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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]

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any T 4 configurations. Here, we continue that study by showing that the differential inclusion does not contain T 5 configurations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a simple, purely stochastic model characterized by two conserved quantities (mass density a and energy density h) which is known to display a condensation transition when h > 2 a 2 : in the localized phase a single site hosts a finite fraction of the whole energy. Its equilibrium properties in the thermodynamic limit are known and in a recent paper [G. Gotti, S. Iubini and P. Politi, Finite-size localization scenarios in condensation transitions, Phys. Rev. E 103 (2021) 052133] we studied the transition for finite systems. Here, we analyze finite-size effects on the energy distribution and on the relaxation dynamics, showing that extremely large systems should be studied in order to observe the asymptotic distribution and even larger systems should be simulated in order to observe the expected relaxation dynamics. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is a more general discrete 2 × 2 matrix spectral problem. Starting from this spectral problem, the positive and negative integrable lattice hierarchies are constructed based on the Tu scheme, then by considering linear combination of the positive and negative lattice hierarchies, we give a more general integrable lattice hierarchy, which can reduce to the well-known Ablowitz–Ladik lattice and the discrete modified Korteweg–de Vries (mKdV) equation. In particular, we obtain some local and nonlocal integrable lattice equations, including reverse-space discrete mKdV equation, reverse-space complex discrete mKdV equation, higher-order discrete mKdV equation, higher-order complex discrete mKdV equation, higher-order reverse-space discrete mKdV equation and higher-order reverse-space complex discrete mKdV equation. In additional, infinitely many conservation laws and Darboux transformation (DT) for the first non-trivial system in the hierarchy are established with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained DT. The results in this paper might be helpful for understanding some physical phenomena. [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 Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation Dtαu = aupu xx + buqu x2 is studied by the symmetry analysis method. Liu et al. obtained two group generators X1 = ∂ ∂x, X2 = 2t α ∂ ∂t + x ∂ ∂x and only one trivial solution u(t,x) = f(t) = C0 Γ(α)tα−1 for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case p = q + 1, we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically. [ABSTRACT FROM AUTHOR]

In this paper, we show that the radius of analyticity σ (t) of solutions to the one-dimensional nonlinear Schrödinger (NLS) equation i ∂ t u + ∂ x 2 u = | u | p − 1 u is bounded from below by c | t | − 2 3 when p > 3 and by c | t | − 4 5 when p = 3 as | t | → + ∞ , given initial data that is analytic with fixed radius. This improves results obtained by Tesfahun [On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. Differential Equations 263(11) (2017) 7496–7512] for p = 3 and Ahn et al. [On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. 40(1) (2020) 423–439] for any odd integers p > 3 , where they obtained a decay rate σ (t) ≥ c | t | − 1 for larger t. The proof of our main theorems is based on a modified Gevrey space introduced in [T. T. Dufera, S. Mebrate and A. Tesfahun, On the persistence of spatial analyticity for the beam equation, J. Math. Anal. Appl. 509(2) (2022) 126001], the local smoothing effect, maximal function estimate of the Schrödinger propagator, a method of almost conservation law, Schrödinger admissibility and one-dimensional Sobolev embedding. [ABSTRACT FROM AUTHOR]

In this work, we derived the (2+1)-dimensional Schrödinger equation from the (2+1)-dimensional Klein–Gordon equation. We also obtained the fractional order form of this equation at the same time so as to discover the connection between them. For the (2+1)-dimensional Klein–Gordon equation, symmetries and conservation laws are pres ented. For different gauge constraint, from the perspective of conservation laws, the corresponding symmetries are obtained. After that, based on the fractional complex transform, soliton solutions of the time fractional (2+1)-dimensional Schrödinger equation are displayed. Some figures are showed behaviors of soliton solutions. It is important to discover the relationships between these equations and to obtain their explicit solutions. These solutions will perhaps provide a theoretical basis for the explanation of complex nonlinear phenomena. From the results of this paper, it is clear that the Lie symmetry method is a particularly important tool for dealing with differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new fractal modified Benjamin–Bona–Mahony equation (MBBME) that can model the long wave in the fractal dispersive media of the optical illusion field based on He's fractal derivative. First, we apply the semi-inverse method (SIM) to develop its fractal generalized variational principle with the aid of the fractal two-scale transforms. The obtained fractal generalized variational principle reveals the conservation laws via the energy form in the fractal space. Second, Wang's Bäcklund transformation-based method, which combines the Bäcklund transformation and the symbolic computation with the ansatz function schemes, is used to study the abundant exact solutions. Some new solutions in the form of the rational function-type, double-exp function-type, Sin-Cos function-type and the Sinh-Cosh function-type are successfully constructed. The impact of the fractal orders on the behaviors of the different solutions is elaborated in detail via the 3D plots, 2D contours and 2D curves, where we can find that: (1) When the fractal order > η , the direction of wave propagation tends to be more vertical to the x -axis, on the other hand, it tends to be more parallel to the x -axis when < η ; (2) The fractal order cannot impact the peak amplitude of the waveform; (3) For the periodic waveform, the fractal orders can affect its period, that is, the period becomes smaller when the fractal order , η < 1. The obtained results show that the proposed methods are effective and powerful, and can construct the abundant exact solutions, which are expected to give some new enlightenment to study the variational theory and traveling wave solutions of the fractal partial differential equations. [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, the symmetry reduction method and Nucci's reduction method are used to obtain exact solutions to the Triki–Biswas equation. Furthermore, the new conservation theorem is utilized for finding the conservation laws of the given model. The conservation laws are derived for each admitted symmetry of the Triki–Biswas equation and satisfy the divergence condition. The 3D, contour and 2D figures are finally plotted to show the dynamics of the obtained exact solutions. [ABSTRACT FROM AUTHOR]

