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## Nonlinear Schrödinger Equation (nonlinear + schrodinger_equation)
## Selected Abstracts## Concentration on curves for nonlinear Schrödinger Equations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2007Manuel Del PinoWe consider the problem where p > 1, , > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let , be a closed curve, nondegenerate geodesic relative to the weighted arc length ,,V,, where , = (p + 1)/(p , 1) , 1/2. We prove the existence of a solution u, concentrating along the whole of ,, exponentially small in , at any positive distance from it, provided that , is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in 3 in the two-dimensional case. © 2006 Wiley Periodicals, Inc. [source] ## Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2009Hassan A. ZedanAbstract In this paper, the method of deriving the Bäcklund transformation from the Riccati form of inverse method is presented for the perturbed nonlinear Schrödinger equation (PNSE). Consequently, the exact solutions for the PNSE can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions that are solutions of the associated AKNS, that is, a linear eigenvalues problem in the form of a system of partial differential equation. Moreover, we construct a new soliton solution from the old one and its wave function. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A dual-reciprocity boundary element solution of a generalized nonlinear Schrödinger equation NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004Whye-Teong AngAbstract A time-stepping dual-reciprocity boundary element method is presented for the numerical solution of an initial-boundary value problem governed by a generalized non-linear Schrödinger equation. To test the method, two specific problems with known exact solutions are solved. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20, 2004. [source] ## Benjamin,Feir instability of Rossby waves on a jet THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 600 2004J. G. EslerAbstract Large-scale waves on the extratropical tropopause have been widely observed to spontaneously organize into groups or wave packets. Here, a simple paradigm for this wave packet formation is presented. Firstly, a weakly nonlinear theory of Rossby wave propagation on a potential-vorticity front, based on small non-dimensional wave amplitude ,, is developed. As is typical for systems allowing conservative one-dimensional wave propagation, the evolution of the wave envelope is governed by the nonlinear Schrödinger equation. The sense of the nonlinearity is consistent with Benjamin,Feir instability, where uniform wave trains are unstable to sideband modulations, leading to the formation of wave packets. Next, numerical results from contour dynamics integrations show that the weakly nonlinear predictions for sideband growth rates are quantitatively accurate up to ,,0.5, and that unstable sideband growth is qualitatively similar, but slower than predicted, at higher values of ,. For ,,0.6 the formation of wave packets leads to wave-breaking, this occuring at much lower initial wave amplitudes than for unmodulated uniform wave trains previously studied. The numerical results reveal that the length and time-scales of the Benjamin,Feir instability are broadly consistent with observed wave packet formation in the extratropics. Copyright © 2004 Royal Meteorological Society. [source] ## Solutions to the nonlinear Schrödinger equation carrying momentum along a curve COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2009Fethi MahmoudiWe prove existence of a special class of solutions to the (elliptic) nonlinear Schrödinger equation ,,2,, + V(x), = |,|p , 1, on a manifold or in Euclidean space. Here V represents the potential, p an exponent greater than 1, and , a small parameter corresponding to the Planck constant. As , tends to 0 (in the semiclassical limit) we exhibit complex-valued solutions that concentrate along closed curves and whose phases are highly oscillatory. Physically these solutions carry quantum-mechanical momentum along the limit curves. © 2008 Wiley Periodicals, Inc. [source] ## Long-time asymptotics of the nonlinear Schrödinger equation shock problem COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2007Robert BuckinghamThe long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrödinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock front at large times. © 2007 Wiley Periodicals, Inc. [source] ## On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2004Alexander TovbisWe calculate the leading-order term of the solution of the focusing nonlinear (cubic) Schrödinger equation (NLS) in the semiclassical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t , 0. We utilize the Riemann-Hilbert problem formulation of the inverse scattering problem to obtain the leading-order term of the solution. Error estimates are provided. © 2004 Wiley Periodicals, Inc. [source] ## Semiclassical limit for the Schrödinger-Poisson equation in a crystal COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2001Philippe BechoucheWe give a mathematically rigorous theory for the limit from a weakly nonlinear Schrödinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner Bloch series" as an adaption of the Wigner series for density matrices related to two different "energy bands." Another essential tool are estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band crossing of the other bands which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation, i.e., we take into account the self-consistent Coulomb interaction. Our results hold also for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors. © 2001 John Wiley & Sons, Inc. [source] ## Erratum: Stabilization of solutions to nonlinear Schrödinger equations, COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2005Scipio CuccagnaNo abstract is available for this article. [source] ## Stabilization of solutions to nonlinear Schrödinger equations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2001Scipio CuccagnaFirst page of article [source] |