Dimensional Soliton System Research Articles

Multi-soliton solutions and fractal structures in a (2+1)-dimensional soliton system

By the projective equation approach and the variable separation method, a series of excitations of the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system is derived. Based on the derived solitary wave excitation, we obtain multi solitons and fractal solitons of the system.

Classical R-matrix theory for bi-Hamiltonian field systems

This is a survey of the application of the classical R-matrix formalism to the construction of infinite-dimensional integrable Hamiltonian field systems. The main point is the study of bi-Hamiltonian structures. Appropriate constructions on Poisson, noncommutative and loop algebras as well as the central extension procedure are presented. The theory is developed for (1 + 1)- and (2 + 1)-dimensional field and lattice soliton systems as well as hydrodynamic systems. The formalism presented contains sufficiently many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite-dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.

New Exact Solutions and Localized Excitations in a (2+1)-Dimensional Soliton System

Starting from a special conditional similarity reduction method, we obtain the reduction equation of the (2+1)-dimensional dispersive long-water wave system. Based on the reduction equation, some new exact solutions and abundant localized excitations are obtained.

Folded Localized Excitations and Chaotic Patterns in a (2+1)-Dimensional Soliton System

Starting from an improved mapping approach and a linear variable separation approach, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for the (2+1)-dimensional breaking soliton system are derived. Based on the derived solitary wave solution, we obtain some special folded localized excitations and chaotic patterns.

Explicit Solutions for a New (2+1)-Dimensional Coupled mKdV Equation

An integrable (2+1)-dimensional coupled mKdV equation is decomposed into two (1+1)-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two (1+1)-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.

Standing solutions of one-dimensional Boiti-Leon-Pempinelli-Spire system localized in space and periodical in time

By means of a special variable separation approach, a common formula with two arbitrary functions has been obtained for suitable physical quantity of (1+1)-dimensional model such as Boiti-Leon-Pempinelli-Spire system. Based on the derived formula, some significant types of solitons such as compacton, peakon, and loop solutions localized in space and periodical in time are simultaneously constructed from the (1+1)-dimensional soliton system by entrancing appropriate piecewise smooth functions and multivalued functions.

The study of soliton fission and fusion in (2+1)-dimensional nonlinear system

By means of a variable separation approach and an extended homogeneous balance method, a general variable separation excitation of a (2+1)-dimensional nonlinear system is derived. Based on the derived solution with arbitrary functions, we reveal soliton fission and fusion phenomena in the (2+1)-dimensional soliton system.

Exact Solutions and Solitons with Fission and Fusion Properties for the Generalized Broer-Kaup System

Starting from a Painlev´e-B¨acklund transformation and a linear variable separation approach, we obtain a quite general variable separation excitation to the generalized (2+1)-dimensional Broer-Kaup (GBK) system. Then based on the derived solution, we reveal soliton fission and fusion phenomena in the (2+1)-dimensional soliton system. - PACS numbers: 05.45.Yv, 03.65.Ge

Solitons with Periodic Behavior in Korteweg-de Vries Type ModelsRelated to Schrödinger System

The linear variable separation approach is successfully extended to (1+1)-dimensional Korteweg-de Vries (KdV) type models related to Schrödinger system. Some significant types of solitons such as compacton, peakon, and loopsolutions with periodic behavior are simultaneously derived from the(1+1)-dimensional soliton system by entrancing appropriate piecewise smooth functionsand multivalued functions.

Peakon, compacton and loop excitations with periodic behavior in KdV type models related to Schrödinger system

In this Letter, the linear variable separation approach is successfully extended to (1+1)-dimensional Korteweg–de Vries (KdV) type models related to Schrödinger system. Some significant types of solitons such as compacton, peakon and loop solutions with periodic behavior are simultaneously derived from the (1+1)-dimensional soliton system by entrancing appropriate piecewise smooth functions and multivalued functions.

Solitons with fission and fusion behaviors in a variable coefficient Broer–Kaup system

Using an extended homogeneous balance approach and a linear variable separation method, a general variable separation excitation of the (2 + 1)-dimensional variable coefficient Broer–Kaup (VCBK) system is derived. Based on the derived solution, we reveal soliton fission and fusion phenomena in the (2 + 1)-dimensional soliton system.

Multi-valued solitary waves in multidimensional soliton systems

Considering that folded phenomena are rather universal in nature and some arbitrary functions can be included in the exact excitations of many (2+1)-dimensional soliton systems, we use adequate multivalued functions to construct folded solitary structures in multi-dimensions. Based on some interesting variable separation results in the literature, a common formula with arbitrary functions has been derived for suitable physical quantities of some significant (2+1)-dimensional soliton systems like the generalized Ablowitz?Kaup?Newell?Segur (GAKNS) model, the generalized Nizhnik?Novikov?Veselov (GNNV) system and the new (2+1)-dimensional long dispersive wave (NLDW) system. Then a new special type of two-dimensional solitary wave structure, i.e. the folded solitary wave and foldon, is obtained. The novel structure exhibits interesting features not found in the single valued solitary excitations.

