Nonlocal Conservation Laws Research Articles

Lagrangian-Eulerian Approach for Nonlocal Conservation Laws

Lagrangian-Eulerian Approach for Nonlocal Conservation Laws

Nonlocal symmetry classifications, nonlocal symmetry reductions, exact solutions and conservation laws of one-dimensional nonlinear Vakhnenko equation

In this paper, the categorization of nonlocal symmetries of the (1+1)-dimensional nonlinear Vakhnenko equation is presented. The governing equation is turned into a set of invertibly equivalent partial differential equations (PDE) via the invertible transformation of the canonical coordinates. The nonlocal symmetries of the aforementioned PDE were derived by establishing an inverse potential system and a locally related subsystem of the system of invertibly equivalent PDEs. In addition, using similarity variables and group invariant solutions connected to nonlocal symmetry, the exact solution of the nonlinear Vakhnenkov equation is achieved. Finally, the conservation laws for the nonlinear Vakhnenko equation were developed using the multiplier method.

Nonlocal symmetry, exact solutions and conservation laws of the (1+1)-dimensional Levi equation

Nonlocal symmetry, exact solutions and conservation laws of the (1+1)-dimensional Levi equation

Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution.

Modified AKNS model, Riccati-type pseudo-potential approach and infinite towers of quasi-conservation laws

In this paper, a dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact nonlocal conservation laws are uncovered in a linear formulation of the system. It is shown that certain modifications of the nonlinear Schrödinger model (MNLS) can be obtained through a reduction process starting from the MAKNS model. So, the novel infinite sets of quasi-conservation laws and related anomalous charges are constructed by an unified and rigorous approach based on the Riccati-type pseudo-potential method, for the standard NLS and modified MNLS cases, respectively. The nonlocal properties, the complete list of towers of infinite number of anomalous charges and the (nonlocal) exact conservation laws of the quasi-integrable systems, such as the deformed Bullough–Dodd, Toda, KdV and SUSY sine-Gordon systems, can be studied in the framework presented in this paper. Our results may find many applications since the AKNS-type system arises in several branches of nonlinear physics such as Bose–Einstein condensation, superconductivity and soliton turbulence.

Cahn–Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions

We consider a doubly nonlocal nonlinear parabolic equation which describes phase segregation of a binary system subject to weak-to-weak interactions [Gal, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018)]. The proposed model reduces to the classical Cahn–Hilliard equation under certain conditions. We establish well-posedness results (based on regular and nonregular mild solutions) along with regularity and long-time results in terms of finite-dimensional attractors. Then we also establish the convergence of (certain) mild solutions to single steady states as time goes to infinity. These results are also supplemented by a handful of (two-dimensional) numerical experiments displaying phase-segregation phenomena with interesting interface morphologies, depending on various choices of the interaction kernels (i.e., Gaussian, logarithmic, Riesz and bimodal potentials). We develop a stable numerical scheme which is able to control the computations under the effect of the double nonlinear convolutions.

Nonlocal scalar conservation laws with discontinuous flux

<abstract><p>We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with nonlocal flux. We approximate the problem adding a viscosity term and we provide $ {{\bf{L}}^\infty} $ and BV estimates for the approximate solutions. We use the doubling of variable technique to prove the stability with respect to the initial data from the entropy condition.</p></abstract>

Lie analysis, conserved vectors, nonlinear self-adjoint classification and exact solutions of generalized $ \left(N+1\right) $-dimensional nonlinear Boussinesq equation

<abstract><p>In this article, the generalized $ \left(N+1\right) $-dimensional nonlinear Boussinesq equation is analyzed via Lie symmetry method. Lie point symmetries of the considered equation and accompanying invariant groups are computed. After transforming the equation into a nonlinear ordinary differential equation (ODE), analytical solutions of various types are obtained using the $ \left(G^\prime/G, 1/G\right) $ expansion method. The concept of nonlinear self-adjointness is used in order to determine nonlocal conservation laws of the equation in lower dimensions. By selecting the appropriate parameter values, the study provides a graph of the solutions to the equation under study.</p></abstract>

Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

On analytic series solutions and conserved fluxes of the time fractional (2+1)-dimensional Burger's system via invariant approach

The article investigates some new properties such as invariant transformations, conserved vectors, and closed-form series solutions of (2+1) dimensional nonlinear coupled Burger system, a model of the evolution of the scaled volume concentration or sedimentation of the two kinds of particles in fluid suspensions of colloids, under the effect of gravity, with fractional temporal evolution by the Lie symmetry method. It is worth noting that the series solutions of (2+1) or higher-dimensional system is not much explored. Moreover, nonlocal conservation laws, one of the important aspects of symmetries, are constructed by the generalized Noether theorem.

