Nonlocal Equation Research Articles

Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal NLS equation via Riemann-Hilbert approach

Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal NLS equation via Riemann-Hilbert approach

The Soliton Solutions for Nonlocal Multi-Component Higher-Order Gerdjikov–Ivanov Equation via Riemann–Hilbert Problem

The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation.

On the separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions

We consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. This model describes phase separation in binary fluid mixtures. Given any global solution (whose existence and uniqueness are already known), we prove the so-called instantaneous and uniform separation property: any global solution with initial finite energy is globally confined (in the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document} metric) in the interval [-1+δ,1-δ]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1+\\delta ,1-\\delta ]$$\\end{document} on the time interval [τ,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\ au ,\\infty )$$\\end{document} for any τ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au >0$$\\end{document}, where δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} only depends on the norms of the initial datum, τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.

Dynamics of Rational and Lump-Soliton Solutions to the Reverse Space-Time Nonlocal Hirota-Maccari System

We mainly construct lump-soliton solutions of the (2 + 1)-dimensional reverse space-time Hirota-Maccari (HM) equation by using the KP hierarchy reduction method. Meanwhile, with the help of a long wave limit, rational solutions to nonlocal HM equation are studied. According to the appropriate parameter selections, these solutions can be divided into two types: line soliton solutions and lump-soliton solutions. Moreover, we obtain one-lump, two-lump and W-type soliton to the nonlocal HM equation. These new lump-soliton solutions expand the structure of nonlocal nonlinear systems and aid in the comprehension of physical phenomena.

On the Doubly Non-local Hele-Shaw–Cahn–Hilliard System: Derivation and 2D Well-Posedness

Starting from a classic non-local (in space) Cahn–Hilliard–Stokes model for two-phase flow in a thin heterogeneous fluid domain, we rigorously derive by mathematical homogenization a new effective mixture model consisting of a coupling of a non-local (in time) Hele-Shaw equation with a non-local (in space) Cahn–Hilliard equation. We then analyse the resulting model and prove its well-posedness. A key to the analysis is the new concept of sigma-convergence in thin heterogeneous domains allowing to pass to the homogenization limit with respect to the heterogeneities and the domain thickness simultaneously.

A perturbative approach to Hölder continuity of solutions to a nonlocal p-parabolic equation

We study local boundedness and Hölder continuity of a parabolic equation involving the fractional p-Laplacian of order s, with 0<s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s<1$$\\end{document}, 2≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le p < \\infty $$\\end{document}, with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side.

An equivalent formulation of Sonine condition

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

Abstract In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ &gt; 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ &gt; 0 fixed and the case ∫ R N u 2 = m &gt; 0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ &gt;}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m &gt; 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg–de Vries equation

This paper proposes a novel nonlocal Hirota system by adding constraints to the coupled Hirota equation, which can be transformed into a nonlocal complex modified Korteweg–de Vries (mKdV) equation with a Galilean transformation. Since the nonlocal Hirota equation and nonlocal complex mKdV equation can be converted into each other, we only need to solve one of them. Starting from the eigenfunctions of the Lax pair and the adjoint Lax pair of the nonlocal complex mKdV equation, the N-fold Darboux transformation (DT) of this equation is constructed. Subsequently, bright soliton solutions, double-hump soliton solutions, breather-like solutions under zero background, and multi-breather solutions under nonzero constant background are obtained for the nonlocal complex mKdV equation. Besides that, the interaction behaviors of the multi-soliton waves are also studied. There are some interesting phenomenons. The breather-II soliton wave, that can be seen as a combination of bright wave and dark wave, will appear. It does not appear alone but in the multi-breather solution.

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

Positive solutions of parameter-dependent nonlocal differential equations with convolution coefficients

In this paper we investigate a parameter-dependent nonlocal differential equations with convolution coefficients. Using the Birkhoff–Kellogg type theorem, existence of positive solutions is established. Under additional growth conditions, we obtain upper and lower bounds for the parameter. An example is also given to illustrate the main results.

Logarithmic or algebraic: Roughening of an active Kardar-Parisi-Zhang surface.

