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In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic (3 + 1)-dimensional nonlinear Schrödinger equation (NLSE), a critical model for understanding wave dynamics under the influence of random perturbations. We apply He’s semi-inverse method to establish a variational formula for the stochastic (3 + 1)-dimensional NLSE and we use this formula to construct conservation laws corresponding to the symmetries of the model. Based on this formulation, a solitary solution can be easily obtained using the Ritz method. We explore approximate solutions of the stochastic (3 + 1)-dimensional NLSE equation by choice of a trial function in the region of the rectangular box in three cases. Subsequently, we approximate the trial function using a piecewise linear ansatz in one case of the two-box potential, followed by an approximation using quadratic polynomials with two free parameters instead of the piecewise linear ansatz. We demonstrate the effectiveness of the analytical approaches, namely, the improved modified extended tanh-function method and He’s semi-inverse variational principle method for seeking more exact solutions via the stochastic (3 + 1)-dimensional NLSE equation. As a result, bright solitons, singular solitons, periodic wave solution and dark solitons solutions are obtained. Furthermore, the dynamical behaviors of the model are discussed by investigating bifurcations at equilibrium points. The paper not only advances the theoretical understanding of the stochastic (3 + 1)D NLSE but also provides practical methodologies for solving this complex equation, offering new insights into wave phenomena influenced by stochastic effects. In addition, we employ the fractional reduced differential transform (F-RDT) method, interpreted through Caputo fractional derivatives, to construct a numerical solution utilizing hyperbolic functions. We further analyze the influence of the fractional order parameter, denoted as α, on the behavior of the numerical solution. This investigation includes a detailed examination of the discrepancies between the exact solitary wave solutions and the corresponding numerical approximations, allowing us to identify the absolute errors. By comparing the two, we provide insights into the accuracy and stability of the numerical approach for solving the studied equation, emphasizing how variations in α impact the precision of the model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to explore Noether symmetries of Bianchi type II universe in the modified teleparallel f(T) gravity, where f(T) is a function of the torsion scalar T. In a physical context, the selection of Noether symmetries provides constraints that are helpful in building conservation laws. Some of the well-known examples of such laws are the conservation of energy and momentum. In this study, initially, we explore certain classes of Bianchi type II universe considering linear as well as nonlinear f(T) gravity models. In view of such models, equations of motion of the f(T) gravity have been solved in twelve cases. In each of the cases, we apply the definition of Noether symmetry which tends to a set of nineteen equations. Upon solving these equations, we reach the fact that the Bianchi type II universe in the background of f(T) gravity admits only four Noether vector fields (NVFs). As a result, the physical entities which we observe are the conservation of linear and generalized momentum. [ABSTRACT FROM AUTHOR]

This paper presents a new stabilised Element-Free Galerkin (EFG) method tailored for large strain transient solid dynamics. The method employs a mixed formulation that combines the Total Lagrangian conservation laws for linear momentum with an additional set of geometric strain measures. The main aim of this paper is to adapt the well-established Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology to the context of EFG, presenting three key contributions. Firstly, a variational consistent EFG computational framework is introduced, emphasising behaviours associated with nearly incompressible materials. Secondly, the suppression of non-physical numerical artefacts, such as zero-energy modes and locking, through a well-established stabilisation procedure. Thirdly, the stability of the SUPG formulation is demonstrated using the time rate of Hamiltonian of the system, ensuring non-negative entropy production throughout the entire simulation. To assess the stability, robustness and performance of the proposed algorithm, several benchmark examples in the context of isothermal hyperelasticity and large strain plasticity are examined. Results show that the proposed algorithm effectively addresses spurious modes, including hour-glassing and spurious pressure fluctuations commonly observed in classical displacement-based EFG frameworks. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to (3 + 1)-dimensional time fractional modified KdV–Zakharov–Kuznetsov equation. All Lie symmetries and the corresponding conserved vectors for the equation are obtained. The one-dimensional optimal system is utilized to reduce the aimed equation with Riemann–Liouville fractional derivative to a low dimensional fractional partial differential equation with Erdélyi–Kober fractional derivative. Then the power series solution of the reduced equation is given. Moreover, some other low dimensional reduced fractional differential equations with Riemann–Liouville fractional derivative are obtained and can be solved by different methods in the literatures herein. [ABSTRACT FROM AUTHOR]

