Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(?2iβx) (or asymptotically periodic, u(x, 0) =α exp(?2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: u_x(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.     

Nonlocality and local causality in the Schrödinger equation with time-dependent boundary conditions

We investigate the nonlocal dynamics of a single particle placed in an infinite well with moving walls. It is shown that in this situation, the Schrödinger equation (SE) violates local causality by causing instantaneous changes in the probability current everywhere inside the well. This violation is formalized by designing a gedanken faster-than-light communication device which uses an ensemble of long narrow cavities and weak measurements to resolve the weak value of the momentum far away from the movable wall. Our system is free from the usual features causing nonphysical violations of local causality when using the (nonrelativistic) SE, such as instantaneous changes in potentials or states involving arbitrarily high energies or velocities. We explore in detail several possible artifacts that could account for the failure of the SE to respect local causality for systems involving time-dependent boundary conditions.     

Darboux transformations for the generalized Schrödinger equation

The intertwining operator technique is applied to the Schrödinger equation with an additional functional dependence h(r) on the right-hand side of the equation. The suggested generalized transformations turn into the Darboux transformations for both fixed and variable values of energy and angular momentum. A relation between the Darboux transformation and supersymmetry is considered.     

Linearizable boundary value problems for the nonlinear Schrödinger equation in laboratory coordinates

Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.     

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.     

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     

On the Schrödinger equation for the supersymmetric FRW model

We consider a time-dependent Schrödinger equation for the Friedmann–Robertson–Walker (FRW) model. We show that for this purpose it is possible to include an additional action invariant under reparametrization of time. The last one does not change the equations of motion for the minisuperspace model, but changes only the constraint. The same procedure is applied to the supersymmetric case.     

Exponentially small adiabatic invariant for the Schrödinger equation

We study an adiabatic invariant for the time-dependent Schrödinger equation which gives the transition probability across a gap from timet to timet. When the hamiltonian depends analytically on time, andt=–,t=+ we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit.Supported by Fonds National Suisse de la Recherche, Grant 2000-5.600     

B-spline solver for one-electron Schrödinger equation

A numerical algorithm for solving the one-electron Schrödinger equation is presented. The algorithm is based on the Finite Element method, and the basis functions are tensor products of univariate B-splines. The application of cubic or higher order B-splines guarantees that the searched solution belongs to a continuous and one time differentiable function space, which is a desirable property in the Kohn–Sham equation context from the Density Functional Theory with pseudopotential approximation. The theoretical background of the numerical algorithm is presented, and additionally, the implementation on parallel computers with distributed memory is described. The current implementation of the algorithm uses the MPI, HYPRE and ParMETIS libraries to distribute matrices on processing units. Additionally, the LOBPCG algorithm from HYPRE library is used to solve the algebraic generalized eigenvalue problem. The proposed algorithm works for any smooth interaction potential, where the domain of the problem is a finite subspace of the ?³ space. The accuracy of the algorithm is demonstrated for a selected interaction potential. In the current stage, the algorithm can be applied to solve the linearized Kohn–Sham equation for molecular systems.     

Lagrangian formulation of the Schrödinger equation

We recast the Schrödinger equation in a new Lagrangian formulation. The equation is —i?dψ (x,t)/dt = Lψ (x,t), whereL is the Lagrangian operator. Expressions forL and ford/dt — ⊥ are derived in terms of coordinate and momentum operators.     

Non-standard finite-difference time-domain method for solving the Schrödinger equation

The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.     

Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Consider the2D defocusing cubic NLSiu _t+u–u|u|²=0 with Hamiltonian . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing ||⁴ by ||⁴ :, is an invariant measure for the appropriately modified equationiu _t + u‒ [u|u ²–2(|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data the solutionu, u(0)=, satisfiesu(t)–e ^it C _Hs (), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).     

Collapse in the nonlinear Schrödinger equation

A new class of exact solutions with a singularity at finite time (collapse) is obtained for the nonlinear Schrödinger equation.     

Exact solutions of the Schrödinger equation for zero energy

Some new exact solutions of the Schrödinger equation for zero energy are presented for certain nontrivial model potentials. Exact expressions for the different scattering lengths are derived and their differences and similarities are worked out. In particular, the respective distributions of the zeros and poles of the scattering lengths are characterized in detail.     

The inhomogeneous Schrödinger equation for the off-shell wave function

A method has been developed for calculating the off-shell wave function by solving the inhomogeneous Schrödinger equation with allowance for the nuclear and Coulomb interactions. The off-shell wave function makes it possible to construct the off-shell scattering amplitude in order to solve the problems for three or more particles. An important application of the method is the Trojan Horse calculations of nuclear reactions that are important in nuclear astrophysics. Specific calculations are performed for neutron and proton scattering on the ⁷Be nucleus. The Woods-Saxon potential is used and the spin-orbital interaction is taken into account.     

Supersymmetry and Darboux transformations for the generalized Schrödinger equation

We construct Darboux transformations for a generalized Schrödinger equation by means of the intertwining operator method. We establish a relation between first-order Darboux transformations, supersymmetry, and factorization of the Hamiltonians that are associated with our generalized Schrödinger equation. Furthermore, our methods allow for the generation of isospectral potentials, where one of the potentials has additional or less bound states than its partner. In the particular case of a conventional Schrödinger equation our generalized Darboux transformations reduce correctly to the well-known expressions.