Spyridon Kamvissis
We study the semiclassical behavior of the focusing nonlinear Schroedinger equation in 1+1-dimensions under discontinuous "barrier" data and we describe the violent oscillations arising in terms of theta functions. The construction of proofs relies on the analysis of the associated Riemann-Hilbert problem.
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Semiclassical states localized on a one-dimensional manifold and governed by the nonlocal NLSE with an anti-Hermitian term
本文提出了一种构造非局部非线性薛定谔方程解的方法，揭示了涡旋态在非平凡几何下的半经典演化行为。
Quasiparticle solutions for the nonlocal NLSE with an anti-Hermitian term in semiclassical approximation
针对含反厄米项的非局域非线性薛定谔方程，基于 Maslov 复胚方法构造半经典渐近解，将问题约化为 ODE 系统，并应用于一维模型，发现长程相互作用可显著改变解的行为。
We consider the one dimensional focusing (cubic) Nonlinear Schrödinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.
A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $z\log z$ type singularity on the jump contour is considered. The jump function and the jump contour are assumed to depend on a vector of external parameters $\vecβ$. We prove that if the RHP has a solution at some value $\vecβ_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $\vecβ_0$ and is smooth in $\vecβ$. This result is applied to the case of semiclassical focusing NLS.
Semiclassical approach to the nonlocal nonlinear Schrödinger equation with a non-Hermitian term
针对含非厄米项的非局域多维非线性薛定谔方程柯西问题，提出基于Maslov方法与辅助动力系统的半经典渐近解构造方法，推导相关算子，并在原子激光模型示例中得到显式解析渐近解。
In this paper we examine the semiclassical behaviour of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic NLS equation, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solu…
In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing t…
We consider the problem of a semiclassical description of quantum chaotic transport, when a tunnel barrier is present in one of the leads. Using a semiclassical approach formulated in terms of a matrix model, we obtain transport moments as power series in the reflection probability of the barrier, whose coefficients are rational functions of the number of open channels M. Our results are therefore valid in the quantum regime and not only when $M\gg 1$. The expressions we arrive at are not identical with the corresponding predictions from random matrix theory, but are in fact much simpler. Both…
The semiclassical limit of the focusing Nonlinear (cubic) Schr\" odinger Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS) system that defines the direct and inverse scattering transforms (IST). In this paper, we derive explicit expressions for the leading order terms of these transforms, which are called semiclassical limits of the direct and inverse scattering transforms. Thus, we establish an explicit connection between the decaying initial data of the form $q(x,0)=A(x)e^{iS(x)}$ and the leading order term of its scattering data. This connection is expressed in ter…
Rigorous derivation of an effective model for periodic Schrödinger equations with linear band crossing of Dirac type
本文通过半经典尺度和多尺度分析，推导出描述NLS在狄拉克点附近谱局域解动力学的非线性狄拉克方程，用于周期性势场中的一维三次薛定谔方程。
We consider the semiclassical limit of nonrelativistic quantum many-boson systems with delta potential in one dimensional space. We prove that time evolved coherent states behave semiclassically as squeezed states by a Bogoliubov time-dependent affine transformation. This allows us to obtain properties analogous to those proved by Hepp and Ginibre-Velo (\cite{Hep}, \cite{GiVe1,GiVe2}) and also to show propagation of chaos for Schrödinger dynamics in the mean field limit. Thus, we provide a derivation of the cubic NLS equation in one dimension.
We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in $1/\hbar$. For a wide variety of incoming wave packets, the leading order tunneling component is Gaussian for sufficiently small $\hbar$. We prove this for both the large time asymptotics and for moderately large values of the time variable.
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.
We consider standing lattice solitons for discrete nonlinear Schrodinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g., the lattice spacing) for which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven a…
The initial value problem of an integrable system, such as the Nonlinear Schr\" odinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent, supplemented by the $g$-function mechanism, is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface $\Rscr=\Rscr(x,t)$ in the spectr…
We consider a generalized nonlinear Schrödinger equation (NLS) with a power nonlinearity |ψ|^2μψ, of focusing type, describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the δpotential of strength αon the line, including as a special case (α=0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (α<0, representing a potential well) and repulsive (α>0, a potential barrier) …
We give the semiclassical asymptotic of barrier-top resonances for Schrödinger operators on ${\mathbb R}^{n}$, $n \geq 1$, whose potential is $C^{\infty}$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained. Our proof is based on a propagation of singularities theorem at a hyperbolic fixed point that we establish here. This last result refines a theorem of the same authors, and its proof follows another approach.
We construct multiple families of solitary standing waves of the discrete cubically nonlinear Schrödinger equation (DNLS) in dimensions $d=1,2$ and $3$. These states are obtained via a bifurcation analysis about the continuum (NLS) limit. One family consists {\it on-site symmetric} (vertex-centered) states; these are spatially localized solitary standing waves which are symmetric about any fixed lattice site. The other spatially localized states are {\it off-site symmetric}. Depending on the spatial dimension, these may be bond-centered, cell-centered, or face-centered. Finally, we show that t…
We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter $h \ll 1$. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intens…
The semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials is studied in a full scaling neighborhood D of the point of gradient catastrophe (x_0,t_0). This neighborhood contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions near the point of gradient catastrophe: i) each spike has the height 3|q_0(x_0,t_0,epsilon)| and uniform shape of the rational brea…
In the paper we revisit the basic problem of tunneling near a nondegenerate global maximum of a potential on the line. We reduce the semiclassical Schrödinger equation to a Weber normal form by means of the Liouville-Green transform. We show that the diffeomorphism which effects this stretching of the independent variable lies in the same regularity class as the potential (analytic or infinitely differentiable) with respect to both variables, i.e., space and energy. We then apply the Weber normal form to the scattering problem for energies near the potential maximum. In particular we obtain a …
We study the orbital stability and instability of single-spike bound states of semiclassical nonlinear Schrodinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develo…
We study a class of discrete focusing nonlinear Schr{ö}dinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the…
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