Yajima-Oikawa System Research Articles

Integrable systems related to Su–Schrieffer–Heeger lattices

We present a novel approach to nonlinear lattices with fermions based on the exact solutions for the equations of motion, where fermions and bosons are treated as Grassmannian anticommuting and commuting classical fields. It is shown that the Su–Schrieffer–Heeger equations of motion can be solved exactly in continuum limit using Hirota bilinear formalism and bright and dark solitons appear for phononic and fermionic fields. The bosonic soliton has a tail due to the presence of fermions and the fermionic density has a strongly localised shape. Moreover, applying the multiple scales method, the emergent Grassmannian amplitude nonlinear equation is the complex modified Korteweg de Vries and at the resonance the fermionic version of Yajima–Oikawa system. As an interesting offshoot this system turns out to be a new completely integrable one having N-dark soliton solution.

Bright-dark soliton complexes in spinor Bose-Einstein condensates

We present bright-dark vector solitons in quasi-one-dimensional spinor (F=1) Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to a completely integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the $m_{F}=+1,-1,0$ spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasi-elastic collisions. It is also shown that, in the presence of a parabolic trap of strength $\omega $, the bright component(s) is (are) guided by the dark one(s), and, as a result, the small-amplitude vector soliton as a whole performs harmonic oscillations of frequency $\omega/ \sqrt{2}$ in the shallow soliton limit. We investigate numerically deviations from this prediction, as the depth of the solitons is increased, as well as when the strength of the spin-dependent interaction is modified.

On an integrable two-dimensional special lattice with self-consistent sources

We propose a two-dimensional special lattice with self-consistent sources which is a coupled higher-dimensional Yajima-Oikawa system. The N -soliton solutions expressed in terms of the Casorati determinants are given through Wronskian technique. The Lax presentation for the special two-dimensional lattice with self-consistent sources is derived from bilinear Biicklund transformation.

Asymptotic reductions of two coupled (2 + 1)-dimensional nonlinear Schrödinger equations: application to Bose–Einstein condensates

We consider a system of two coupled (2 + 1)-dimensional nonlinear Schrodinger equations, describing two-component disc-shaped Bose–Einstein condensates. We present three different asymptotic reductions of this system. In particular, we derive the Mel'nikov system, the Yajima–Oikawa system as well as the Davey–Stewartson system (the latter is found as a special case of the Djordjevic–Redekopp system). Conditions for integrability of the reduced systems, their soliton solutions and the asymptotic relevance of such solutions to the original system are also discussed. Numerical results pertaining to the reduction to the Davey–Stewartson system are found to be in good agreement with the analytical predictions.

Multilinear forms associated with the Yajima-Oikawa system

It is well known that the Kadomtsev-Petviashvili hierarchy (KP hierarchy) is compatible with generalized k-constraints of the type ( L k ) − = q∂ −1 r. Those constraints are leading to integrable systems in 1 + 1 dimensions such as the AKNS system ( k = 1) or the Yajima-Oikawa system ( k = 2). The effect of the 1-constraint on the τ-function of the KP hierarchy has been studied: it adds trilinear equations to the set of bilinear ones. In this paper we shall derive a quattro and a quintolinear equation for the τ-function of the KP hierarchy under the 2-constraint.

Darboux transforms, deep reductions and solitons

Explicit formalisms for deep reductions of matrix differential equations and Darboux covariance properties are presented. The matrix 2*2 spectral problem of the second order is considered. This problem with the reduction constraints being imposed on the potentials is the first equation of the Lax representation of the Hirota-Satsuma system (1981). The two reduction cases are treated with the help of the bilinear delta -forms. The covariance of these forms with respect to the Darboux transforms under restrictions gives rise to explicit formulas of N-soliton solutions. In particular the two-parameter soliton solutions of the Hirota-Satsuma system are obtained. The specific feature of such solutions evolution is that the singularity appears in some parameter region. The Yajima-Oikawa system (1976) is given as an example of the technique application to a 3*3 spectral problem.

Particle-Like Excitations in Many Component Magnon-Phonon Systems

It is shown that many sublattice isotropic XY chain with magnon-phonon interactions at the long-wave approximations may be described by generalized Zakharov's system with U(p, q) isogroup. Two-sublattice XY chain through the Jordan-Wigner transformations reduced to Su-Schriffer-Heeger coupled electron-phonon model in the quasi-one-dimensional conductor polyacetylene (CH)x theory. The Hamiltonian structure of U(p, q) Zakharov's system and its "ultrarelativistic" limit (i.e., the Yajima-Oikawa system with U(p, q) isogroup) are studied. The linear problem and the generating functionals for infinite series of additional integrals of motion for U(p, q) Yajima-Oikawa system are constructed. Four types of soliton solutions under different boundary conditions and appropriate integrals of particles number and energy are found. The quasistationary and ultra-relativistic limits are discussed.