Einstein-æther scalar–tensor cosmology

We propose an Einstein-æther scalar–tensor cosmological model. In particular, in the scalar–tensor Action Integral, we introduce the æther field with æther coefficients to be functions of the scalar field. This cosmological model extends previous studies on Lorentz-violating theories. For a spatially flat Friedmann–Lemaître–Robertson–Walker background space, we write the field equations which are of second order with dynamical variables the scale factor and the scalar field. The physical evolution of the field equations depends upon three unknown functions which are related to the scalar–tensor coupling function, the scalar field potential, and the æther coefficient functions. We investigate the existence of analytic solutions for the field equations and the integrability properties according to the existence of linear in the momentum conservation laws. We define a new set of variables in which the dynamical evolution depends only upon the scalar field potential. Furthermore, the asymptotic behavior and the cosmological history are investigated where we find that the theory provides inflationary eras similar to that of scalar–tensor theory but with Lorentz-violating terms provided by the æther field. Finally, in the new variables, we found that the field equations are integrable due to the existence of nonlocal conservation laws for arbitrary functional forms of the three free functions.

On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments seem to suggest that the solutions of the nonlocal problems converge to the entropy admissible solution of the conservation law in the singular local limit. However, recent analytical results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii) convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytical results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that Lax–Friedrichs type schemes may provide the wrong intuition and erroneously suggest that the solutions of the nonlocal problems converge to the entropy admissible solution of the conservation law in cases where this is ruled out by analytical results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide an intuition more in accordance with the analytical results.

Asymptotically compatible approximations of linear nonlocal conservation laws with variable horizon

AbstractWe consider a quadrature‐based finite difference discretization of one‐dimensional scalar linear nonlocal conservation laws. The range of nonlocal interactions is allowed to vary in the spatial domain. We are particularly concerned with the convergence of the discrete approximation both in the nonlocal setting and in the local limit as the horizon parameter approaches zero. We present the first complete proof of the convergence of numerical discretization to both the nonlocal regime and the local limit of all feasible kernels, which in particular, establishes the asymptotically compatibility of the numerical scheme. We also present numerical results to demonstrate the effect of the variable horizon on the wave propagation described by the nonlocal model.

Nonlinear self-adjointness and conservation laws for the modified dissipative hyperbolic geometric flow equation

The nonlinear self-adjointness of the modified dissipative hyperbolic geometric flow equation is studied. Furthermore, based on the method of Ibragimov, we can obtain conservation laws and nonlocal conservation laws for the modified dissipative hyperbolic geometric flow equation.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

Singular limits with vanishing viscosity for nonlocal conservation laws

We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal impact and the viscosity parameter simultaneously tend to zero (under a suitable balance condition), the solution of the nonlocal problem converges to the entropy solution of the corresponding local conservation law. The key ideas of our proof are to observe that the nonlocal term satisfies a third-order equation with diffusive and dispersive effects and to deduce a suitable energy estimate on the nonlocal term. The convergence result is then based on the compensated compactness theory.

Kinetic Solutions for Nonlocal Stochastic Conservation Laws

Abstract This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation.

Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.

Second-Order Accurate TVD Numerical Methods for Nonlocal Nonlinear Conservation Laws

We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced by Du, Huang, and LeFloch. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper from Du, Huang, and LeFloch concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that, if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.

Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs

Dynamical PDEs that have a spatial divergence form possess conservation laws that involve an arbitrary function of time. In one spatial dimension, such conservation laws are shown to describe the presence of an x -independent source/sink; in two and more spatial dimensions, they are shown to produce a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing non-local conservation laws and symmetries to be found for a given dynamical PDE. This type of potential system has a different form and different gauge freedom compared to potential systems that arise from ordinary conservation laws. Second, when a topological charge arises from a conservation law whose conserved density is non-trivial off of solutions to the dynamical PDE, then this relation yields a constraint on initial/boundary data for which the dynamical PDE will be well posed. Several examples of nonlinear PDEs from applied mathematics and integrable system theory are used to illustrate these results.