Nonlocal Conservation Laws Research Articles

Local limit of nonlocal traffic models: Convergence results and total variation blow-up

Consider a nonlocal conservation law where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from 0. We also exhibit a counter-example showing that, if the initial datum attains the value 0, then there are severe obstructions to a convergence proof.

Existence and uniqueness results for a class of nonlocal conservation laws by means of a Lax–Hopf-type solution formula

We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.

Nonlocal Conservation Laws of PDEs Possessing Differential Coverings

In his 1892 paper, L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings, we show that this observation is of a quite general nature. We describe the procedures to construct such conservation laws and present a number of illustrative examples.

Optical soliton group invariant solutions by optimal system of Lie subalgebra with conservation laws of the resonance nonlinear Schrödinger equation

In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws.

Local discontinuous Galerkin method for a nonlocal viscous conservation laws

SummaryThe purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L2 norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.

Formulation of a maximum principle satisfying a numerical scheme for traffic flow models

We consider a non-local traffic flow model with Arrhenius look-ahead dynamics. In recent times, a maximum principle satisfying local conservation framework has been getting much attention, yet conventional numerical approximation scheme may lead to a breakdown of the maximum principle. In this paper, we construct a maximum principle satisfying a numerical scheme for a class of non-local conservation laws and present numerical simulations for the traffic flow models. The technique and the idea developed in this work are applicable to a large class of non-local conservation laws.

Nonlocal conservation laws, nonlocal symmetries and exact solutions of an integrable soliton equation

Abstract We compute nonlocal symmetries and obtain kink type soliton solutions to an integrable soliton equation. We construct a tree of nonlocally related partial differential equations (PDEs) of integrable soliton equation which consists of potential systems arising from conservation laws based method and inverse potential systems (IPS) from symmetry based method. We show that the integrable soliton equation admits two nonlocal symmetries among these one of which results from potential system and the other from locally related subsystem of IPS. We propose a systematic procedure to obtain exact solutions of a given PDE system using nonlocal symmetry which arises from IPS or its locally related subsystem. Finally, we obtain exact solutions to integrable soliton equation using nonlocal symmetries and discuss the physical behavior of the explicit solution graphically.

Wave breaking in a class of non-local conservation laws

For models describing water waves, Constantin and Escher [3]'s works have long been considered as the cornerstone method for proving wave breaking phenomena. Their rigorous analytic proof shows that if the lowest slope of flows can be controlled by its highest slope initially, then the wave-breaking occur for the Whitham-type equation. Since this breakthrough, there have been numerous refined wave-breaking results established by generalizing the kernel which describes the dispersion relation of water waves. In this work, from a rich class of non-local conservation laws, a Riccati-type system that governs the flow's gradient is extracted and investigated. The system's leading coefficient functions are allowed to change their values and signs over time as opposed to the ones in many of other previous works are fixed constants. The blow-up analysis is illustrated via the Whitham-type equation with nonlinear drift.

Nonlocally related systems, nonlocal symmetry reductions and exact solutions for one-dimensional macroscopic production model

We perform the classification of nonlocal symmetries and obtain some exact solutions to the system of nonlinear partial differential equations (PDEs), which describe a one-dimensional macroscopic production model. We obtain infinitely many nonlocal conservation laws, which lead to producing new potential systems. Further, we construct a tree of nonlocally related PDE systems using local conservation laws and symmetry-based method. The governing system of PDEs admits nonlocal symmetries from potential systems as well as inverse potential systems. Using these nonlocal symmetries, we obtain some implicit solutions and one arbitrary family of solutions that can be reduced to explicit form by some particular choice of the arbitrary function. Further, the physical behavior of different explicit solutions is shown graphically.

Conserved quantities in parity-time symmetric systems

Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics are determined by its global, local, or accidental symmetries. They were instrumental to advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems. Recently, a special class of these systems, called parity-time ($\mathcal{PT}$) symmetric systems, has been intensely explored for their remarkable properties that are absent in their closed counterparts. A complete characterization and observation of conserved quantities in these systems and their consequences is still lacking. Here we present a complete set of conserved observables for a broad class of $\mathcal{PT}$-symmetric Hamiltonians and experimentally demonstrate their properties using single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-order exceptional point, we measure its four conserved quantities and demonstrate their consequences. Our results spell out non-local conservation laws in non-unitary dynamics and provide key elements that will underpin self-consistent analysis of non-Hermitian quantum many-body systems that are forthcoming.

Quasi-integrable KdV models, towers of infinite number of anomalous charges and soliton collisions

We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for N −soliton solution. Some deformations of the KdV model are performed through the Riccati-type pseudo-potential approach, and infinite number of exact non-local conservation laws is provided using a linear formulation of the deformed model. In order to check the degrees of modifications of the charges around the soliton interaction regions, we compute numerically some representative anomalies, associated to the lowest order quasi-conservation laws, depending on the deformation parameters {ϵ1, ϵ2}, which include the standard KdV (ϵ1 = ϵ2 = 0), the regularized long-wave (RLW) (ϵ1 = 1, ϵ2 = 0), the modified regularized long-wave (mRLW) (ϵ1 = ϵ2 = 1) and the KdV-RLW (KdV-BBM) type (ϵ2 = 0, ≠ = {0, 1}) equations, respectively. Our numerical simulations show the elastic scattering of two and three solitons for a wide range of values of the set {ϵ1, ϵ2}, for a variety of amplitudes and relative velocities. The KdV-type equations are quite ubiquitous in several areas of non-linear science, and they find relevant applications in the study of General Relativity on AdS3, Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.

Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

This paper focuses on the numerical approximation of the solutions of a class of non-local systems in one space dimension, arising in traffic modeling. We propose alternative simple schemes by splitting the non-local conservation laws into two different equations, namely the Lagrangian and the remap steps. We provide some properties and estimates recovered by approximating the problem with the Lagrangian-antidiffusive remap (L-AR) scheme, and we prove the convergence to weak solutions in the scalar case. Finally, we show some numerical simulations illustrating the efficiency of the L-AR schemes in comparison with classical first- and second-order numerical schemes.

A non-local traffic flow model for 1-to-1 junctions

We present a model for a class of non-local conservation laws arising in traffic flow modelling at road junctions. Instead of a single velocity function for the whole road, we consider two different road segments, which may differ for their speed law and number of lanes (hence their maximal vehicle density). We use an upwind type numerical scheme to construct a sequence of approximate solutions, and we provide uniformL∞and total variation estimates. In particular, the solutions of the proposed model stay positive and below the maximum density of each road segment. Using a Lax–Wendroff type argument and the doubling of variables technique, we prove the well-posedness of the proposed model. Finally, some numerical simulations are provided and compared with the corresponding (discontinuous) local model.

Riccati-type pseudo-potentials, conservation laws and solitons of deformed sine-Gordon models

Deformed sine-Gordon (DSG) models ∂ξ∂ηw+ddwV(w)=0, with V(w) being the deformed potential, are considered in the context of the Riccati-type pseudo-potential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for the DSG model and an associated infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have recently been defined as quasi-integrable in the anomalous zero-curvature approach (Ferreira and Zakrzewski, 2011 [1]), possess new towers of infinite number of quasi-conservation laws. We compute numerically the first sets of non-trivial and independent charges (beyond energy and momentum) of the DSG model: the two third order conserved charges and the two fifth order asymptotically conserved charges in the pseudo-potential approach, and the first four anomalies of the new towers of charges, respectively. We consider kink-kink, kink-antikink and breather configurations for the Bazeia et al. potential Vq(w)=64q2tan2⁡w2(1−|sin⁡w2|q)2(q∈R), which contains the usual SG potential V2(w)=2[1−cos⁡(2w)]. The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.

Nonlocal conservation laws with time delay

This study considers nonlocal conservation laws in which the velocity depends nonlocally on the solution not in real time but in a time-delayed manner. Nonlocal refers to the fact that the velocity of the conservation law depends on the solution integrated over a specific area in space. In every model modelling human’s behavior a time delay as reaction/response time is crucial. We distinguish a so called nonlocal classical delay model where only the velocity of the conservation law is delayed from a more realistic nonlocal delay model where also a shift backwards in space is considered. For both models we show existence and uniqueness of the solutions and study their analytical properties. We also present a direct application in traffic flow modelling. We also show that for delay approaching zero, the solutions of the considered delayed models converge to the solutions of the non-delayed models in the proper topology. Finally, a comprehensive numerical study illustrating exemplary the impacts of delay and nonlocality are presented and compared with nonlocal models without delay, as well as the corresponding local models.

Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation

We study the Veronese web equation uyutx+λuxuty−(λ+1)utuxy=0 and using its isospectral Lax pair construct two infinite series of nonlocal conservation laws. In the infinite differential coverings associated to these series, we describe the Lie algebras of the corresponding nonlocal symmetries. Finally, we construct a recursion operator and explore its action on nonlocal shadows. The operator provides a new shadow which serves as a master-symmetry.

Integrability of Anti-Self-Dual Vacuum Einstein Equations with Nonzero Cosmological Constant: An Infinite Hierarchy of Nonlocal Conservation Laws

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.

On approximation of local conservation laws by nonlocal conservation laws

We show that for monotone initial datum the solution of nonlocal conservation laws converges to the entropy solution of the corresponding local conservation laws when the nonlocal reach tends to zero. This particularly covers the principle cases of conservation laws: shocks and rarefactions. The considered problem is addressed by studying the Entropy of the nonlocal conservation laws in the limit and by exploiting the semi-explicit solution formula developed in [30].

On invariant analysis and conservation laws for degenerate coupled multi-KdV equations for multiplicity $$l = 3$$ l = 3

The degenerate coupled multi-KdV equations for coupled multiplicity l=3 are studied. The equations also known as three fields Kaup-Boussinesq equations are considered for invariant analysis and conservation laws. The classical Lie's symmetry method is used to analyze the symmetries of equations. Based on Killing's form which is invariant of adjoint action, the full classification for Lie algebra is presented. Further, one-dimensional optimal group classification is used to obtain invariant solutions. Besides this, using general theorem proved by Ibragimov we find several nonlocal conservation laws for these equations. The conserved currents obtained in this work can be useful for the better understanding of some physical phenomena modeled by the underlying equations.

Maximum Principle Satisfying CWENO Schemes for Nonlocal Conservation Laws

Central WENO schemes are a natural candidate for higher-order schemes for non-local conservation laws, since the underlying reconstructions do not only provide single point values of the solution but a complete (high-order) reconstruction in every time step, which is beneficial to evaluate the integral terms. Recently, in [C. Chalons et al., SIAM J. Sci. Comput., 40(1), A288-A305], Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes have been proposed to obtain high-order approximations for a certain class of non-local conservation laws. In contrast to their schemes, the presented CWENO approach neither requires a very restrictive CFL condition (as the DG methods) nor an additional reconstruction step (as the FV-WENO schemes). Further, by making use of the well-known linear scaling limiter of [X. Zhang and C.-W. Shu, J. Comput. Phys., 229, p. 3091-3120], our CWENO schemes satisfy a maximum principle for suitable non-local conservation laws.