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.

Chaos and Fractals in a (2+1)-Dimensional Soliton System

Considering that there are abundant coherent soliton excitations in highdimensions, we reveal a novel phenomenon that the localized excitationspossess chaotic and fractal behaviour in some (2+1)-dimensional solitonsystems. To clarify the interesting phenomenon, we take the generalized(2+1)-dimensional Nizhnik-Novikov-Vesselov system as a concrete example.A quite general variable separation solutions of this system is derivedvia a variable separation approach first, then some new excitations likechaos and fractals are derived by introducing some types of lower-dimensional chaotic and fractal patterns.

Localized excitations in (2+1)-dimensional systems.

By means of a special variable separation approach, a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various (2+1)-dimensional models such as the Davey-Stewartson (DS) model, the Nizhnik-Novikov-Veselov (NNV) system, asymmetric NNV equation, asymmetric DS equation, dispersive long wave equation, Broer-Kaup-Kupershmidt system, long wave-short wave interaction model, Maccari system, and a general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system. Selecting the arbitrary functions appropriately, one may obtain abundant stable localized interesting excitations such as the multidromions, lumps, ring soliton solutions, breathers, instantons, etc. It is shown that some types of lower dimensional chaotic patterns such as the chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns, chaotic dromion patterns, fractal lump patterns, and fractal dromion patterns may be found in higher dimensional soliton systems. The interactions between the traveling ring type soliton solutions are completely elastic. The traveling ring solitons pass through each other and preserve their shapes, velocities, and phases. Some types of localized weak solutions, peakons, are also discussed. Especially, the interactions between two peakons are not completely elastic. After the interactions, the traveling peakons also pass through each other and preserve their velocities and phases, however, they completely exchange their shapes.

Explicit solutions for some (2+1)-dimensional nonlinear evolution equations

Some (2+1)-dimensional nonlinear evolution equations, including the Kadomtsev-Petviashvili II (KPII) equation and the modified Kadomtsev-Petviashvili (KP) equation, are decomposed into two (1+1)-dimensional soliton systems in the coupled KdV hierarchy. With the help of the decomposition and the Darboux transformation, some explicit solutions of these (2+1)-dimensional nonlinear evolution equations such as new soliton solutions of the KPII equation and the modified KP equation are obtained.

Decomposition of the (2 + 1)-dimensional Gardner equation and its quasi-periodic solutions

To decompose the (2 + 1)-dimensional Gardner equation, an isospectralproblem and a corresponding hierarchy of (1 + 1)-dimensional solitonequations are proposed. The (2 + 1)-dimensional Gardner equation isseparated into the first two non-trivial (1 + 1)-dimensional solitonsystems in the hierarchy, and in turn into two new compatible Hamiltoniansystems of ordinary differential equations. Using the generating functionflow method, the involutivity and the functional independence of theintegrals are proved. The Abel-Jacobi coordinates are introduced tostraighten out the associated flows. The Riemann-Jacobi inversionproblem is discussed, from which quasi-periodic solutions of the (2 + 1)-dimensional Gardner equation are obtained by resorting to theRiemann theta functions.

Abundant New Explicit Exact Soliton-Like Solutions and Painlevé Test for the Generalized Burgers Equation in (2+1)-Dimensional Space**The project supported by the National Key Basic Research Development Program Foundation of China under Grant No. G1998030600, the Higher Education Doctoral Foundation of China under Grant No. 98014119 and National Natural Science Foundation of China under Grant No. 10072013

We obtain Bäcklund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2+1)-dimensional space, , by using the extended homogeneous balance method. As is well known, the introduction of the concept of dromions (the exponentially localized solutions in (2+1)-dimensional space) has triggered renewed interest in (2+1)-dimensional soliton systems. The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself. In addition, it is shown that the equation passes Painlevé test.

The potential constraints of the complex (1+2)-dimensional soliton systems and related Hamiltonian equations

Potential complete constraints (symmetric and nonsymmetric) of the complex KP and MKP systems are derived by using the general theory for a nonlinear equation to be a Hamiltonian system. All the complex Hamiltonian equations corresponding to this kind of constraint are obtained, which contain the famous nonlinear Schrödinger equations and several new Hamiltonian equations. In addition, the classical Kaup–Newell system and the Heisenberg system are also led by a new reduction approach from the MKP system.

Inelastic collision and switching of coupled bright solitons in optical fibers

By constructing the general six-parameter bright two-soliton solution of the integrable coupled nonlinear Schr\"odinger equation (Manakov model) using the Hirota method, we find that the solitons exhibit certain novel inelastic collision properties, which have not been observed in any other $(1+1)$-dimensional soliton system so far. In particular, we identify the exciting possibility of switching solitons between modes by changing the phase. However, the standard elastic collision property of solitons is regained with specific choices of parameters.