Strong and weak orders in averaging for SPDEs

We show an averaging result for a system of stochastic evolution equations of parabolic type with slow and fast time scales. We derive explicit bounds for the approximation error with respect to the small parameter defining the fast time scale. We prove that the slow component of the solution of the system converges towards the solution of the averaged equation with an order of convergence 1/2 in a strong sense–approximation of trajectories–and 1 in a weak sense–approximation of laws. These orders turn out to be the same as for the SDE case.     

KP reductions and various soliton solutions to the Fokas–Lenells equation under nonzero boundary condition

In this paper, we clarify the connection of the Fokas–Lenells (FL) equation to the Kadomtsev–Petviashvili (KP)–Toda hierarchy by using a set of bilinear equations as a bridge and confirm multidark soliton solution to the FL equation previously given by Matsuno (J. Phys. A 2012 45 (475202). We also show that the set of bilinear equations in the KP–Toda hierarchy can be generated from a single discrete KP equation via Miwa transformation. Based on this finding, we further deduce the multibreather and general rogue wave solutions to the FL equation. The dynamical behaviors and patterns for both the breather and rogue wave solutions are illustrated and analyzed.     

Complete classification of local conservation laws for generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation

In this paper, we consider nonlinear multidimensional Cahn–Hilliard and Kuramoto–Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these two equations to which we refer to as the generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation. For an arbitrary number of spatial independent variables, we present a complete list of cases when the latter equation admits nontrivial local conservation laws of any order, and for each of those cases, we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. In particular, we show that the original Kuramoto–Sivashinsky equation admits no nontrivial local conservation laws, and find all nontrivial local conservation laws for the Cahn–Hilliard equation.     

Fisher–KPP equation on the Heisenberg group

In this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow-up solutions. Moreover, we extend the obtained results to the time-fractional Fisher–KPP equation on the Heisenberg group.     

Miura transformation for the “good” Boussinesq equation

It is well known that each solution of the modified Korteveg–de Vries (mKdV) equation gives rise, via the Miura transformation, to a solution of the Korteveg–de Vries (KdV) equation. In this work, we show that a similar Miura-type transformation exists also for the “good” Boussinesq equation. This transformation maps solutions of a second-order equation to solutions of the fourth-order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann–Hilbert problems and this is in fact how we construct the new transformation.     

Some exact and approximate solutions to a generalized Maxwell–Cattaneo equation

The simple heat conduction equation in one-space dimension does not have the property of a finite speed for information transfer. A partial resolution of this difficulty can be obtained within the context of heat conduction by the introduction of a partial differential equation (PDE ) called the Maxwell–Cattaneo (M-C) equation, elsewhere called the damped wave equation, a special case of the telegraph equation. We construct a generalization to the M-C equation by allowing the relaxation time parameter to be a function of temperature. In the balance of the paper, we present a variety of special exact and approximate solutions to this nonlinear PDE .     

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

Stochastic averaging principle is a powerful tool for studying qualitative analysis of multiscale stochastic dynamical systems. In this paper, we will establish an averaging principle for stochastic reaction‐diffusion‐advection equations with slow and fast time scales. Under suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation.     

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.     

Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case

We consider a system of ordinary first-order differential equations. The right-hand sides of the system are proportional to a small parameter and depend almost periodically on fast time and periodically on slow time. With this system, we associate the system averaged over fast time. We assume that the averaged system has a structurally unstable periodic solution. We prove a theorem on the existence and stability of almost periodic solutions of the original system. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 451–456, March, 1998.     

Cramér's asymptotics in systems with fast and slow motions

This paper is devoted to the averaging principle for stochastic systems with slow and intermixing fast motions. Here we (i) prove the existence of the Cramér type asymptotics for the probabilities of large deviations from an averaged motion, which implies the central limit theorem, and (ii) develop an analytic procedure for computation of this asymptotics. We use general apparatus of superregular perturbations of fiber ergodic semigroups to investigate two systems in the same way. The first of them is a cascade in which slow motions are determined by a vector field depending both on slow and fast variables, and fast motions compose a Markov chain depending on the slow variable. The second is a process defined by a system of two stochastic differential equations.     

