We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit, and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

More and more experimental evidence demonstrates that the slip boundary condition plays an important role in the study of nano- or micro-scale fluid. We propose a homogenization approach to study the effective slippage problem. We show that the effective slip length obtained by homogenization agrees with the results obtained by the traditional method in the literature for the simplest Stokes flow; then we use our approach to deal with two examples which seem quite hard by other analytical methods. We also include some numerical results to validate our analytical results.

We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leqslant d\leqslant 4$). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\geqslant 2$. As a consequence, we obtain global strong solutions (for $d\geqslant 2$) with infinitely growing high Sobolev norms $H^{s}$.

In this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforthscheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.

We define and investigate, via numerical analysis, a one dimensional toy-model of a cloud chamber. An energetic quantum particle, whose initial state is a superposition of two identical wave packets with opposite average momentum, interacts during its evolution and exchanges (small amounts of) energy with an array of localized spins. Triggered by the interaction with the environment, the initial superposition state turns into an incoherent sum of two states describing the following situation: or the particle is going to the left and a large number of spins on the left side changed their states, or the same is happening on the right side. This evolution is reminiscent of what happens in a cloud chamber where a quantum particle, emitted as a spherical wave by a radioactive source, marks its passage inside a supersaturated vapour-chamber in the form of a sequence of small liquid bubbles arranging themselves around a possible classical trajectory of the particle.

We investigate the decay for |x|→∞ of weak Sobolev-type solutions of semilinear nonlocal equations Pu = F(u). We consider the case when P = p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ), and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation

for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

We compare six fixed-stepsize fourth-order numerical methods for a number of test problems described by a system of coupled Korteweg–de Vries equations. Particular attention is paid to the ability of these methods to preserve fixed points (solitary waves) and the invariants of the system, and establishing to what extent the conservation of integral invariants is indicative of the solution error for these methods.

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

We introduce a novel solution concept, denoted ${\it\alpha}$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa–Holm system on the line with vanishing asymptotics. All the ${\it\alpha}$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

This paper presents a model of a 1D–1D dynamic multi-structure, supporting propagation of a transition wave. It is used to explain the recent phenomenon of the collapse of the San Saba bridge. An analytical model is supplied with illustrative numerical simulations.

We study the propagation of wave packets for a one-dimensional system of two coupled Schrödinger equations with a cubic nonlinearity, in the semiclassical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an avoided crossing: at one given point, the gap between them reduces as the semiclassical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point the nonlinearity’s role is negligible at leading order, and the transition probabilities can be computed with the linear Landau–Zener formula.

This paper is concerned with the Cauchy problem for a nonlinear Schrödinger equation with a harmonic potential and exponential growth nonlinearity in two space dimensions. In the defocusing case, global well-posedness is obtained. In the focusing case, existence of nonglobal solutions is discussed via potential-well arguments.

In this paper, we consider the existence of multi-soliton structures for the nonlinear Klein–Gordon (NLKG) equation in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{R}^{1+d}$ . We prove that, independently of the unstable character of NLKG solitons, it is possible to construct a $N$ -soliton family of solutions to the NLKG equation, of dimension $2N$ , globally well defined in the energy space $H^1\times L^2$ for all large positive times. The method of proof involves the generalization of previous works on supercritical Nonlinear Schrödinger (NLS) and generalized Korteweg–de Vries (gKdV) equations by Martel, Merle, and the first author [R. Côte, Y. Martel and F. Merle, Rev. Mat. Iberoam. 27 (1) (2011), 273–302] to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis, Shatah, and Strauss [J. Funct. Anal. 74 (1) (1987), 160–197] and Duyckaerts and Merle [Int. Math. Res. Pap. IMRP (2008), Art ID rpn002] to the case of boosted solitons, and provide new solutions to be studied using the recent work of Nakanishi and Schlag [Zurich Lectures in Advanced Mathematics, vi+253 pp (European Mathematical Society (EMS), Zürich, 2011)] theory.

In this paper, we obtain the well posedness of the linear stochastic Korteweg–de Vries equation by the Galerkin method, and then establish the Carleman estimate, leading to the unique continuation property (UCP) for the linear stochastic Korteweg–de Vries equation. This UCP cannot be obtained from the classical Holmgren uniqueness theorem.

We establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

The aim of this paper is to obtain a new unique continuation property (UCP) for the Korteweg–de Vries equation posed on a finite interval. Compared with the previous UCP, we need fewer conditions on the solution. For this purpose, we have to establish a global Carleman estimate for the Korteweg–de Vries equation.

We propose and study a number of layer methods for Navier‒Stokes equations (NSEs) with spatial periodic boundary conditions. The methods are constructed using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the layer methods are nevertheless deterministic.

A second-order in time finite-difference scheme using a modified predictor–corrector method is proposed for the numerical solution of the generalized Burgers–Fisher equation. The method introduced, which, in contrast to the classical predictor–corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.

We prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrödinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces and H10 of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r(2−N)/2 and of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.