An inequality relating averages of generalized correlations to susceptibilities for Gaussian field distributions is presented. This applied random-field systems prove under the assumption a continuous transition (tree level) decoupling quenched two-point function. By only power-law divergence, lower bound $\ensuremath{\eta}$ obtained. It rules out possibility that some recent experimental and numerical results reflect equilibrium properties near transition.

The nonlinear Schrödinger equation (NLSE) with a random potential is motivated by experiments in optics and atom paradigm for the competition between randomness nonlinearity. analysis of NLSE (Anderson like) has been done at various levels control: numerical, analytical rigorous. Yet this model presents us highly inconclusive often contradictory picture. We will describe main recent results obtained field propose list specific problems to focus on, which we hope enable resolve these...

The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time goes to infinity) a linear combination N non-colliding

Abstract We prove dispersive estimates for the time‐dependent Schrödinger equation with a charge transfer Hamiltonian. As by‐product we also obtain another proof of asymptotic completeness wave operators model established earlier by K. Yajima and J. M. Graf. consider more general matrix non‐self‐adjoint problem. This appears naturally in study nonlinear multisoliton systems is specifically motivated problem stability states equation. © 2004 Wiley Periodicals, Inc.

We give a new derivation of the minimal velocity estimates [27] for unitary evolutions with some optimal estimates. Let H and A be selfadjoint operators on Hilbert space H. The starting point is Mourre's inequality which supposed to hold in form sense spectral subspaceof interval. second assumption that multiple commutators are well-behaved Then we show that, dense set allm contained subspace up an error order t-m norm. apply this general result case where Schrödinger operator Rn dilation...

We study, theoretically and experimentally, the nonlinear dynamics of a wave packet launched inside trap potential. Increasing power transforms its from linear tunneling through potential barrier, to soliton tunneling, eventually, above well-defined threshold, ejection trap.

Abstract We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein–Gordon equation with spatially localized, variable coefficient quadratic nonlinearity and non-generic linear potential. The purpose this work is continue investigation occurrence novel modified scattering that involves logarithmic slow-down decay rate along certain rays. This phenomenon ultimately caused by threshold resonance operator. It was previously uncovered for special case zero potential in...

We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time interactions localized waves in this setting. This problem arises when studying perturbations around (possibly non-decaying) special solutions PDEs, trying control projection onto continuous spectrum radiative interactions. One our main tools is Fourier transform adapted Schrödinger operator <inline-formula content-type="math/mathml">...

The semilinear wave equation on the (outer) Schwarzschild manifold is studied. We prove local decay estimates for general (non-radial) data, deriving a-priori Morawetz type estimates.