This paper investigates the geometry of compact stellar objects via Noether symmetry strategy in the framework of curvature-matter coupled gravity. For this purpose, we assume the specific model of this theory to evaluate Noether equations, symmetry generators and corresponding conserved parameters. We use conserved parameters to examine some fascinating attributes of the compact objects for suitable values of the model parameters. It is analyzed that compact objects in this theory depend on the conserved quantities and model parameters. We find that the obtained solutions provide the viability of this process as they are compatible with the astrophysical data. [ABSTRACT FROM AUTHOR]

In this paper, we intend to establish a rigorous mathematical relation between the Gibbons–Hawking–York (GHY) boundary term and the conserved symmetric current which arises due to Noether's second theorem. The GHY term is necessary to ensure that the equations of motion in general relativity are well-posed, and it is a key ingredient in the thermodynamics of black holes and the derivation of the Bekenstein–Hawking entropy. Noether's second theorem on the other hand relates the symmetries of the action to conserved quantities of the system. In this paper, we prove that the connection between the GHY term and Noether's second theorem can be established by considering a diffeomorphism that preserves the boundary condition of the surface integral. Additionally, we also derive the associated conserved current and charge. [ABSTRACT FROM AUTHOR]

This paper investigates the (2+1)-dimensional variable coefficients Kundu–Mukherjee–Nasker equation which obtained by some transformations. The infinitesimal generators of the equation are obtained by the Lie group method. Next, the representative elements of the one-dimensional subalgebras optimal system are determined with adjoint representation. Based on the optimal system, (1+1)-dimensional partial differential equations can be obtained by similarity reductions. With the help of (G ′ / G) -expansion method, several exact solutions including hyperbolic function solutions, trigonometric function solutions and rational function solutions of variable coefficients Kundu–Mukherjee–Nasker equation are obtained. Finally, the conservation laws of the variable coefficients Kundu–Mukherjee–Nasker equation can be obtained by using the conservation theorem proposed by Ibragimov. [ABSTRACT FROM AUTHOR]

We consider the follow-the-leader particle approximation scheme for a 1 d scalar conservation law with non-negative compactly supported L ∞ initial datum and with a C 1 concave flux, which is known to provide convergence towards the entropy solution ρ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ρ n is a "discrete version of an entropy condition"; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f (ρ) / ρ) x ≤ 1 / t of said condition allows to select a unique weak solution, despite (f (ρ) / ρ) x ≤ 1 / t is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition f ′ (ρ) x ≤ 1 / t. Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case f (ρ) = ρ (A − ρ γ) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and "sharp") Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases "alternative" ones) of all steps of the convergence of the particle scheme. [ABSTRACT FROM AUTHOR]

The Zakharov–Kuznetsov (ZK) equation is extensively investigated in this study. This equation is a two-dimensional (2D) generalization of the well-known Korteweg–de Vries (KdV) equation, which is spatially limited as the 1D model of weakly nonlinear waves in shallow water. In this paper, we use three different methods to solve the conservation laws of it. First, the direct construction method is used to calculate the multipliers of the objective equation, and the conservation laws can be obtained by using the multipliers. Then its strict self-adjoint property is verified, and its conservation laws are solved by Ibragimov's method. Finally, the conservation laws of the target equation are solved by Noether's theorem. Then we calculate some exact solutions of the ZK equation by the extended Kudryashov method and seek a few solutions in terms of hyperbolic tangent functions and exp (ξ). Meanwhile, the conservation law analysis of the target equation is carried out, and its Lie point symmetries, reduced order form and invariant solution are determined. In the end, the Hamiltonian structure of the target equation, the generalized pre-symplectic that maps symmetries into adjoint-symmetries and some of its soliton solutions are calculated. [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]

