Rigorous Derivation of the Cubic NLS in Dimension One

We derive rigorously the one-dimensional cubic nonlinear Schrödinger equation from a many-body quantum dynamics. The interaction potential is rescaled through a weak-coupling limit together with a short-range one. We start from a factorized initial state, and prove propagation of chaos with the usual two-step procedure: in the former step, convergence of the solution of the BBGKY hierarchy associated to the many-body quantum system to a solution of the BBGKY hierarchy obtained from the cubic NLS by factorization is proven; in the latter, we show the uniqueness for the solution of the infinite BBGKY hierarchy.     

Exact Solutions of the Two-Dimensional Cubic-Quintic Nonlinear Schrdinger Equation with Spatially Modulated Nonlinearities

Applying the similarity transformation,we construct the exact vortex solutions for topological charge S ≥ 1 and the approximate fundamental soliton solutions for S = 0 of the two-dimensional cubic-quintic nonlinear Schrdinger equation with spatially modulated nonlinearities and harmonic potential.The linear stability analysis and numerical simulation are used to exam the stability of these solutions.In different profiles of cubic-quintic nonlinearities,some stable solutions for S ≥ 0 and the lowest radial quantum number n = 1 are found.However,the solutions for n ≥ 2 are all unstable.     

Asymptotic Analysis of the Spatially Homogeneous Boltzmann Equation: Grazing Collisions Limit

In the present work, we consider the asymptotic problem of the spatially homogeneous Boltzmann equation when almost all collisions are grazing, that is, the deviation angle $\theta $ of the collision is limited near zero (i.e., $\theta \le \epsilon $ ). We show that by taking the proper scaling to the cross-section which was used in [37], that is, assuming $$\begin{aligned} B^\epsilon ( v-v_{*},\sigma )=2(1-s)|v-v_*|^{\gamma }\epsilon ^{-3}\sin ^{-1}\theta \left( \frac{\theta }{\epsilon }\right) ^{-1-2s}\mathrm {1}_{\theta \le \epsilon }, \end{aligned}$$ where $\theta = \langle \theta ={\frac{\upsilon -\upsilon _*}{|\upsilon -\upsilon _*|}}.\sigma \rangle , $ the solution $f^\epsilon $ of the Boltzmann equation with initial data $f_0$ can be globally or locally expanded in some weighted Sobolev space as $$\begin{aligned} f^\epsilon = f+ O(\epsilon ), \end{aligned}$$ where the function $f$ is the solution of Landau equation, which is associated with the grazing collisions limit of Boltzmann equation, with the same initial data $f_0$ . This gives the rigorous justification of the Landau approximation in the spatially homogeneous case. In particular, if taking $\gamma =-3$ and $s=1-\epsilon $ in the cross-section $B^\epsilon $ , we show that the above asymptotic formula still holds and in this case $f$ is the solution of Landau equation with the Coulomb potential. Going further, we revisit the well-posedness problem of the Boltzmann equation in the limiting process. We show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution $f^\epsilon $ . Thanks to the weak convergence results on the grazing collisions limit in [37], in other words, we establish a unified framework to establish the well-posedness results for both Boltzmann and Landau equations.     

The Scaling Limit of the KPZ Equation in Space Dimension 3 and Higher

We study in the present article the Kardar–Parisi–Zhang (KPZ) equation

$$\begin{aligned} \partial _t h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta (t,x), \qquad (t,x)\in \mathbb {R}_+\times \mathbb {R}^d \end{aligned}$$

in \(d\ge 3\) dimensions in the perturbative regime, i.e. for \(\lambda >0\) small enough and a smooth, bounded, integrable initial condition \(h_0=h(t=0,\cdot )\). The forcing term \(\eta \) in the right-hand side is a regularized space-time white noise. The exponential of h—its so-called Cole-Hopf transform—is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer and Magnen (Commun Math Phys 162(1):85–121, 1994). Standard large deviation estimates for \(\eta \) make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution h may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards–Wilkinson model (\(\lambda =0\)) with renormalized coefficients \(\nu _{eff}=\nu +O(\lambda ^2),D_{eff}=D+O(\lambda ^2)\).

     

Singular Limit of a Reaction-Diffusion Equation with a Spatially Inhomogeneous Reaction Term

We study reaction-diffusion equations with a spatially inhomogeneous reaction term. If the coefficient of these reaction term is much larger than the diffusion coefficient, a sharp interface appears between two different phases. We show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations. In particular, we show that the same rich spatial patterns observed for a chemotaxis-growth model can be realized by a system without a drift term.     

Singular Limit of a p-Laplacian Reaction-Diffusion Equation with a Spatially Inhomogeneous Reaction Term

We study singular limit of a p-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term. The coefficient of the reaction term is much larger than the diffusion coefficient and sharp interfaces appear between two phases. We show by matched asymptotic expansions that the limit equation (interface equation) is a mean curvature flow with drift terms, similar to the case p = 2.     

On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice ${h\mathbb{Z}}$ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}$ with the fractional Laplacian (?Δ) ^α as dispersive symbol. In particular, we obtain that fractional powers ${\frac{1}{2} < \alpha < 1}$ arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian ?Δ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.     

