共查询到20条相似文献,搜索用时 15 毫秒
1.
Joel L. Lebowitz Harvey A. Rose Eugene R. Speer 《Journal of statistical physics》1988,50(3-4):657-687
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. 相似文献
2.
《Physica D: Nonlinear Phenomena》1988,31(1):78-102
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. 相似文献
3.
The nonlinear Schrödinger equation for the ground-state wave function of an inhomogeneous boson system is derived in the self-consistent Hartree–Fock approximation without the use of the formalism of anomalous averages. The results obtained correspond to the Gross–Pitaevskii equation for the Bose–Einstein condensate wave function when using the delta-shaped boson interaction potential. 相似文献
4.
J. Pade 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2009,53(1):41-50
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. 相似文献
5.
A new class of exact solutions with a singularity at finite time (collapse) is obtained for the nonlinear Schrödinger equation. 相似文献
6.
We consider global solutions of the nonlinear Schrödinger equation
相似文献
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8.
《Physica A》1995,214(4):619-628
We propose a new class of nonlinear homogeneous extension of the Doebner-Goldin Schrödinger equation, valid for arbitrary representations and operators, chosen in accordance with the investigated physical problem. We verify that the nonlinearity simulates an environment, thence, the new model leads to simple exact solutions as, for instance, the time-dependent squeezed coherent states and a special class of stationary states that we call pseudothermal, reached after relaxation. We illustrate the use of the new equation with applications to problems such as, the relaxation of a two-level or spin-1/2 system, and of the harmonic oscillator (HO) or equivalently, the emission-absorption process of photons in an electromagnetic cavity. Furthermore, in order to compare solutions for the HO example we introduce two different representations in the new equation, one continuous (positional representation) and the other discrete (Fock states). 相似文献
9.
《Physics letters. A》2019,383(36):126028
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. 相似文献
10.
《Physica D: Nonlinear Phenomena》1988,32(1):83-106
Using the multiscale approach of Zakharov and Kuznetsov it is shown that the nonlinear Schrödinger periodic scattering data is related to the Korteweg-de Vries periodic scattering data via an average over the Korteweg-de Vries carrier oscillation. This allows a complete elucidation of the physical meaning of the nonlinear Schrödinger scattering data, conservation laws, theta function solutions and reality constraint. 相似文献
11.
J. Bourgain 《Communications in Mathematical Physics》1994,166(1):1-26
In this paper we continue some investigations on the periodic NLSEiu
u
+iu
xx
+u|u|
p-2
(p6) started in [LRS]. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measrue
(after suitableL
2-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations. 相似文献
12.
《Physica D: Nonlinear Phenomena》1988,32(2):210-226
For the cubic Schrödinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d = 2 tod > 2. 相似文献
13.
E.V. Doktorov 《The European Physical Journal B - Condensed Matter and Complex Systems》2002,29(2):227-231
We argue that the integrable modified nonlinear Schr?dinger equation with the nonlinearity dispersion term is the true starting
point to analytically describe subpicosecond pulse dynamics in monomode fibers. Contrary to the known assertions, solitons
of this equation are free of self-steepening and the breather formation is possible.
Received 29 September 2001 / Received in final form 25 January 2002 Published online 2 October 2002
RID="a"
ID="a"doktorov@dragon.bas-net.by 相似文献
14.
Yu. N. Ovchinnikov 《Journal of Experimental and Theoretical Physics》2003,96(5):975-981
It is shown that one of the conditions for a weakly collapsing solution with zero energy produces an infinite number of functionals I N identically vanishing on the regular solutions to the corresponding differential equation. On the parameter plane {A, C1}, there are at least two singular lines. Along one of these lines (A/C1=1/6), are located weakly collapsing solutions with zero energy. It is assumed that, along the second line (A/C1=αc), another family of weakly collapsing solutions with zero energy is located. In the domain of large values of the parameters C1, α=A/C1, there exists a domain of an intermediate asymptotic form, where the amplitude of oscillations of the function U grows in a large domain relative to the ξ coordinate. 相似文献
15.
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17.
S. Murugesh Radha Balakrishnan 《The European Physical Journal B - Condensed Matter and Complex Systems》2002,29(2):193-196
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 相似文献
18.
The generic asymptotic behavior of a three-parameter weakly collapsing solution of a nonlinear Schrödinger equation is examined. A discrete set of zero-energy states is shown to exist. In the (A, C 1) parameter space, there are two close lines along which the amplitude of oscillating terms is exponentially small in the parameter C 1. 相似文献
19.
Nauman Raza Sultan Sial Shahid S. Siddiqi Turab Lookman 《Journal of computational physics》2009,228(7):2572-2577
In this the window of the Sobolev gradient technique to the problem of minimizing a Schrödinger functional associated with a nonlinear Schrödinger equation. We show that gradients act in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite-difference and finite-element settings in a computationally efficient way to find minimum energy states of Schrödinger functionals. 相似文献
20.
《Physics letters. A》2002,295(4):192-197
By introducing the concept of differential equations with “given curvature condition”, we show that the modified nonlinear Schrödinger equations for κ=1 and −1 are, respectively, gauge equivalent to the modified HF model and the modified M-HF model. As a consequence, soliton solutions to the modified HF model are constructed. 相似文献
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