This paper is devoted to a hyperbolic system of nonlinear conservation laws, that is, the pressure system independent of density and energy from the Liou–Steffen flux-splitting scheme on the compressible Euler equations. First, the one-dimensional Riemann problem is solved with eight kinds of structures. Second, the two-dimensional Riemann problem is discussed; the solution reveals a variety of geometric structures; by the generalized characteristic analysis method and studying the pointwise interactions of waves, we construct 29 kinds of structures of solution consisting of shocks, rarefaction waves and contact discontinuities; the theoretical analysis is confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation (gKDKK) is investigated. Based on Bell polynomial theory, the bilinear form, Bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the equation are obtained. Lump solution and half periodic kink solution are obtained by combining the test function with bilinear form. Furthermore, with the help of the variable separation method, we obtain some new compound solutions composed of exponential function, trigonometric function, hyperbolic function, rational function and Jacobi elliptic function in various forms. Using computer software to draw the three-dimensional diagram and profile of the solutions, the dynamic properties of the solutions are analyzed. [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, the nonlinear (1 + n) -dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (g-CH-KP) equation is examined using Lie theory. Lie point symmetries of the equation are computed using MAPLE software and are generalized for the case of any dimension. Moreover, the equation is transformed into a nonlinear ordinary differential equation using the Abelian subalgebra. The nonlinear self-adjoint classification of the equation under consideration is accomplished with the help of which conservation laws for a particular dimension are calculated. Moreover, the new extended algebraic approach is used to compute a wide range of solitonic structures using different set of parameters. Graphic description of some specific applicable solutions for certain physical parameters is portrayed. [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]

In this paper, a (2 + 1) -dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the ( G ′ G) -expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated. [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]

In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative. [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]

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen. [ABSTRACT FROM AUTHOR]

Under investigations into this paper is a higher-dimensional model, namely the time fractional (3 + 1) -dimensional Korteweg–de Vries (KdV)-type equation, which can be usually used to express shallow water wave phenomena. At the beginning, the symmetry of the time fractional (3 + 1) -dimensional KdV-type equation via the group analysis scheme is obtained. The definition of the fractional derivative in the sense of the Riemann–Liouville is considered. Then, the one-parameter Lie group and invariant solutions of this considered equation are constructed. Subsequently, we applied a direct method to construct the optimal system of one-dimensional of this considered equation. Next, this considered higher-dimensional model can be reduced into the lower-dimensional fractional differential equations (FDEs) with the help of the three-parameter and two-parameter Erdélyi–Kober fractional differential operators (FDOs). Lastly, conservation laws of this discussed equation by using a new conservation theorem are also found. A series of results of the above obtained can provide strong support for us to reveal the mysterious veil of this viewed equation. [ABSTRACT FROM AUTHOR]

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach. [ABSTRACT FROM AUTHOR]

The scope of this work is based on symmetry analysis of perturbed fractional Boussinesq equation. For beginning the group formalism, invariance properties and conservation laws of the nonlinear perturbed fractional Boussinesq equation have been explored generally. This method was first described by Lukashchuk [Commun. Nonlinear Sci. Numer. Simul. 68 (2019) 147–159]. The key subject is that when the order of fractional derivative in a fractional differential equation (FDE) is nearly integers, we can approximate it to a perturbed integer-order differential equation with a small perturbation parameter. For obtaining the results, perturbed and unperturbed symmetries are computed. Then, the methodology of reduction is applied for finding some new solutions by the symmetry operators of the equation. These solutions are obtained by the similarity transformations of the symmetries. Another exact solution will be found with constructing one-dimensional optimal system of the symmetries. Finally, the meaning of nonlinear self-adjointness concept is attended in order to find conservation laws with informal Lagrangians. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis has been proposed by utilizing the Lie group of continuous point transformation for obtaining the new exact soliton solutions of the (2+1)-dimensional Hirota–Maccari system. Lie infinitesimals and possible geometric vector fields are obtained by applying the third-order prolongation on this system. Also, their commutative product and adjoint relations have been presented in Tables 1 and 2. By considering the resulting symmetries, one-dimensional optimal system of Lie subalgebra is obtained. Meanwhile, the Hirota–Maccari system is reduced to a system of ordinary differential equations with the help of optimal subalgebras. Furthermore, the simplest equation method has been used to obtain the abundant exact soliton solutions of the reduced system. At last, conservation laws of Hirota–Maccari system have been extracted by utilizing the generalized "new conservation theorem" invoked by Ibragimov. [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]

In this work, we study the (2+1)-dimensional nonlinear Schrödinger-type equation that is related to many physical phenomena in nonlinear optical fibers and water waves. Some properties of the (2+1)-dimensional nonlinear Schrödinger-type equation are considered. We determine the infinitesimal generators, an optimal system and a commutator table of the Lie algebra by using Lie symmetry analysis. Also the conservation laws of the equation are obtained using the new conservation theorem proposed by Ibragimov. [ABSTRACT FROM AUTHOR]

