共查询到20条相似文献,搜索用时 15 毫秒
1.
Gino Biondini Daniel K. Kraus Barbara Prinari 《Communications in Mathematical Physics》2016,348(2):475-533
We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as \({x\to\pm\infty}\). The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n-th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation. 相似文献
2.
Cornelis van der Mee 《Journal of Nonlinear Mathematical Physics》2013,20(2):233-264
In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed. 相似文献
3.
The inverse scattering transform for the nonlinear Schrödinger equation in normal dispersion with non-vanishing boundary values is re-examined using an affine parameter to avoid double-valued functions. An operable algebraic procedure is developed to evaluate dark multi-soliton solutions. The dark two-soliton solution is given explicitly as an example, and is verified by direct substitution. The additional motion of the soliton center is given by its asymptotic behavior. 相似文献
4.
Xiang-Yao Wu Bai-Jun Zhang Xiao-Jing Liu Li Xiao Yi-Heng Wu Yan Wang Qing-Cai Wang Shuang Cheng 《International Journal of Theoretical Physics》2010,49(10):2437-2445
We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak. 相似文献
5.
We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter. 相似文献
6.
In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed. 相似文献
7.
W.?K.?Abou Salem J.?Fr?hlich I.?M.?Sigal 《Communications in Mathematical Physics》2009,291(1):151-176
We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed ||v|| of the solitons, we show that, up to times of order ||v|| after the collision, the solitons preserve their shape (in L 2-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential. 相似文献
8.
Jinping Tian 《Fiber and Integrated Optics》2013,32(2):101-110
The generalized form with varying coefficients for nonlinear Schrödinger equation including fourth-order dispersion and quintic nonlinearity is presented in this article. The exact bright, dark, and combined solitonlike solutions were given by taking proper ansatz into account. The different forms of dispersion functions were considered to investigate the pulse's evolution or dispersion managements in optical fiber. 相似文献
9.
The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using localized initial conditions, the time-averaged occupation probability of the initial site is investigated which is a function of the degree of nonlinearity and the linear coupling strengths. The self-trapping transition occurs at larger values of the nonlinearity parameter as the NNN coupling strength of the lattice increases for fixed size. Furthermore, given NNN coupling strength, the self-trapping properties for different sizes are considered which are some different from the case with general nearest neighbor (NN) interaction. 相似文献
10.
11.
Marcel Guardia 《Communications in Mathematical Physics》2014,329(1):405-434
Fix s > 1. Colliander et al. proved in (Invent Math 181:39–113, 2010) the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus whose s-Sobolev norm undergoes arbitrarily large growth as time evolves. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in Guardia and Kaloshin (Growth of Sobolev norms in the cubic defocusing Nonlinear Schrödinger Equation. To appear in the Journal of the European Mathematical Society, 2012). 相似文献
12.
We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation i q t + q xx ? i q 2 q ? x + 1 2 | q | 4 q = 0 $$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0 $$ with step-like initial data q ( x , 0 ) = 0 $q(x,0)=0$ for x ≤ 0 $x \leqslant 0$ and q ( x , 0 ) = A e ? 2 iBx $q(x,0)=A\mathrm {e}^{-2iBx}$ for x > 0 $x>0$ , where A > 0 $A>0$ and B ∈ ? $B\in \mathbb R$ are constants. We show that there are three regions in the half-plane { ( x , t ) | ? ∞ < x < ∞ , t > 0 } $\{(x,t) | -\infty <x<\infty , t>0\}$ , on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x > ? 4 tB $x>-4tB$ , a plane wave region: x > ? 4 t B + 2 A 2 B + A 2 4 $x<-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )$ , an elliptic region: ? 4 t B + 2 A 2 B + A 2 4 > x > ? 4 tB $-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )<x<-4tB$ . Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method. 相似文献
13.
We consider the propagation of wave packets for the nonlinear Schrödinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets. 相似文献
14.
This paper obtains the travelling wave solutions of the nonlinear Schrödinger’s equation with higher order polynomial law nonlinearity. The doubly periodic wave solution of this equation is obtained. The numerical simulation is also included. 相似文献
15.
Frank Hövermann Herbert Spohn Stefan Teufel 《Communications in Mathematical Physics》2001,215(3):609-629
We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time
dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined
by the semiclassical dynamics.
Received: 7 February 2000 / Accepted: 7 July 2000 相似文献
16.
In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrdinger(NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics. 相似文献
17.
We prove a dynamical localization in the nonlinear Schrödinger equation with a random potential for times of the order of O(β ?2), where β is the strength of the nonlinearity. 相似文献
18.
Conservation Laws for the Derivative Nonlinear Schroedinger Equation with Non-vanishing Boundary Conditions
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Conservation laws for the derivative nonlinear Schr6dinger equation with non-vanishing boundary conditions are derived, based on the recently developed inverse scattering transform using the affine parameter technique. 相似文献
19.
Fangfang Fu Linghua Kong & Lan Wang 《advances in applied mathematics and mechanics.》2009,1(5):699-710
In this paper, we establish a family of symplectic integrators for a class
of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization.
Then we apply the symplectic Euler method to the Hamiltonian system.
It is demonstrated that the scheme not only preserves symplectic geometry structure
of the original system, but also does not require to resolve coupled nonlinear
algebraic equations which is different from the general implicit symplectic schemes.
The linear stability of the symplectic Euler scheme and the errors of the numerical
solutions are investigated. It shows that the semi-explicit scheme is conditionally
stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests
suggest that the symplectic integrators are more effective than non-symplectic ones,
such as backward Euler integrators. 相似文献
20.
We develop a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data. As an example we consider the cubic nonlinear Schrödinger equation.
Mathematics Subject Classifications (2000) 35Q53, 58B99.The work is partially supported by NSF grant DMS-9971834. 相似文献