Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians

Abstract

We construct one-soliton solutions for the nonlinear Schrödinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the complete (super) integrability of generalized harmonic oscillators. The soliton-wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schrödinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlevé transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons and Jacobi elliptic and second Painlevé transcendental solutions for several variable Hamiltonians that are important for research in nonlinear optics, plasma physics, and Bose-Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.

On a hidden symmetry of quantum harmonic oscillators

“…The same is true for the probability distribution of the particle linear momentum due to the Heisenberg Uncertainty Principle [27]. These effects, quite possibly, can be observed experimentally, say in Bose condensates, if the nonlinearity of the Gross-Pitaevskii equation is turned off by the Feshbach resonance [11], [19], [32], [52], [56], [60]. A more elementary example is an electron moving in a uniform magnetic field.…”

On Linearized Korteweg-de Vries Equations

“…As mentioned before we are working with an ansatz substitution which is a modified version of the one used in (CorderoSoto, 2008;Suazo, 2010;Suazo, 2011). The first thing to notice about their work is the splitting form of the solution ψ(x, y, t) into A(t) and e S (x,y,t) , where S (x, y, t) can be seen as a polynomial function on x and y with time-dependent coefficients.…”

“…This substitution lead to an ordinary differential equations system that has been solved in (Cordero-Soto, 2008); they have used this ansatz in (Suazo, 2010;Suazo, 2011) where they work a quadratic and nonlinear quadratic Hamiltonian.…”

“…We observed that this technique will not be successful in general nonlinear differential equations, but in (E. Suazo, et al, 2010) the authors applied it to a nonlinear case. The nonlinear term has the form |ψ(x, y, t)| 2 ψ(x, y, t).…”

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

“…For the latter case, in [36], [37] and [38], multiparameter solutions in the spirit of Marhic in [30] have been presented. The parameters for the Riccati system arose originally in the process of proving convergence to the initial data for the Cauchy initial value problem Equation (1) with h(t) = 0 and in the process of finding a general solution of a Riccati system [38] and [39].…”