Multi-symplectic variational integrators for the Gross–Pitaevskii equations in BEC

In this paper, a variational integrator is constructed for Gross–Pitaevskii equations in Bose–Einstein condensate. The discrete multi-symplectic geometric structure is derived. The discrete mass and energy conservation laws are proved. The numerical tests show the effectiveness of the variational integrator, and the performance of the proved discrete conservation law.     

Solitary waves for the Klein–Gordon–Dirac model

We study the existence of solitary waves for non-autonomous Klein–Gordon–Dirac equations with a subcritical nonlinear term via variational methods. The problem is strongly indefinite and lacks compactness. To overcome these difficulties, we use the linking and concentration-compactness arguments.     

Automatic structures for subsemigroups of Baumslag–Solitar semigroups

This paper studies automatic structures for subsemigroups of Baumslag–Solitar semigroups (that is, semigroups presented by 〈x,y∣(yx ^m ,x ⁿ y)〉 where $m,n \in\mathbb {N}$ ). A geometric argument (a rarity in the field of automatic semigroups) is used to show that if m>n, all of the finitely generated subsemigroups of this semigroup are (right-) automatic. If m<n, all of its finitely generated subsemigroups are left-automatic. If m=n, there exist finitely generated subsemigroups that are not automatic. An appendix discusses the implications of these results for the theory of Malcev presentations. (A Malcev presentation is a special type of presentation for semigroups embeddable into groups.)     

Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.     

Stability of singular waves for Dullin–Gottwald–Holm equation

In this paper, the asymptotic stability of the solutions near the explicit singular waves of Dullin–Gottwald–Holm equation is studied based on the commutator estimate, the semi-group theory of linear operator and the Banach fixed point theorem.     

Traveling waves for a boundary reaction–diffusion equation

We prove the existence of a traveling wave solution for a boundary reaction–diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction–diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).     

Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.     

Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

Algebraic traveling waves for the generalized Newell–Whitehead–Segel equation

In this paper we provide the only possible algebraic traveling wave solutions for the celebrated general Newell–Whitehead–Segel equation.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.     

Travelling waves for a non-local Korteweg–de Vries–Burgers equation

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.     

An Adaptive Interacting Wang–Landau Algorithm for Automatic Density Exploration

While statisticians are well-accustomed to performing exploratory analysis in the modeling stage of an analysis, the notion of conducting preliminary general-purpose exploratory analysis in the Monte Carlo stage (or more generally, the model-fitting stage) of an analysis is an area that we feel deserves much further attention. Toward this aim, this article proposes a general-purpose algorithm for automatic density exploration. The proposed exploration algorithm combines and expands upon components from various adaptive Markov chain Monte Carlo methods, with the Wang–Landau algorithm at its heart. Additionally, the algorithm is run on interacting parallel chains—a feature that both decreases computational cost as well as stabilizes the algorithm, improving its ability to explore the density. Performance of this new parallel adaptive Wang–Landau algorithm is studied in several applications. Through a Bayesian variable selection example, we demonstrate the convergence gains obtained with interacting chains. The ability of the algorithm’s adaptive proposal to induce mode-jumping is illustrated through a Bayesian mixture modeling application. Last, through a two-dimensional Ising model, the authors demonstrate the ability of the algorithm to overcome the high correlations encountered in spatial models. Supplemental materials are available online.     

Solitary waves for the perturbed nonlinear Klein–Gordon equation

In this work, we study the perturbed nonlinear Klein–Gordon equation. We shall use the sech-ansätze method to derive the solitary wave solutions of this equation.     

Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

Conservation of energy for the Euler–Korteweg equations

In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.