Oktay K. Pashaev

Resonance NLS solitons as black holes in Madelung fluid

Envelope solitons of the Nonlinear Schrodinger equation (NLS) under quantum potentials influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s 1, to the reaction–diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1+1)-dimensional Jackiw–Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space–time metric. The space–time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.

On the integrability and isotopic structure of the one-dimensional Hubbard model in the long wave approximation

Abstract The Lax representation for the continuum limit of the Hubbard model is found. The lagrangian of the system is shown to be invariant under four parametric U(1, 1) internal symmetry group transformations. The group properties and the set of solutions generated by this group are considered. A possible generalization of the model considered is presented for the case of the U(p, q) group.

Integrable dissipative structures in the gauge theory of gravity

The Jackiw - Teitelboim gauge formulation of (1 + 1)-dimensional gravity allows us to relate different gauge-fixing conditions to integrable hierarchies of evolution equations. We show that the equations for the zweibein fields can be written as a pair of time-reversed evolution equations of the reaction - diffusion type, admitting dissipative solutions. The spectral parameter for the related Lax pair appears as the constant-valued spin connection associated with the SO(1,1) gauge symmetry. Spontaneous breaking of the non-compact symmetry and irreversible evolution are discussed.

Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons

Abstract This paper is devoted to the semiclassical limit of the one-dimensional Schrodinger equation with current nonlinearity and Sobolev regularity, before shocks appear in the limit system. In this limit, the modified Euler equations are recovered. The strictly hyperbolicity and genuine nonlinearity are proved for the limit system wherever the Riemann invariants remain distinct. The dispersionless equation and its deformation which is the quantum potential perturbation of JNLS equation are also derived.

Dynamics of multidimensional solitons

Abstract The N 2 -soliton solution of the Davey-Stewartson equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The solution can simulate quantum effects as ineleastic scattering, fusion and fission, creation and annihilation.

DEGENERATE FOUR-VIRTUAL-SOLITON RESONANCE FOR THE KP-II

We propose a method for solving the (2+1)-dimensional Kadomtsev-Petviashvili equation with negative dispersion (KP-II) using the second and third members of the disipative version of the AKNS hierarchy. We show that dissipative solitons (dissipatons) of those members yield the planar solitons of the KP-II. From the Hirota bilinear form of the SL(2, ℝ) AKNS flows, we formulate a new bilinear representation for the KP-II, by which we construct one- and two-soliton solutions and study the resonance character of their mutual interactions. Using our bilinear form, for the first time, we create a four-virtual-soliton resonance solution of the KP-II, and we show that it can be obtained as a reduction of a four-soliton solution in the Hirota-Satsuma bilinear form for the KP-II.

BLACK HOLES AND SOLITONS OF THE QUANTIZED DISPERSIONLESS NLS AND DNLS EQUATIONS

The classical dynamics of non-relativistic particles are described by the Schrodinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrodinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.

Continual classical Heisenberg models defined on graded su(2,1) and su(3) algebras

Continual integrable Heisenberg models are constructed on real subalgebras of the superalgebra spl(2/1). Two Heisenberg models are shown to exist on the compact subalgebra uspl(2/1)≊su(2/1). One of these, SU(2/1)/S(U(2)×U(1)), is gauge equivalent to SU(2) nonlinear vector Schrodinger equation (NLSE) expressed in odd Grassman variables, the other, SU(2/1)/S(L(1/1)×U(1)), to ‘‘super’’ NLSE which is invariant under global supersymmetry transformations of SL(1/1). Also constructed are a Heisenberg model on the noncompact subalgebra ospu(1,1/1), with higher nonlinearities, and its gauge equivalent analog. Hamiltonian structure and classical solutions are studied and the possible connection of the given models with a version of the Hubbard one is discussed.

Vortex images and q-elementary functions

In the present paper, the problem of vortex images in the annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, and are located at sites of the q-lattice, where a dimensionless parameter q = r22/r21 is given by the square ratio of the cylinder radii. The resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. Our approach in this paper provides an efficient path to rediscover known solutions for the vortex–cylinder pair problem and yields new solutions as well. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we show that our solution coincides with the known one, obtained before by elliptic functions. The Schottky–Klein prime function for the annular domain is factorized explicitly in terms of q-exponents. The Hamiltonian, the Kirchhoff–Routh and the Green functions are constructed. As a new application of our approach, the uniformly rotating exact N-vortex polygon solutions with the rotation frequency expressed in terms of q-logarithms at Nth roots of unity are found. In particular, we show that a single vortex orbits the cylinders with constant angular velocity, given as the q-harmonic series. Vortex images in two particular geometries with only one cylinder as the q → ∞ limit are studied in detail.

On the gauge equivalence of the Landau-Lifshitz and the nonlinear Schrödinger equations on symmetric spaces

Abstract The gauge equivalence between a generalized Heisenberg spin chain (G/H) in the classical and continuum limit and the nonlinear Schrodinger equation (NLSE), with special attention to noncompact groups, is established. It has been demonstrated that noncompact groups allow a richer spectrum of possible reductions of the Heisenberg system to the NLSE. Some specialities of the model with nontrivial boundary conditions are discussed. The gauge equivalence between single-axis anisotropic Landau-Lifshitz equations (LLE) and isotropic LLE is briefly discussed.