Bethe ansatz equations for the multicomponent nonlinear Schrödinger model with supermatrices

The Bethe ansatz equations for the multicomponent nonlinear Schrödinger model with supermatrices are obtained by using the quantum inverse scattering transform.     

Quantum Gelfand-Levitan equations for the multicomponent nonlinear Schrödinger model with supermatrices

The quantum Gelfand-Levitan equations for the multicomponent nonlinear Schrödinger model with supermatrices are derived, and the commutation relations for scattering data operators which are needed for calculating Green functions and correlation functions are given.     

Pointwise bounds for Schrödinger eigenstates

In a different paper we constructed imaginary time Schrödinger operatorsH _q=–1/2+V acting onL ^q( ⁿ ,dx). The negative part of typical potential functionV was assumed to be inL +L ^q for somep>max{1,n/2}. Our proofs were based on the evaluation of Kac's averages over Brownian motion paths. The present paper continues this study: using probabilistic techniques we prove pointwise upper bounds forL ^q-Schrödinger eigenstates and pointwise lower bounds for the corresponding groundstate. The potential functionsV are assumed to be neither smooth nor bounded below. Consequently, our results generalize Schnol's and Simon's ones. Moreover probabilistic proofs seem to be shorter and more informative than existing ones.Laboratoire de Mathématiques de Marseille associé au C.N.R.S. L.A.225     

Fermionic perturbation theory for the statistical mechanics of the nonlinear Schrödinger model

A fermionic perturbation theory is developed for the statistical mechanics of the nonlinear Schrödinger model. The theory is based on an interacting-fermion picture of the Bethe wave function. The inner product of the Bethe wave function is explicitly evaluated, and a simple graphical representation of it is given. The basic equations obtained for the free energy agree with those of Yang and Yang. In particular, the present theory gives a clear-cut meaning to the function of Yang and Yang: It represents a fermion energy at finite temperatures.     

Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model

The truncated forced nonlinear Schr?dinger (NLS) model is known to mimic well the forced NLS solutions in the regime at which only one linearly unstable mode exists. Using a novel framework in which a hierarchy of bifurcations is constructed, we analyze this truncated model and provide insights regarding its global structure and the type of instabilities which appear in it. In particular, the significant role of the forcing frequency is revealed and it is shown that a parabolic resonance mechanism of instability arises in the relevant parameter regime of this model. Numerical experiments demonstrating the different types of chaotic motion which appear in the model are provided.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Modified Schrödinger dynamics with attractive densities

The linear Schrödinger equation does not predict that macroscopic bodies should be located at one place only, or that the outcome of a measurement shoud be unique. Quantum mechanics textbooks generally solve the problem by introducing the projection postulate, which forces definite values to emerge during measurements; many other interpretations have also been proposed. Here, in the same spirit as the GRW and CSL theories, we modify the Schrödinger equation in a way that efficiently cancels macroscopic density fluctuations in space. Nevertheless, we do not assume a stochastic dynamics as in GRW or CSL theories. Instead, we propose a deterministic evolution that includes an attraction term towards the averaged density in space of the de Broglie-Bohm position of particles, and show that this is sufficient to ensure macroscopic uniqueness and compatibility with the Born rule. The state vector can then be seen as directly related to physical reality.     

Quasilinear theory for the nonlinear Schrödinger equation with periodic coefficients

The nonlinear Schrödinger equation with periodic coefficients is analyzed under the condition of large variation in the local dispersion. The solution after n periods is represented as the sum of the solution to the linear part of the nonlinear Schrödinger equation and the nonlinear first-period correction multiplied by the number of periods n. An algorithm for calculating the quasilinear solution with arbitrary initial conditions is proposed. The nonlinear correction to the solution for a sequence of Gaussian pulses is obtained in the explicit form.     

Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling

We summarize the most featured items characterizing the semi-discrete nonlinear Schrödinger system with background-controlled inter-site resonant coupling. The system is shown to be integrable in the Lax sense that make it possible to obtain its soliton solutions in the framework of properly parameterized dressing procedure based on the Darboux transformation. On the other hand the system integrability inspires an infinite hierarchy of local conservation laws some of which were found explicitly in the framework of generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem and it permits the Hamiltonian formulation accompanied by the highly nonstandard Poisson structure. The nonzero background level of concomitant fields mediates the appearance of an additional type of inter-site resonant coupling and as a consequence it establishes the triangular-lattice-ribbon spatial arrangement of location sites for the basic field excitations. Adjusting the background parameter we are able to switch over the system dynamics between two essentially different regimes separated by the critical point. The system criticality against the background parameter is manifested both indirectly by the auxiliary linear spectral problem and directly by the nonlinear dynamical equations themselves. The physical implications of system criticality become evident after the rather sophisticated canonization procedure of basic field variables. There are two variants of system standardization equal in their rights. Each variant is realizable in the form of two nonequivalent canonical subsystems. The broken symmetry between canonical subsystems gives rise to the crossover effect in the nature of excited states. Thus in the under-critical region the system support the bright excitations in both subsystems, while in the over-critical region one of subsystems converts into the subsystem of dark excitations.     

