Fully three dimensional breather solitons can be created using Feshbach resonance

We investigate the stability properties of breather solitons in a three-dimensional Bose-Einstein Condensate with Feshbach Resonance Management of the scattering length and con ned only by a one dimensional optical lattice. We compare regions of stability in parameter space obtained from a fully 3D analysis with those from a quasi two-dimensional treatment. For moderate con nement we discover a new island of stability in the 3D case, not present in the quasi 2D treatment. Stable solutions from this region have nontrivial dynamics in the lattice direction, hence they describe fully 3D breather solitons. We demonstrate these solutions in direct numerical simulations and outline a possible way of creating robust 3D solitons in experiments in a Bose Einstein Condensate in a one-dimensional lattice. We point other possible applications.Comment: 4 pages, 4 figures; accepted to Physical Review Letter

Spontaneous symmetry breaking of gap solitons in double-well traps

We introduce a two dimensional model for the Bose-Einstein condensate with both attractive and repulsive nonlinearities. We assume a combination of a double well potential in one direction, and an optical lattice along the perpendicular coordinate. We look for dual core solitons in this model, focusing on their symmetry-breaking bifurcations. The analysis employs a variational approximation, which is verified by numerical results. The bifurcation which transforms antisymmetric gap solitons into asymmetric ones is of supercritical type in the case of repulsion; in the attraction model, increase of the optical latttice strength leads to a gradual transition from subcritical bifurcation (for symmetric solitons) to a supercritical one.Comment: 6 pages, 5 figure

Design and Control of Libration Point Spacecraft Formations

The article of record as published may be located at http://dx.doi.org/10.2514/6.2004-4786Proceedings of AIAA Guidance, Navigation, and Control Conference ; Paper no. AIAA 2004-4786, Providence, Rhode Island, Aug. 16-19 2004We investigate the concurrent problem of orbit design and formation control around a libration point. The problem formulation is based on a framework of multi-agent, nonlinear optimal control. The optimality criterion is fuel consumption modeled as the L1-norm of the control acceleration. Fuel budgets are allocated by isoperimetric constraints. The nonsmooth optimal control problem is discredited using DIDO, a software package that implements the Legendre pseudospectral method. The discretized problem is solved using SNOPT, a sequential quadratic programming solver. Among many, one of the advantages of our approach is that we do not require linearization or analytical results; consequently, the inherent nonlinearities associated with the problem are automatically exploited. Sample results for formations about the Sun-Earth L2 point in the 3-body circular restricted dynamical framework are presented. Globally optimal solutions for relaxed and almost periodic formations are presented for both a large separation constraint (about a third to half of orbit size), and a small separation constraint (a few hundred km or about 5_10_6 of orbit size).N

Young diagrams and N-soliton solutions of the KP equation

We consider $N$-soliton solutions of the KP equation, (-4u_t+u_{xxx}+6uu_x)_x+3u_{yy}=0 . An $N$-soliton solution is a solution $u(x,y,t)$ which has the same set of $N$ line soliton solutions in both asymptotics $y\to\infty$ and $y\to -\infty$. The $N$-soliton solutions include all possible resonant interactions among those line solitons. We then classify those $N$-soliton solutions by defining a pair of $N$-numbers $({\bf n}^+,{\bf n}^-)$ with ${\bf n}^{\pm}=(n_1^{\pm},...,n_N^{\pm}), n_j^{\pm}\in\{1,...,2N\}$, which labels $N$ line solitons in the solution. The classification is related to the Schubert decomposition of the Grassmann manifolds Gr$(N,2N)$, where the solution of the KP equation is defined as a torus orbit. Then the interaction pattern of $N$-soliton solution can be described by the pair of Young diagrams associated with $({\bf n}^+,{\bf n}^-)$. We also show that $N$-soliton solutions of the KdV equation obtained by the constraint $\partial u/\partial y=0$ cannot have resonant interaction.Comment: 22 pages, 5 figures, some minor corrections and added one section on the KdV N-soliton solution

Higher-order-in-spin interaction Hamiltonians for binary black holes from source terms of Kerr geometry in approximate ADM coordinates

The Kerr metric outside the ergosphere is transformed into ADM coordinates up to the orders $1/r^4$ and $a^2$, respectively in radial coordinate $r$ and reduced angular momentum variable $a$, starting from the Kerr solution in quasi-isotropic as well as harmonic coordinates. The distributional source terms for the approximate solution are calculated. To leading order in linear momenta, higher-order-in-spin interaction Hamiltonians for black-hole binaries are derived.Comment: REVTeX4, 20 pages, typos corrected in Eq. (124) and (130

Nonlinear self-adjointness and conservation laws

The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the strict self-adjointness and quasi self-adjointness introduced earlier by the author. It is shown that the equations possessing the nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint. For example, the heat equation $u_t - \Delta u = 0$ becomes strictly self-adjoint after multiplying by $u^{-1}.$ Conservation laws associated with symmetries can be constructed for all differential equations and systems having the property of nonlinear self-adjointness

Degenerate Four Virtual Soliton Resonance for KP-II

By using disipative version of the second and the third members of AKNS hierarchy, a new method to solve 2+1 dimensional Kadomtsev-Petviashvili (KP-II) equation is proposed. We show that dissipative solitons (dissipatons) of those members give rise to the real solitons of KP-II. From the Hirota bilinear form of the SL(2,R) AKNS flows, we formulate a new bilinear representation for KP-II, by which, one and two soliton solutions are constructed and the resonance character of their mutual interactions is studied. By our bilinear form, we first time created four virtual soliton resonance solution for KP-II and established relations of it with degenerate four-soliton solution in the Hirota-Satsuma bilinear form for KP-II.Comment: 10 pages, 5 figures, Talk on International Conference Nonlinear Physics. Theory and Experiment. III, 24 June-3 July, 2004, Gallipoli(Lecce), Ital

Classification of the line-soliton solutions of KPII

In the previous papers (notably, Y. Kodama, J. Phys. A 37, 11169-11190 (2004), and G. Biondini and S. Chakravarty, J. Math. Phys. 47 033514 (2006)), we found a large variety of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation. The line-soliton solutions are solitary waves which decay exponentially in $(x,y)$-plane except along certain rays. In this paper, we show that those solutions are classified by asymptotic information of the solution as $|y| \to \infty$. Our study then unravels some interesting relations between the line-soliton classification scheme and classical results in the theory of permutations.Comment: 30 page

Ladder operators for subtle hidden shape invariant potentials

Ladder operators can be constructed for all potentials that present the integrability condition known as shape invariance, satisfied by most of the exactly solvable potentials. Using the superalgebra of supersymmetric quantum mechanics we construct the ladder operators for two exactly solvable potentials that present a subtle hidden shape invariance.Comment: 9 pages, based on the talk given at International Conference Progress in Supersymmetric Quantum Mechanics (PSQM03), Valladolid, Spain, 15-19 July, 2003, to appear in a Special Issue of J. Phys. A: Math. Ge

A BPS Interpretation of Shape Invariance

We show that shape invariance appears when a quantum mechanical model is invariant under a centrally extended superalgebra endowed with an additional symmetry generator, which we dub the shift operator. The familiar mathematical and physical results of shape invariance then arise from the BPS structure associated with this shift operator. The shift operator also ensures that there is a one-to-one correspondence between the energy levels of such a model and the energies of the BPS-saturating states. These findings thus provide a more comprehensive algebraic setting for understanding shape invariance.Comment: 15 pages, 2 figures, LaTe