m→0, those periodic solutions degenerate as the corresponding solitary wave solutions, shock wave solutions or trigonometric function solutions.

Variable separation solutions obtained from Darboux Transformations for the asymmetric Nizhnik–Novikov–Veselov system

The use of a seed solution with some arbitrary functions for the asymmetric Nizhnik–Novikov–Veselov system in the first step Darboux transformation yields the variable separable solutions with two space-variable separated functions. The more variable separated functions which are not arbitrary can be introduced by using the Darboux transformation repeatedly. The Nth step Darboux transformation (for arbitrary N) with arbitrary number of space–variable separated functions is explicitly written down by means of the Pfaffian. The ”universal” variable separation formula which is valid for a diversity of (2 + 1)-dimensional integrable systems can be obtained from a particular reduction of the solutions constructed from the second step Darboux transformation. A new saddle-type ring soliton solution with completely elastic interaction and nonzero phase shifts is also studied in this paper.