Controlled Generation of Dark Solitons with Phase Imprinting

The generation of dark solitons in Bose-Einstein condensates with phase imprinting is studied by mapping it into the classic problem of a damped driven pendulum. We provide simple but powerful schemes of designing the phase imprint for various desired outcomes. We derive a formula for the number of dark solitons generated by a given phase step, and also obtain results which explain experimental observations.Comment: 4pages, 4 figure

Differential constraints for the Kaup -- Broer system as a reduction of the 1D Toda lattice

It is shown that some special reduction of infinite 1D Toda lattice gives differential constraints compatible with the Kaup -- Broer system. A family of the travelling wave solutions of the Kaup -- Broer system and its higher version is constructed.Comment: LaTeX, uses IOP styl

One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schr\"odinger Equation and Solution of Bogoliubov Equation in These Systems

In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schr\"odinger equation, and solve the Bogoliubov equation of these systems. A map of spin-$n$ BEC is constructed from the $2^n$-dimensional spinor representation of irreducible tensor operators of $so(2n+1)$. Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.Comment: 2.1 pages, JPSJ shortnote style. Published version. Note and reference adde

Gurevich-Zybin system

We present three different linearizable extensions of the Gurevich-Zybin system. Their general solutions are found by reciprocal transformations. In this paper we rewrite the Gurevich-Zybin system as a Monge-Ampere equation. By application of reciprocal transformation this equation is linearized. Infinitely many local Hamiltonian structures, local Lagrangian representations, local conservation laws and local commuting flows are found. Moreover, all commuting flows can be written as Monge-Ampere equations similar to the Gurevich-Zybin system. The Gurevich-Zybin system describes the formation of a large scale structures in the Universe. The second harmonic wave generation is known in nonlinear optics. In this paper we prove that the Gurevich-Zybin system is equivalent to a degenerate case of the second harmonic generation. Thus, the Gurevich-Zybin system is recognized as a degenerate first negative flow of two-component Harry Dym hierarchy up to two Miura type transformations. A reciprocal transformation between the Gurevich-Zybin system and degenerate case of the second harmonic generation system is found. A new solution for the second harmonic generation is presented in implicit form.Comment: Corrected typos and misprint

Completely integrable models of non-linear optics

The models of the non-linear optics in which solitons were appeared are considered. These models are of paramount importance in studies of non-linear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency, and parametric interaction of three waves. At the present time there are a number of the theories based on completely integrable systems of equations, which are both generations of the original known models and new ones. The modified Korteweg-de Vries equation, the non- linear Schrodinger equation, the derivative non-linear Schrodinger equation, Sine-Gordon equation, the reduced Maxwell-Bloch equation, Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are gradually putting together a list of soliton equations, which are usually to be found in non-linear optics theory.Comment: Latex, 17 pages, no figures, submitted to Pramana

Second harmonic generation: Goursat problem on the semi-strip and explicit solutions

A rigorous and complete solution of the initial-boundary-value (Goursat) problem for second harmonic generation (and its matrix analog) on the semi-strip is given in terms of the Weyl functions. A wide class of the explicit solutions and their Weyl functions is obtained also.Comment: 20 page

Theory of Pump Depletion and Spike Formation in Stimulated Raman Scattering

By using the inverse spectral transform, the SRS equations are solved and the explicit output data is given for arbitrary laser pump and Stokes seed profiles injected on a vacuum of optical phonons. For long duration laser pulses, this solution is modified such as to take into account the damping rate of the optical phonon wave. This model is used to interprete the experiments of Druhl, Wenzel and Carlsten (Phys. Rev. Lett., (1983) vol. 51, p. 1171), in particular the creation of a spike of (anomalous) pump radiation. The related nonlinear Fourier spectrum does not contain discrete eigenvalue, hence this Raman spike is not a soliton.Comment: LaTex file, includes two figures in LaTex format, 9 page

Zero curvature representation for a new fifth-order integrable system

In this brief note we present a zero-curvature representation for one of the new integrable system found by Mikhailov, Novikov and Wang in nlin.SI/0601046.Comment: 2 pages, LaTeX 2e, no figure

On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe

Dynamical Evolution of Boson Stars II: Excited States and Self-Interacting Fields

The dynamical evolution of self-gravitating scalar field configurations in numerical relativity is studied. The previous analysis on ground state boson stars of non-interacting fields is extended to excited states and to fields with self couplings. Self couplings can significantly change the physical dimensions of boson stars, making them much more astrophysically interesting (e.g., having mass of order 0.1 solar mass). The stable ($S$) and unstable ($U$) branches of equilibrium configurations of boson stars of self-interacting fields are studied; their behavior under perturbations and their quasi-normal oscillation frequencies are determined and compared to the non-interacting case. Excited states of boson stars with and without self-couplings are studied and compared. Excited states also have equilibrium configurations with $S$ and $U$ branch structures; both branches are intrinsically unstable under a generic perturbation but have very different instability time scales. We carried out a detailed study of the instability time scales of these configurations. It is found that highly excited states spontaneously decay through a cascade of intermediate states similar to atomic transitions.Comment: 16 pages+ 13 figures . All figures are available at http://wugrav.wustl.edu/Paper