Local dynamics of an equation with large delay and distributed deviation of the space variable

We study the local dynamics in a neighborhood of an equilibrium state for an equation with delay in time and integral distribution in the space variable. In the close-to-critical cases depending on the relation between the small parameters, we construct special equations—quasinormal forms, describing the behavior of solutions to the initial problem.     

Extremal equilibria for reaction-diffusion equations in bounded domains and applications

We show the existence of two special equilibria, the extremal ones, for a wide class of reaction-diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation.     

Periodic problem for an evolution equation with a quadratic and a cubic nonlinearity

We study conditions for the existence of a solution of a periodic problem for a model nonlinear equation in the spatially multidimensional case and consider various types of large time asymptotics (exponential and oscillating) for such solutions. The generalized Kolmogorov-Petrovskii-Piskunov equation, the nonlinear Schrödinger equation, and some other partial differential equations are special cases of this equation. We analyze the solution smoothing phenomenon under certain conditions on the linear part of the equation and study the case of nonsmall initial data for a nonlinearity of special form. The leading asymptotic term is presented, and the remainder in the asymptotics of the solution is estimated in a spatially uniform metric.     

A numerical method for finding soliton solutions in nonlinear differential equations

An iterative method for finding soliton solutions to Korteweg-de Vries and sine-Gordon equations, and for nonlinear Schrodinger equations with cubic nonlinearity, is proposed. In addition, the existence of soliton solutions depending on control parameters is investigated for femtosecond pulse propagation problems in media with cubic nonlinearity. In contrast to other approaches to finding soliton solutions, the proposed method has a high convergence rate, can be applied to multidimensional problems, and does not require special selection of the initial approximation.     

Inverse Scattering Transform for the Nonlocal Reverse Space–Time Nonlinear Schrödinger Equation

Nonlocal reverse space–time equations of the nonlinear Schrödinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse scattering transform (IST) for rapidly decaying initial conditions was constructed. Here, we present the IST for the reverse space–time NLS equation with nonzero boundary conditions (NZBCs) at infinity. The NZBC problem is more complicated because the branching structure of the associated linear eigenfunctions is complicated. We analyze two cases, which correspond to two different values of the phase at infinity. We discuss special soliton solutions and find explicit one-soliton and two-soliton solutions. We also consider spatially dependent boundary conditions.     

A Numerical Method for Computing Time‐Periodic Solutions in Dissipative Wave Systems

A numerical method is proposed for computing time‐periodic and relative time‐periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi‐Rayleigh quotients, so that the resulting integrodifferential equation is for the time‐periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton‐conjugate‐gradient iterations. The proposed method applies to both stable and unstable time‐periodic solutions; its numerical accuracy is spectral; it is fast‐converging; its memory use is minimal; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto–Sivashinsky equation and the cubic‐quintic Ginzburg–Landau equation, whose time‐periodic or relative time‐periodic solutions with spatially periodic or spatially localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in the Appendices to illustrate the simple implementation of the proposed method.     

Planforms in two and three dimensions

Summary When solving systems of PDE with two space dimensions it is often assumed that the solution is spatially doubly periodic. This assumption is usually made in systems such as the Boussinesq equation or reaction-diffusion equations where the equations have Euclidean invariance. In this article we use group theoretic techniques to determine a large class of spatially doubly periodic solutions that are forced to existence near a steady-state bifurcation from a translation-invariant equilibrium.This type of bifurcation problem has been considered by many authors when studying a number of different systems of PDE. Typically, these studies focus at the beginning on equilibria that are spatially periodic with respect to a fixed planar lattice type-such as square or hexagonal. Our focus is different in that we attempt to find all spatially periodic equilibria that bifurcate on all lattices. This point of view leads to some technical simplifications such as being able to restrict to translation free irreducible representations.Of course, many of the types of solutions that we find are well-known-such as hexagon and roll solutions on a hexagonal lattice. This coordinated group theoretic approach does lead, however, to solutions which seem not to have been discussed previously (antisquare solutions on a square lattice) as well as to a more complete classification of the symmetry types of possible solutions. Moreover, our methods extend to triply periodic solutions of PDE with three spatial variables. Some of these results, namely those concerned with primitive cubic lattices, are presented here. The complete results on triply periodic solutions may be found in [6, 7].In honor of Klaus Kirchgässner on the occasion of his sixtieth birthdayResearch supported in part by NSF/DARPA (DMS-8700897) and by the Texas Advanced Research Program (ARP-1100).     

Dynamics of the logistic delay equation with a large spatially distributed control coefficient

The local dynamics of the logistic delay equation with a large spatially distributed control coefficient is asymptotically studied. The basic bifurcation scenarios are analyzed depending on the relations between the parameters of the equation. It is shown that the equilibrium states can lose stability even for asymptotically small values of the delay parameter. The corresponding critical cases can have an infinite dimension. Special nonlinear parabolic equations are constructed whose nonlocal dynamics determine the local behavior of solutions to the original boundary value problem.     