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities c 1 and c 2 and the parameters of nonlinearity a 1 and a 2 on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation. [ABSTRACT FROM AUTHOR]

In this work, one-point Lie symmetry method is applied to time fractional super KdV equation in order to obtain similarity variables and similarity transformations with Riemann–Liouville derivative. These transformations reduce the governing equation to an ordinary differential equation of fractional order. A new and effective conservation theorem based on Noether's theorem is used to obtain conserved vectors. Then, we construct power series solutions for the reduced time fractional ordinary differential equation and prove that the solutions are convergent. Lastly, some interesting graphs are given to explain physical behaviors. [ABSTRACT FROM AUTHOR]

The current study examines the geometry of static wormholes with anisotropic matter distribution in the context of modified f () gravity. We consider the well-known Noether and conformal symmetries, which help in investigating wormholes in f () gravity. For this purpose, we develop symmetry generators associated with conserved quantities by taking into consideration the f () gravity model. Moreover, we use the conservation relationship gained from the classical Noether method and conformal Killing symmetries to develop the metric potential. These symmetries provide a strong mathematical background to investigate wormhole solutions by incorporating some suitable initial conditions. The obtained conserved quantity performs a significant role in defining the essential physical characteristics of the shape-function and energy conditions. Further, we also describe the stability of obtained wormholes solutions by employing the equilibrium condition in modified f () gravity. It is observed from graphical representation of obtained wormhole solutions that Noether and conformal Killing symmetries provide the results with physically accepted patterns. [ABSTRACT FROM AUTHOR]

In this work, new (3 + 1) -dimensional Korteweg–de Vries (KdV) equation and modified KdV (mKdV) equation as well as the corresponding fractional forms are presented. These two equations are derived for the first time relying on the extended (3 + 1) -dimensional zero curvature equation. In addition, symmetries and conservation laws are displayed. Meanwhile, one-soliton solution for the time fractional of (3 + 1) -dimensional KdV equation and mKdV equation is obtained. [ABSTRACT FROM AUTHOR]

We prove the convergence of the Godunov-type scheme for a scalar conservation law in one space dimension with possibly infinitely many spatial discontinuities, which may have accumulation points. In contrast to the study appearing in Ghoshal, Jana and Towers [Convergence of a Godunov scheme to an Audusse–Perthame adapted entropy solution for conservation laws with BV spatial flux, Numer. Math. 146(3) (2020) 629–659], we do not restrict the flux to be unimodal and allow for the case where the flux has degeneracies due to which the corresponding singular map may not be invertible and hence, the so-called singular mapping technique is not applicable. We present two proofs of convergence depending on the nature of the flux. For fluxes which are piecewise constant in the space variable, convergence is established by proving BV bounds on the finite volume approximations away from the set of spatial discontinuity. On the other hand, if the fluxes are of Panov-type, i.e. A (x , u) = g (β (x , u)) , we prove the convergence by establishing a novel β − TVD property, which in turn implies the existence of BV bounds on the solution. We present numerical examples that illustrate the theory. [ABSTRACT FROM AUTHOR]

We estimate the L p distance between piecewise constant solutions to the Cauchy problem of scalar conservation laws and propose a sufficient condition for having an L p contraction of such solutions. Moreover, we prove that there exist L p contractive solutions on a set of all monotone bounded initial functions to the Cauchy problem of scalar conservation laws with convex or concave flux functions. [ABSTRACT FROM AUTHOR]

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure). [ABSTRACT FROM AUTHOR]

The Riemann problem for a hyperbolic system of conservation laws belonging to Temple class is considered within frame of α -solutions based on the concept of solutions defined in the setting of a distributional product. We obtain all the possible discontinuous solutions within a convenient space of distributions that contains Dirac delta measures and discontinuous functions. In addition, an interesting discontinuous solution is found for certain Riemann initial conditions, in which the Dirac delta measure is involved in both of the two state variables simultaneously. As an application, a detailed example for the mentioned system but involving a linear damping term is provided, in which the trajectories of discontinuous solutions including contact discontinuity, shock wave and delta shock wave change in exponential forms. [ABSTRACT FROM AUTHOR]

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics. [ABSTRACT FROM AUTHOR]

The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed. [ABSTRACT FROM AUTHOR]

The unsmooth boundary will greatly affect the motion morphology of ion-acoustic waves in plasma, so a modified equal width equation with fractal derivatives is proposed. The fractal variational formulation of the problem is established by using the semi-inverse method, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation. Then He's variational method based on the variational theory and Ritz-like method, combined with the two-scale transform is used to seek the periodic and solitary wave solutions of the fractal modified equal width equation. The obtained results show that the variational method is simple but powerful, which is expected to open some new perspectives toward the study of traveling wave theory in fractal space. [ABSTRACT FROM AUTHOR]