Nonlocal Evolution Equations Research Articles

A direct approach for solving the cubic Szegö equation

AbstractWe study the cubic Szegö equation which is an integrable nonlinear nondispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues.

A BDF2 ADI difference scheme for a three-dimensional nonlocal evolution equation with multi-memory kernels

A BDF2 ADI difference scheme for a three-dimensional nonlocal evolution equation with multi-memory kernels

On stability and regularity for subdiffusion equations involving delays

We study a class of nonlocal evolution equations involving time-varying delays which is employed to depict subdiffusion processes. The global solvability, stability and regularity are shown by using the resolvent theory, nonlocal Halanay inequality, fixed point argument and embeddings of fractional Sobolev spaces. Our result is applied to the nonlocal Fokker–Planck model with nonlinear force fields.

Global regularity for critical SQG in bounded domains

AbstractWe prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

Non-local time evolution equation with singular integral and its application to traffic flow model

We consider an integro-differential equation model for traffic flow, which extends the Burgers equation model. To derive exact solutions to the equation, we first examine the features of integrable integro-differential equations and find that they are obtained through the residue formula from integrable equations in complex domains. Using this approach, we construct several new integrable equations with double singular integrals and elliptic singular integrals. Then, we discuss the traffic model and show that it exhibits interactions between free-flow and congested regions, as well as the deadlock phenomenon, depending on the non-locality parameter.

Non-autonomous fractional nonlocal evolution equations with superlinear growth nonlinearities

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator −A(⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space Xpυ (0⩽υ<1, 1<p<∞).

Total controllability of nonlocal semilinear functional evolution equations with non-instantaneous impulses

Total controllability of nonlocal semilinear functional evolution equations with non-instantaneous impulses

Theta-type convolution quadrature OSC method for nonlocal evolution equations arising in heat conduction with memory

Theta-type convolution quadrature OSC method for nonlocal evolution equations arising in heat conduction with memory

CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

Traveling Waves for a Sign-Changing Nonlocal Evolution Equation with Delayed Nonlocal Response

Traveling Waves for a Sign-Changing Nonlocal Evolution Equation with Delayed Nonlocal Response

Smooth dynamics properties of a non-local evolution equation

Smooth dynamics properties of a non-local evolution equation

Dissipative light bullets in a doped and weakly nonlocal optical fiber

The letter introduces an extended (3+1)-dimensional [(3+1)D] nonlocal cubic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in optical fiber amplifiers under the interplay between dopants and a spatially nonlocal nonlinear response. The model equation includes the effects of fiber dispersion, linear gain, nonlinear loss, fiber nonlinearity, atomic detuning, linear and nonlinear diffractive transverse effects, and nonlocal nonlinear response. A system of coupled ordinary differential equations for the amplitude, temporal, and spatial pulse widths and position of the pulse maximum, unequal wavefront curvatures, chirp parameters, and phase shift is derived using the variational technique. A stability criterion is established, where a domain of dissipative parameters for stable steady-state solutions is found. Direct integration of the proposed nonlocal evolution equation is performed, which allows us to investigate the evolution of the Gaussian beam along a doped nonlocal optical fiber, showing stable self-organized dissipative spatiotemporal light bullets.

Approximate solvability method for nonlocal impulsive evolution equation

Abstract In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of C 0-semigroup.

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

A class of stochastic nonlocal evolution equations with nonlocal initial conditions

In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in [Formula: see text] under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms.

Nonlocal partial fractional evolution equations with state dependent delay

In this work, we propose sufficient conditions guaranteeing an existence result of mild solutions by using the nonlinear Leray-Schauder alternative in Banach spaces combined with the semigroup theory for the class of Caputo partial semilinear fractional evolution equations with finite state-dependent delay and nonlocal conditions.

A bilateral preconditioning for an L2-type all-at-once system from time-space non-local evolution equations with a weakly singular kernel

A bilateral preconditioning for an L2-type all-at-once system from time-space non-local evolution equations with a weakly singular kernel

Non-Local Vectorial Internal Variables and Generalized Guyer-Krumhansl Evolution Equations for the Heat Flux.

In this paper, we ask ourselves how non-local effects affect the description of thermodynamic systems with internal variables. Usually, one assumes that the internal variables are local, but that their evolution equations are non-local, i.e., for instance, that their evolution equations contain non-local differential terms (gradients, Laplacians) or integral terms with memory kernels. In contrast to this typical situation, which has led to substantial progress in several fields, we ask ourselves whether in some cases it would be convenient to start from non-local internal variables with non-local evolution equations. We examine this point by considering three main lengths: the observation scale R defining the elementary volumes used in the description of the system, the mean free path l of the microscopic elements of the fluid (particles, phonons, photons, and molecules), and the overall characteristic size L of the global system. We illustrate these ideas by considering three-dimensional rigid heat conductors within the regime of phonon hydrodynamics in the presence of thermal vortices. In particular, we obtain a generalization of the Guyer-Krumhansl equation, which may be of interest for heat transport in nanosystems or in systems with small-scale inhomogeneities.

Global Existence and Finite-Time Blow-Up for a Nonlinear Nonlocal Evolution Equation

We study the Cauchy problem for a nonlinear nonlocal evolution equation arising in the study of oceanic flows in equatorial regions. We present a well-posedness result and show that while some initial data develop into solutions that exist for all times, others lead to blow-up in finite time.

Stability analysis for a class of semilinear nonlocal evolution equations

Stability analysis for a class of semilinear nonlocal evolution equations