Generalization of a Relaxation Scheme for Systems of Forced Nonlinear Hyperbolic Conservation Laws with Spatially Dependent Flux Functions

A generalization of a finite difference method for calculating numerical solutions to systems of nonlinear hyperbolic conservation laws in one spatial variable is investigated. A previously developed numerical technique called the relaxation method is modified from its initial application to solve initial value problems for systems of nonlinear hyperbolic conservation laws. The relaxation method is generalized in three ways herein to include problems involving any combination of the following factors: systems of nonlinear hyperbolic conservation laws with spatially dependent flux functions, nonzero forcing terms, and correctly posed boundary values. An initial value problem for the forced inviscid Burgers' equation is used as an example to show excellent agreement between theoretical solutions and numerical calculations. An initial boundary value problem consisting of a system of four partial differential equations based on the two-layer shallow-water equations is solved numerically to display a more general applicability of the method than was previously known.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Asymptotically recurrent solutions ofβ-differential equations

In the paper methods from the theory of extensions of dynamical systems are used to studyβ-differential equations whose solutions possess the uniqueness property and depend continuously on the initial data and on the right-hand side of the equation. The Zhikov-Bronshtein theorems concerning asymptotically almost periodic solutions of ordinary differential equations are extended toβ-differential equations (in particular, to total differential equations). Along with asymptotic almost periodicity, we also consider asymptotic recurrence, weak asymptotic distality, and asymptotic distality. To the equations we associate dynamical systems generated by the space of the right-hand sides and the spaces of the solutions and of the initial data of solutions of the equation. Generally, the phase semigroups of the dynamical systems are not locally compact. Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 837–851, June, 2000.     

Numerical study of the asymptotics of the second Painlevé equation by a functional fitting method

The Painlevé equations arise as reductions of the soliton equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation and so on. In this study, we are concerned with numerical approximation of the asymptotics of solutions of the second Painlevé equation on pole‐free intervals along the real axis. Classical integrators such as high order Runge–Kutta schemes might be expensive to simulate oscillation, decay and blow‐up behaviours depending on initial conditions. However, a lower order functional fitting method catches all kinds of solutions even for relatively large step sizes. Copyright © 2013 John Wiley & Sons, Ltd.     

On difference equations with powers as solutions and their connection with invariant curves

We look for nonlinear systems of difference equations which have power functions as solutions and we find a connection to the theory of invariant curves and to self-reversed polynomials. In particular, we construct algebraic solutions of a special equation of invariant curves.     

Deformations of algebraic curves and integrable nonlinear evolution equations

A brief exposition of applications of the methods of algebraic geometry to systems integrable by the IST method with variable spectral parameters is presented. Usually, theta-functional solutions for these systems are generated by some deformations of algebraic curves. The deformations of algebraic curves are also related with theta-functional solutions of Yang-Mills self-duality equations which contain special systems with a variable spectral parameter as a special reduction. Another important situation in which the deformations of algebraic curves naturally occur is the KdV equation with string-like boundary conditions. Most important concrete examples of systems integrable by the IST method with variable spectral parameter having different properties within a framework of the behavior of moduli of underlying curves, analytic properties of the Baker-Akhiezer functions, and the qualitative behavior of the solutions, are vacuum axially symmetric Einstein equations, the Heisenberg cylindrical magnet equation, the deformed Maxwell-Bloch system, and the cylindrical KP equation.Dedicated to the memory of J.-L. Verdier     

Irregular modulation of non-linear Alfvén/Beltrami wave coupled with an ion-sound wave

A singularity of a system of differential equations may produce “intrinsic” solutions that are independent of initial or boundary conditions—such solutions represent “irregular behavior” uncontrolled by external conditions. In the recently formulated non-linear model of Alfvén/Beltrami waves [Commum Nonlinear Sci Numer Simulat 17 (2012) 2223], we find a singularity occurring at the resonance of the Alfvén velocity and sound velocity, from which pulses bifurcate irregularly. By assuming a stationary waveform, we obtain a sufficient number of constants of motion to reduce the system of coupled ordinary differential equations (ODEs) into a single separable ODE that is readily integrated. However, there is a singularity in the separable equation that breaks the Lipschitz continuity, allowing irregular solutions to bifurcate. Apart from the singularity, we obtain solitary wave solutions and oscillatory solutions depending on control parameters (constants of motion).     

Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term

We study crystalline driven curvature flow with spatially nonuniform driving force term. We assume special monotonicity properties of the driving term, which are motivated by our previous work on Berg's effect. We consider special initial data which we call ‘bent rectangles.’ We prove existence of solutions for a generic forcing term as well as generic subclass of bent rectangles. We show the initially flat facets may begin to bend, provided, loosely speaking, they are too large. Moreover, depending on the initial configuration we notice instantaneous loss of regularity of the moving curve.     

On the exact number of solutions of certain linearized equations

In this note we have revisited some of the results of Trachtenberg (On the cross-correlation functions of maximal linear sequences, Ph.D. thesis, University of Southern California, Los Angeles, 1970), which are directly related with the number of solutions of some special linearized polynomials over finite fields. In some cases we give improvements. Also, we give some results on the exact number of solutions of certain linearized equations depending on the coefficients of that equation.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation Via the Modified Simple Equation Method

In this article, the modified simple equation method (MSE) is used to acquire exact solutions to nonlinear evolution equations (NLEEs) namely the Zakharov- Kuznetsov Benjamin-Bona-Mahony equation and the Kadomtsov-Petviashvilli Benjamin- Bona-Mahony equation which have widespread usage in modern science. The MSE method is ascending and useful mathematical tool for constructing exact traveling wave solutions to NLEEs in the field of science and engineering. By means of this method we attained some significant solutions with free parameters and for special values of these parameters, we found some soliton solutions derived from the exact solutions. The solutions obtained in this article have been shown graphically and also discussed physically.