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Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension d and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties. [ABSTRACT FROM AUTHOR]

The fractal calculus has gained more widespread attention in the last years. The fractal variational principle plays a major role in the fractal travelling wave theory of the fractal PDEs. This paper develops the fractal generalized variational principle (GVP) of the fractal Gardner equation by virtue of the Semi-inverse method (SIM) for the first time. On the other hand, we also discuss and verify the fractal GVP via the fractal two-scale transform (FTST) from another dimension field. The extracted fractal GVP shows the conservation laws through the energy form in the fractal space, and can be manipulated to explore the fractal solitary wave properties. [ABSTRACT FROM AUTHOR]

This paper investigates the (3 + 1) -dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the (3 + 1) -dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie's continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver's standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov's generalized theorem and quasi-self-adjointness condition. [ABSTRACT FROM AUTHOR]

In this paper, we studied the time fractional generalized (2 + 1) -dimensional Zakharov–Kuznetsov (ZK) equation with single-power law nonlinearity, which can be used to describe weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions. Through the fractional Lie transformation group idea, the symmetries, similarity reduction, one-parameter Lie transformation group and conservation laws of these considered equations were obtained. In addition, some new exact solutions were also constructed for them. These novel results provide a new perspective for studying the evolutionary mechanism of these considered models. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive study on the (4 + 1) -Fokas equation that includes new solutions and conservation laws of the equation. Specifically, for the first time, new shrunken-periodic and fascinating interactions on the bright soliton background have been investigated. Furthermore, a bright, dark, dipole soliton, tripole soliton, multipole soliton, periodic, doubly-periodic, damped-periodic, kink, lump, rogue wave, breather and their interactions have been attained and graphically illustrated. In addition, a significant interaction between four parabolic-periodic solitary waves is first detected and emphasized by harnessing 3D and contour plots. On the other hand, the latter has been compared to an experimentally-induced Moiré effect from a monitor screen camera capture. These outstanding results are achieved using the Lie symmetry approach. The multiple reduction process of the (4 + 1) -Fokas equation through the Lie algebra spanned elements, and the corresponding optimal system guarantees 23 solutions in general forms. To determine the physical meaning of the latter, an appropriate ansatz for the arbitrary functions and parameters has been proposed and listed in the corresponding tables, which leads to the various types of solitary wave solutions. Finally, the conserved vectors for the underlying equation are derived using the multiplier method. [ABSTRACT FROM AUTHOR]

In this paper, the integrability of a generalized (2 + 1)-dimensional equation is investigated by using the Bell polynomials and Hirota bilinear method. By appropriate transformation, the bilinear equation is constructed. Some soliton solutions are yielded via the Hirota bilinear method. Then the evolution and collision of the soliton solutions are discussed. Starting from the two-field condition, the bilinear Bäcklund transformation is derived. Linearization of the -polynomials-typed Bäcklund transformation leads to the Lax pair. Besides, infinite conservation laws are derived by constructing the Riccati-type equation and devergence-type equation. [ABSTRACT FROM AUTHOR]

In this paper, using the Lie symmetry method, we obtain the Lie symmetry group of the potential modified Korteweg–de Vries (potential mKdV) equation, and we construct the optimal system of one-dimensional subalgebras for the Lie symmetry algebra of this equation. Furthermore, similarity reductions related to the infinitesimal generators of the Lie symmetry group are obtained. Also, we construct new conservation laws for this equation by the scaling method. This method is algorithmic and based on the linear algebra tools and calculus of variations. The density of the conservation laws is constructed using the Euler operator and scaling symmetry of the equation and also the associated flux is calculated by the homotopy operator. This density-flux pair renders a conservation law for the potential mKdV equation. We specify the density-flux pairs of conservation laws in ranks 4, 8, 12, 16 and 20 for the potential mKdV equation. [ABSTRACT FROM AUTHOR]

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters. [ABSTRACT FROM AUTHOR]

The vector Lakshmanan–Porsezian–Daniel (vLPD) equations are mainly studied in this paper. The Darboux-dressing transformation (DDT) and infinitely-many conservation laws of the vLPD equations are constructed by the Lax pair. Furthermore, one soliton and bound-state solitons solutions of the vLPD equations are obtained by the DDT method. The obtained graphs can directly reflect the dynamic behavior of the aforementioned solutions. [ABSTRACT FROM AUTHOR]

In this paper, the conservation laws, significant symmetries' application, and traveling wave solutions are obtained for Burger–Fisher equation (BFE). Conservation laws have a great importance for partial and fractional differential equations and their solutions, especially in physics implementations. The conservation theorem and partial Noether approach are implemented for conservation laws for this equation, and the extended sinh-Gordon expansion method (esGEM) is presented for new solitary wave solutions. All obtained conservation laws are trivial conservation laws. The new and comprehensive solitary wave solutions of the equation by the esGEM are also obtained. [ABSTRACT FROM AUTHOR]

The convection–dispersion equation has always been a classic equation for studying pollutant migration models. There are certain deviations in scientific research because of the existence of the impurity of the medium and the nonsmooth boundary. In this paper, we introduced the one-dimensional convection–dispersion equation with fractal derivatives in fractal space, and established the fractal variational formula of the equation through the semi-inverse method. The fractal variational formula we have obtained can provide the conservation laws in an energy form in the fractal space and possible solution structures of the given equation. An analytical solution is obtained through the two-scale transform method and Laplace transform. [ABSTRACT FROM AUTHOR]