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V. S. Gerdjikov

Publications and source records attributed to V. S. Gerdjikov.

30 records · Page 2Linked to original sources
arXiv2003

Real Hamiltonian forms of Hamiltonian systems

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero--Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.

nlin.SI↗
arXiv2003

Soliton generation of two-component Bose-Einstein condensates in optical lattices

Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.

cond-mat.soft↗
arXiv2001

N-wave interactions related to simple Lie algebras. Z_2- reductions and soliton solutions

The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analyzed. The Zakharov- Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one soliton solutions of the corresponding N- wave equations and their reductions are studied. We show that to each soliton solution one can relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave equations related to so(5) which find applications in Stockes-anti-Stockes wave generation.

nlin.SI↗
arXiv2001

Reductions of N-wave interactions related to low-rank simple Lie algebras. I: Z_2- reductions

The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem. We show how the second order reductions of the N-wave interactions related to low-rank simple Lie algebras g can be embedded also in the Weyl group of g. This allows us to display along with the well known ones a number of new types of integrable N-wave systems. Some of the reduced systems find applications to nonlinear optics.

nlin.SI↗
arXiv2001

Adiabatic Interaction of N Ultrashort Solitons: Universality of the Complex Toda Chain Model

Using the Karpman-Solov'ev method we derive the equations for the 2-soliton adiabatic interaction for solitons of the modified nonlinear Schrodinger equation (MNSE). Then we generalize these equations to the case of N interacting solitons with almost equal velocities and widths. On the basis of this result we prove that the N MNSE-soliton train interaction (N>2) can be modeled by the completely integrable complex Toda chain (CTC). This is an argument in favor of universality of the complex Toda chain which was previously shown to model the soliton train interaction for nonlinear Schrodinger solitons. The integrability of the CTC is used to describe all possible dynamical regimes of the N-soliton trains which include asymptotically free propagation of all N solitons, N-soliton bound states, various mixed regimes, etc. It allows also to describe analytically the manifolds in the 4N-dimensional space of initial soliton parameters which are responsible for each of the regimes mentioned above. We compare the results of the CTC model with the numerical solutions of the MNSE for 2 and 3-soliton interactions and find a very good agreement.

nlin.SI↗
arXiv2000

N-soliton train interactions and perturbed complex Toda chain in nonlinear optics. Adiabatic and non-adiabtic aspects

Our previous results on the N-soliton interaction in the adiabatic approximation have been extended. It is shown that the complex Toda chain (CTC) model is an universal one in the sense that it describes the N-soliton train interactions for all NLEE from the NLS hierarchy. We derive the perturbed CTC system and show that the small perturbations affect only the center of mass motion and the global phase of the N-soliton train. A special reduction of CTC known as the Toda chain with indefinite metric describes the interaction of the sine-Gordon solitons and anti-solitons. The peculiarities of the interactions in the non-adiabatic cases are outlined.

nlin.SI↗
arXiv2000

Dynamical models of adiabatic $N $-soliton interactions

The adiabatic N-soliton train interactions for the scalar nonlinear Schrodinger (NLS) equation and its perturbed versions are well studied. Here we briefly outline how they can be generalized for the higher NLS-type equations and for the multicomponent NLS equations. It is shown that in all these cases the complex Toda chain plays fundamental role.

nlin.SI↗
arXiv1999

The complex Toda chains and the simple Lie algebras - solutions and large time asymptotics -- II

We propose a compact and explicit expression for the solutions of the complex Toda chains related to the classical series of simple Lie algebras g. The solutions are parametrized by a minimal set of scattering data for the corresponding Lax matrix. They are expressed as sums over the weight systems of the fundamental representations of g and are explicitly covariant under the corresponding Weyl group action. In deriving these results we start from the Moser formula for the A_r series and obtain the results for the other classical series of Lie algebras by imposing appropriate involutions on the scattering data. Thus we also show how Moser's solution goes into the one of Olshanetsky and Perelomov. The results for the large-time asymptotics of the A_r -CTC solutions are extended to the other classical series B_r - D_r. We exhibit also some `irregular' solutions for the D_{2n+1} algebras whose asymptotic regimes at t ->\pm\infty are qualitatively different. Interesting examples of bounded and periodic solutions are presented and the relations between the solutions for the algebras D_4, B_3 and G_2 $ are analyzed.

solv-int↗
arXiv1998

The Complex Toda Chains and the Simple Lie Algebras - Solutions and Large Time Asymptotics

The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymptotically free propagation (the only possible regime for the RTC), CTC allow bound state regimes, various intermediate regimes when one (or several) group(s) of particles form bound state(s), singular and degenerate solutions. These results can be used e.g., in describing the soliton interactions of the nonlinear Schroedinger equation. Explicit expressions for the solutions in terms of minimal sets of scattering data are proposed for all classical series B_r - D_r.

solv-int↗
arXiv1997

Criterion and Regions of Stability for Quasi-Equidistant Soliton Trains

Using the complex Toda chain (CTC) as a model for the propagation of the N-soliton pulse trains of the nonlinear Schrodinger (NLS) equation, we predict the asymptotic behavior of these trains. The following asymptotic regimes are stable: (i)~asymptotically free propagation of all N solitons; (ii)~bound state regime where the N solitons may move quasi-equidistantly (QED); and (iii)~various different combinations of (i) and (ii). For N=2 and 3 we determine analytically the set of initial soliton parameters corresponding to each of these regimes. We find excellent agreement between the solutions of CTC and NLS for all regimes and propose realistic choices for the sets of amplitudes, for which the solitons propagate QED to very large run lengths. This is of importance for optical fiber communication.

solv-int↗