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

In this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two-parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t.     

On a modification of the FitzHugh-Nagumo neuron model

A singularly perturbed system of ordinary differential equations with a fast and a slow variable is proposed, which is a modification of the well-known FitzHugh-Nagumo model from neuroscience. The existence and stability of a nonclassical relaxation cycle in this system are studied. The slow component of the cycle is asymptotically close to a discontinuous function, while the fast component is a δ-like function.     

An application of a qd-type discrete hungry Lotka–Volterra equation over finite fields to a decoding problem

In this paper, the decoding problem for multiple Bose–Chaudhuri–Hocquenghem (BCH)-Goppa codes over the same finite field is investigated. A new iterative decoding algorithm is proposed based on the quotient difference (qd)-type discrete hungry Lotka–Volterra equation over finite fields. Compared with certain existing algorithms, the proposed algorithm manifests its advantage in computational complexity. A few of examples are presented to demonstrate its efficiency.     

Symmetries of the DΔmKP hierarchy and their continuum limits

In the recent paper [Stud. App. Math. 147 (2021), 752], squared eigenfunction symmetry constraint of the differential-difference modified Kadomtsev–Petviashvili (DΔmKP) hierarchy converts the DΔmKP system to the relativistic Toda spectral problem and its hierarchy. In this paper, we introduce a new formulation of independent variables in the squared eigenfunction symmetry constraint, under which the DΔmKP system gives rise to the discrete spectral problem and a hierarchy of the differential-difference derivative nonlinear Schrödinger equation of the Chen–Lee–Liu type. In addition, by introducing nonisospectral flows, two sets of symmetries of the DΔmKP hierarchy and their algebraic structure are obtained. We then present a unified continuum limit scheme, by which we achieve the correspondence of the mKP and the DΔmKP hierarchies and their integrable structures.     

The KP limit of a reduced quantum Euler–Poisson equation

In this paper, we consider the derivation of the Kadomtsev–Petviashvili (KP) equation for cold ion-acoustic wave in the long wavelength limit of a two-dimensional reduced quantum Euler–Poisson system under different scalings for varying directions in the Gardner–Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter $H>0$. The KP-I is derived for $H>2$, KP-II for , and the dispersiveless KP (dKP) equation for the critical case $H=2$. The rigorous proof for these limits is given in the well-prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner–Morikawa transform. The results can be generalized in several directions.     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

Random attractor of stochastic Zakharov lattice system

In this paper, we consider a lattice system of stochastic Zakharov equation with white noise. We first show that the solutions of the system determine a continuous random dynamical system with random absorbing set. And then we prove the random  asymptotic compact on the random absorbing set. Finally, we obtain the existence of a random attractor for the system.     

A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation

In the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step-type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.     

Dynamics for a quantum parliament

In this paper, we propose a dynamical approach based on the Gorini–Kossakowski–Sudarshan–Lindblad equation for a problem of decision making. More specifically, we consider what was recently called a quantum parliament, asked to approve or not a certain law, and we propose a model of the connections between the various members of the parliament, proposing in particular some special form of the interactions giving rise to a collaborative or noncollaborative behavior.     

Doubly localized two-dimensional rogue waves generated by resonant collision in Maccari system

In this paper, the Maccari system is investigated, which is viewed as a two-dimensional extension of nonlinear Schrödinger equation. We derive doubly localized two-dimensional rogue waves on the dark solitons of the Maccari system with Kadomtsev–Petviashvili hierarchy reduction method. The two-dimensional rogue waves include line segment rogue waves and rogue-lump waves, which are localized in two-dimensional space and time. These rogue waves are generated by the resonant collision of rational solitary waves and dark solitons, the whole process of transforming elastic collision into resonant collision is analytically studied. Furthermore, we also discuss the local characteristics and asymptotic properties of these rogue waves. Simultaneously, the generating conditions of the line segment rogue wave and rogue-lump wave are also given, which provides the possibility to predict rogue wave. Finally, a new way to obtain the high-order rogue waves of the nonlinear Schrödinger equation are given by proper reduction from the semi-rational solutions of the Maccari system.