The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painleve test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Towards Noncommutative Integrable Systems

We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1) and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed.Comment: 13 pages, 1 figure, LaTeX; v2: typos corrected, comments and references added; v3: comments and references added, version to appear in Phys. lett.

Static Hopfions in the extended Skyrme-Faddeev model

We construct static soliton solutions with non-zero Hopf topological charges to a theory which is an extension of the Skyrme-Faddeev model by the addition of a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled non-linear partial differential equations in two variables by a successive over-relaxation (SOR) method. We construct numerical solutions with Hopf charge up to four, and calculate their analytical behavior in some limiting cases. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms. Their energies and sizes tend to zero as that combination approaches a particular special value. We calculate the equivalent of the Vakulenko and Kapitanskii energy bound for the theory and find that it vanishes at that same special value of the coupling constants. In addition, the model presents an integrable sector with an infinite number of local conserved currents which apparently are not related to symmetries of the action. In the intersection of those two special sectors the theory possesses exact vortex solutions (static and time dependent) which were constructed in a previous paper by one of the authors. It is believed that such model describes some aspects of the low energy limit of the pure SU(2) Yang-Mills theory, and our results may be important in identifying important structures in that strong coupling regime.Comment: 22 pages, 42 figures, minor correction

Vortices in the extended Skyrme-Faddeev model

We construct analytical and numerical vortex solutions for an extended Skyrme-Faddeev model in a (3 + 1) dimensional Minkowski space-time. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction makes use of an ansatz, invariant under the joint action of the internal SO(2) and three commuting U(1) subgroups of the Poincare group, and which reduces the equations of motion to an ordinary differential equation for a profile function depending on the distance to the x(3) axis. The vortices have finite energy per unit length, and have waves propagating along them with the speed of light. The analytical vortices are obtained for a special choice of potentials, and the numerical ones are constructed using the successive over relaxation method for more general potentials. The spectrum of solutions is analyzed in detail, especially its dependence upon special combinations of coupling constants.FAPESP (Brazil)FAPESP (Brazil)UK Engineering and Physical Sciences Research CouncilUK Engineering and Physical Sciences Research CouncilCNPq (Brazil)CNPq-Brazi

Numerical Nahm transform for 2-caloron solutions

A new numerical method for performing the Nahm transform for charge $k=2$ caloron is presented. The Weyl equations with boundary impurities are solved directly and the determination of the appropriate basis to the linear system is established. The action densities of the 2-calorons with 10 moduli parameters are shown.Comment: 8 pages, 4 figures, 1 table, v2: coincides with Phys.Lett.B versio

N-Soliton Solutions to a New (2 + 1) Dimensional Integrable Equation

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $\phi_{xt} + \phi_{xxxz}/4 + \phi_x \phi_{xz} + \phi_{xx} \phi_z/2 + \partial_x^{-1} \phi_{zzz}/4 = 0$. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations.Comment: 7 pages, uses ioplppt.st

Noncommutative Burgers Equation

We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach.Comment: 24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato theory and the integrability added, references added, version to appear in J. Phys.