Development of the Evans Wave Equation in the Weak-Field Limit: The Electrogravitic Equation

The Evans wave equation [1-3] is developed in the weak-field limit to give the Poisson equation and an electrogravitic equation expressing the electric field strength E in terms of the acceleration g due to gravity and a fundamental scalar potential ⁽⁰⁾ with the units of volts (joules per coulomb). The electrogravitic equation shows that an electric field strength can be obtained from the acceleration due to gravity, which in general relativity is non-Euclidean spacetime. Therefore an electric field strength can be obtained, in theory, from scalar curvature R. This inference is supported by recent experimental data from the patented motionless electromagnetic generator [5].     

Quantum Invariants for Solving the Time-Dependent Schrödinger Equation in One Dimension

Extending the work of Lewis and Leach on classical invariants for solving the classical equation of motion in one-dimensional system, the quantum invariants in polynomial form of momentum are obtained. The involved Hamiltonian is time-dependent and quadratic in momentum.     

The Vortex-Wave Equation with a Single Vortex as the Limit of the Euler Equation

In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti (Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland Delta Ser., Amsterdam, North-Holland, pp. 79–95, 1991), in the case of a single point vortex moving in an ambient vorticity. We consider a sequence of solutions for the Euler equation in the plane corresponding to initial data consisting of an ambient vorticity in L ¹ ∩ L ^∞ and a sequence of concentrated blobs which approach the Dirac distribution. We introduce a notion of a weak solution of the Vortex-Wave equation in terms of velocity (or primitive variables) and then show, for a subsequence of the blobs, the solutions of the Euler equation converge in velocity to a weak solution of the Vortex-Wave equation.     

The Mott Problem in One Dimension

In the α decay of a nucleus, the tracks left in the medium by the α particle are linear, even though its initial wave function is spherically symmetric. Understanding this quantum phenomenon has been called “the Mott problem”, ever since Mott’s fundamental paper on the subject (Mott in Proc. R. Soc. London Ser. A 126:79 1929). Here we study a one dimensional version of the Mott problem. The particle emitted in the decay is represented as a superposition of waves, one traveling to the left, the other to the right. The atoms with which the particle interacts are modeled as two level systems. The wave equation obeyed by the particle is taken to be the massless Dirac equation. For a certain space-time structure for the particle-atom interaction, it is possible to derive an explicit space-time solution for the entire system, for an arbitrary number of atoms. In the one dimensional solution, the coherent superposition of right and left-moving wave packets leaves behind tracks of excited atoms. The Mott problem on the nature of the tracks left behind is addressed using the reduced density matrix, defined by taking the trace over all particle degrees of freedom. It is found that the reduced density matrix is the incoherent sum of two terms, one involving excited atoms only on the right; the other involving excited atoms only on the left, implying that tracks will show excited atoms on one side or the other. In one dimension, tracks which involve excited atoms exclusively on one side or the other are the analog of straight tracks in three dimensions.     

The Bose Molecule in One Dimension

We give the Green function, momentum distribution, two-particle correlation function, and structure factor for the bound state of N indistinguishable bosons with an attractive delta-function interaction in one dimension, and an argument showing that this boson “molecule” has no excited states other than dissociation into separated pieces.     

Dirac Equation in the Spatially Flat Friedmann Model

The unitary transformation which diagonalizesthe field-free Dirac Hamiltonian in the spatially flatFriedmann-Robertson-Walker metric is analyzed, and apair of simultaneous first-order nonlinear differential equations is derived for the two parameters(two angles) that characterize the transformation. Theequations are solved approximately for a test particlewhose kinetic energy is small compared to its mass energy, and minimum-uncertainty wave packetsare constructcd from the solutions. It is found thatgeneral relativity limits the quantum mechanical spreadof the wave packets, but forces then to expand with the expanding space, as if they were embeddedin it. The massless Dirac equation is solved exactly forthe two-component neutrino spinor, and yieldsgeneralized nonspreading wave packets which display no quantum mechanical spread at all, but areconstrained to expand with the expanding space as theyfollow null geodesics.     

Long-Time Asymptotics for the Focusing NLS Equation with Time-Periodic Boundary Condition on the Half-Line

We consider the focusing nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time- periodic, of the form \({a{\rm e}^{\i\alpha} {\rm e}^{2\i\omega t}}\) . The goal of this paper is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem. We show that for \({\omega < -3a^2}\) the solution of the IBV problem has different asymptotic behaviors in different regions. In the region \({x > 4bt}\) , where \({b\mathop{:=} \sqrt{(a^2-\omega)/2} > 0}\) , the solution takes the form of the Zakharov-Manakov vanishing asymptotics. In a region of type \({4bt-\frac{N+1}{2a} {\rm log} t < x < 4bt}\) , where N is any integer, the solution is asymptotic to a train of asymptotic solitons. In the region \({4(b-a\sqrt2)t < x < 4bt}\) , the solution takes the form of a modulated elliptic wave. In the region \({0 < x < 4(b-a\sqrt2)t}\) , the solution takes the form of a plane wave.     

The Critical Attractive Random Polymer in Dimension One

A polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e ^– for every self-intersection and e ^/(2d) for every self-contact to the probability of an n-step simple random walk on ^d , where , >0 are parameters. It is known that for d=1 and > the chain collapses down to finitely many sites, while for d=1 and < it spreads out ballistically. Here we study for d=1 the critical case = corresponding to the collapse transition and show that the end-to-end distance runs on the scale _n = (log n)^–1/4. We describe the asymptotic shape of the accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size _n times a constant. Moreover, we derive the asymptotics of the partition function.     

Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exist a family of solutions to the Boltzmann equation globally in time for small Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number ε tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.