The Kardar-Parisi-Zhang (KPZ) equationsets the universality class for growing and roughening of nonequilibrium surfaces without any conservation law and nonlocal effects. We argue here that the KPZ equationcan be generalized by including a symmetry-permitted nonlocal nonlinear term of active origin that is of the same order as the one included in the KPZ equation. Including this term, the 2D active KPZ equationis stable in some parameter regimes, in which the interface conformation fluctuations exhibit sublogarithmic or superlogarithmic roughness, with nonuniversal exponents, giving positional generalized quasi-long-ranged order. For other parameter choices, the model is unstable, suggesting a perturbatively inaccessible algebraically rough interface or positional short-ranged order. Our model should serve as a paradigmatic nonlocal growth equation.

On the weak Harnack estimate for nonlocal equations

On the weak Harnack estimate for nonlocal equations

Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy

We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class Cloc1,α , for some constant α∈(0,1) . In addition, under suitable conditions on degree of the operator σ, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.

Finite element modeling of the rotational dynamics of a single magnetic particle in a strong magnetic field and liquid medium

We have performed a comparative computational study of magnetic particle alignment in a strong magnetic field and ambient viscous fluid using the first-principles Navier–Stokes equation as well as numerical modeling using the finite element method (FEM). FEM solution has been compared with the solution of the unsteady rotations of a magnetic particle in a viscous fluid during the alignment process described with nonlocal integro-differential equations (IDEs) for torque and angular velocity. The assumption of nonlocality comes from the history term of acceleration torque with a non-Basset kernel function, which has its origin in the presence of vortices in the flow of a particle fluid environment due to its unsteady rotation and ambient fluid inertia and friction. The flow vortices of the ambient fluid are explicitly shown in the solution of the FEM model. Moreover, the solution of the time evolution of the alignment angle and angular velocity for the FEM model are in good agreement with an IDE model which we have recently developed. The obtained results are justification for interchangeability of the FEM and IDE models, which may have important consequences for large-scale simulations of magnetic microparticles.

Temporal evolution of sound pulse in shallow water under conditions of horizontal refraction. Vertical modes and space-time horizontal rays

As is known, to describe horizontal refraction in a shallow water waveguide, the “vertical modes and horizontal rays” approach (Burridge &amp; Weinberg, 1974) is used, within which the two-dimensional eikonal equation for the modal amplitude includes the refractive index determined by the eigenvalues depending on the horizontal coordinates (x,y) and frequency (f). In other words, we have an effective two-dimensional inhomogeneous dispersive medium, to describe the propagation of signals in which, the authors propose to use the so-called space-time rays (Rytov, 1940, Connor &amp; Felsen, 1974, Hillion, 1993) applied for the horizontal plane (STHR). The set of STHRs in a given situation is constructed for each mode. Within the framework of this approach, it is possible to obtain the evolution of the pulses of each mode from the solution of the nonstationary eikonal equation in the horizontal plane or from the corresponding Hamilton-Jacobi equations, obtained using non-local integral wave equation. Examples of constructing STHR in some typical situations (in particular in a wedge) are considered, some specific effects during signal propagation are considered, in particular, the so-called the pulse front tilt, previously discovered in laser optics (Bor &amp; Rasz, 1985). Work was supported by ISF, grant 946/20.

Spherically symmetric evolution of self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

A nonlocal finite-dimensional integrable system related to the nonlocal mKdV equation

Предложена иерархия нелокального модифицированного уравнения Кортевега-де Фриза. Путем наложения связей получены нелокальные конечномерные интегрируемые системы со структурой Ли-Пуассона. С помощью преобразования координат нелокальные гамильтоновы системы Ли-Пуассона сводятся к нелокальным каноническим гамильтоновым системам, имеющим стандартную симплектическую структуру. С помощью нелокальных конечномерных интегрируемых систем получены параметрические решения нелокального модифицированного уравнения Кортевега-де Фриза и обобщенного нелокального нелинейного уравнения Шредингера. На основе теории Гамильтона получены координаты типа действие-угол и задачи обращения, связанные с гамильтоновыми системами Ли-Пуассона.

Unraveling forward and backward source problems for a nonlocal integrodifferential equation: A journey through operational calculus for Dzherbashian‐Nersesian operator

AbstractThis article primarily aims at introducing a novel operational calculus of Mikusiński's type for the Dzherbashian‐Nersesian operator. Using this calculus, we are able to derive exact solutions for the forward and backward source problems of a differential equation that features Dzherbashian‐Nersesian operator in time and intertwined with nonlocal boundary conditions. The initial condition is expressed in terms of Riemann‐Liouville integral. Solutions are presented using Mittag‐Leffler (ML) type functions. The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are developed.