Focusing and multi-focusing solutions of the nonlinear Schrödinger equation

A method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic Schrödinger equations in two and three dimensions and describe their universal properties. The same method is applied to simulate the multi-focusing phenomena produced by simple models of saturating nonlinearities.     

The Hamiltonian dynamics of the soliton of the discrete nonlinear Schrödinger equation

Hamiltonian equations are formulated in terms of collective variables describing the dynamics of the soliton of an integrable nonlinear Schrödinger equation on a 1D lattice. Earlier, similar equations of motion were suggested for the soliton of the nonlinear Schrödinger equation in partial derivatives. The operator of soliton momentum in a discrete chain is defined; this operator is unambiguously related to the velocity of the center of gravity of the soliton. The resulting Hamiltonian equations are similar to those for the continuous nonlinear Schrödinger equation, but the role of the field momentum is played by the summed quasi-momentum of virtual elementary system excitations related to the soliton.     

New geometries associated with the nonlinear Schrödinger equation

We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr?dinger equation (NLS). We show that any given solution of the NLS gets associated with three distinct space curve evolutions. The tangent vector of the first of these curves, the binormal vector of the second and the normal vector of the third, are shown to satisfy the integrable Landau-Lifshitz (LL) equation = ×, ( = 1). These connections enable us to find the three surfaces swept out by the moving curves associated with the NLS. As an example, surfaces corresponding to a stationary envelope soliton solution of the NLS are obtained. Received 5 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: radha@imsc.ernet.in     

Statistical mechanics of the nonlinear Schrödinger equation

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian $$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$ is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL ² norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.     

Solitons for the cubic-quintic nonlinear Schrödinger equation with Raman effect in nonlinear optics

Considering the ultrashort optical soliton propagation in the non-Kerr media, the cubic-quintic nonlinear Schrödinger equation with Raman effect is studied through the dependent variable transformation and Hirota method. Based on symbolic computation, the bilinear form, the explicit one- and two-soliton solutions for the equation are presented. The constraint parametric condition for the existence of soliton solutions is also derived. Propagation characteristics and interaction behaviors of the solitons are graphically shown and discussed: (1) Overtaking elastic interactions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) propagation in parallel of the two solitons.     

On the finite-time behaviour for nonlinear Schrödinger equations

Consider the nonlinear Schrödinger equationu _t –iu=f(u). Forf(u)=±|u|^1+p , ±i|u|^1+p , ±u|u| ^p (p>0), and the Dirichlet boundary or nonlinear boundary (including the Neumann boundary and the Robin boundary) conditions, we establish the local estimates for the timet to the solutions of the initial-boundary value problems. Being based up on these estimates, we investigate the blowing-up properties of the solutions.Research supported in part by the Youth Foundation of Sichuan Education Committee and the Natural Science Foundation of China     

The fractional discrete nonlinear Schrödinger equation

We examine a fractional version of the discrete nonlinear Schrödinger (dnls) equation, where the usual discrete laplacian is replaced by a fractional discrete laplacian. This leads to the replacement of the usual nearest-neighbor interaction to a long-range intersite coupling that decreases asymptotically as a power-law. For the linear case, we compute both, the spectrum of plane waves and the mean square displacement of an initially localized excitation in closed form, in terms of regularized hypergeometric functions, as a function of the fractional exponent. In the nonlinear case, we compute numerically the low-lying nonlinear modes of the system and their stability, as a function of the fractional exponent of the discrete laplacian. The selftrapping transition threshold of an initially localized excitation shifts to lower values as the exponent is decreased and, for a fixed exponent and zero nonlinearity, the trapped fraction remains greater than zero.     

On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrödinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the system’s Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to N?≈?70 sites. An advantage of the latter methods is the better conservation of the system’s second integral, i.e. the wave packet’s norm.