On the statistical derivation of the Schrödinger equation

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of simple systems—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrödinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrödinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a point of a continuous medium is described by the classical electron radiusr _e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr _e and the corresponding time interval _e =r _e /c play the role of hidden parameters in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient = _e ^–1 , the presence of which reflects loss of information on smoothing over the volume of a point of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length _C and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the collision integrals being expressed in terms of the correlation functions of these fluctuations.In memory of Dmitrii Nikolaevich ZubarevMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 3–26, October, 1993.     

Strichartz estimates and the nonlinear Schr?dinger equation on manifolds with boundary

We establish Strichartz estimates for the Schr?dinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key ${L^{4}_{t}L^{\infty}_x}$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schr?dinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schr?dinger equations on general compact manifolds with boundary.     

A note on the illposedness for anisotropic nonlinear Schrödinger equation

In this short note, we show the illposedness of anisotropic Schroedinger equation in L^2 if the growth of nonlinearity is larger than a threshold power pc which is also the critical power for blowup, as Fibich, Ilan and Schochet have pointed out recently. The illposedness in anisotropic Sobolev space Hk,d-d^2s,s where 0 〈 s 〈 sc, sc =d/2-k/4-2/p-1, and the illposedness in Sobolev space of negative order H^s, s 〈 0 are also proved.     

On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity

We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic–quintic nonlinear Schrödinger equation with a periodic potential.     

Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity

We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.     

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ?1,K?1,s>1, we construct smooth initial data u ₀ with \(\|u_{0}\|_{{H}^{s}}<\delta\), so that the corresponding time evolution u satisfies \(\|u(T)\|_{{H}^{s}}>K\) at some time T. This growth occurs despite the Hamiltonian’s bound on \(\|u(t)\|_{\dot{H}^{1}}\) and despite the conservation of the quantity \(\|u(t)\|_{L^{2}}\).The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems.     

Allowable graphs of the nonlinear Schrödinger equation and their applications

We construct the definition of allowable graphs of the nonlinear Schrödinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving 3-dimensional blocks of the normal form of the nonlinear Schrödinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schrödinger equation.     

Explicit breather solution of the nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We present a one-line closed-form expression for the three-parameter breather of the nonlinear Schrödinger equation. This provides an analytic proof of...     

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrödinger equation

A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrödinger equation into a linear algebraic system. This method is developed by re placing the time and the space partial derivatives by parametric finite-difference re placements and the nonlinear term by an appropriate parametric linearized scheme based on Taylor’s expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given.     

Bilinear eigenfunction estimates and the nonlinear Schr?dinger equation on surfaces

We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere and more generally on Zoll surfaces, we prove that, for s>1/4, NLS is uniformly well-posed in H^s, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces.

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Résumé On étudie léquation de Schrödinger non linéaire (NLS) sur une surface compacte. Sur la sphère et plus généralement sur toute surface de Zoll, on démontre que pour s>1/4, NLS est uniformément bien posée dans H^s, ce qui est optimal sur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

     

Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients

We study the internal stabilization of the higher order nonlinear Schrödinger equation with constant coefficients. Combining multiplier techniques, a fixed point argument and nonlinear interpolation theory, we can obtain the well-posedness. Then, applying compactness arguments and a unique continuation property, we prove that the solution of the higher-order nonlinear Schrödinger equation with a damping term decays exponentially.     

The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.     

Analytical solutions of nonlinear Schrödinger equation with distributed coefficients

We combine the F-expansion method with the homogeneous balance principle to build a strategy to find analytical solitonic and periodic wave solutions to a generalized nonlinear Schrödinger equation with distributed coefficients, linear gain/loss, and nonlinear gain/absorption. In the case of a dimensionless effective Gross–Pitaevskii equation which describes the evolution of the wave function of a quasi-one-dimensional cigar-shaped Bose–Einstein condensate, the building strategy is applied to generate analytical solutions.