Higher-dimensional nonlinear models can describe more complex evolutionary mechanisms. In this paper, we considered the time fractional (3 + 1)-dimensional modified extended Zakharov–Kuznetsov equation with the sense of the Riemann–Liouville fractional derivative in plasma physics. In the first place, the existence of symmetry of this studied equation through the symmetry scheme was proved. Then, the optimal system to the time fractional (3 + 1)-dimensional modified extended Zakharov–Kuznetsov equation was also constructed. Subsequently, the time fractional higher-dimensional equation was reduced into the lower-dimensional fractional differential equation with the help of the Erdélyi–Kober fractional operators. Last, some conservation laws by using a new conservation theorem were also given. These novel results provide a window for us to discover this high-dimensional nonlinear equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of scalar viscous conservation laws via the structure of $ \omega $-limit sets. We show that $ \omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ \omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics. [ABSTRACT FROM AUTHOR]

This paper explores the evolutionary behavior of some specific waves for a Generalized Hirota Bilinear (GHB) equation with applications in modern sciences. More precisely, the linear superposition principle is first adopted to construct the resonant multi-wave of the GHB equation. Through the resonant N-wave and some particular computations, positive multi-complexiton to the governing model is then extracted with the use of computational packages. Furthermore, after deriving nonclassical Lie symmetries and their corresponding invariant solutions, Conservation Laws (CLs) for the GHB equation are formally constructed based on a general method developed by Ibragimov. The propagation dynamics of resonant double- and triple-waves as well as positive single- and double-complexitons are examined in detail by considering several case studies. The present results demonstrate how adjusting the nonlinear parameter can be used to control specific waves in nonlinear systems. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators. [ABSTRACT FROM AUTHOR]

Wave propagation in the form of fronts or kinks, a common occurrence in a wide range of physical phenomena, is studied in the context of models defined by their Hamiltonian structure. Motivated, for dispersive wave evolution equations such as a strongly nonlinear model of two‐layer internal waves in the Boussinesq limit, by the symmetric properties of a class of front‐propagating solutions, known as conjugate states or solibores, a generalized formulation based purely on the dispersionless reduction of a system is introduced, and a class of undercompressive shock solutions, here referred to as "Hamiltonian shocks," is defined. This analysis determines whether a Hamiltonian shock, representing locally a kink for the parent dispersive equations, will interact with a sufficiently smooth background wave without inducing loss of regularity, which would take the form of a classical dispersive shock for the parent equations. This property is also related to an infinitude of conservation laws, drawing a parallel to the case of completely integrable systems. [ABSTRACT FROM AUTHOR]

Optical fibers are used in the communications, biological sensors and chemical sensors. We investigate a variable-coefficient modified Hirota equation for the amplification or absorption of pulses propagating in an inhomogeneous optical fiber. With respect to the complex envelope of the optical field, we construct the infinitely-many conservation laws based on the existing Lax pair. According to the existing Darboux transformation, we derive the three-soliton solutions, the higher-order breather solutions and breather-to-soliton transition condition. Amplitudes of the two solitons change after the interaction, while velocities of them are unchanged via asymptotic analysis. When $ P(z)=0 $ P (z) = 0 , interactions among the three parabolic or wavy solitons, interaction between the two parabolic or wavy or crooked breathers, and interactions among the three parabolic and wavy breathers are presented, where $ P(z) $ P (z) is related to the nonlinear focus length. Velocities of three solitons or two crooked breathers with $ P(z)\neq 0 $ P (z) ≠ 0 are different from those with $ P(z)=0 $ P (z) = 0. Based on the breather-to-soliton transition condition, when $ P(z)=0 $ P (z) = 0 , parabolic or wavy multi-peak and M-shaped solitons are presented; when $ P(z)\neq 0 $ P (z) ≠ 0 , the crooked periodic wave and anti-dark soliton are shown. [ABSTRACT FROM AUTHOR]

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of α which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail. [ABSTRACT FROM AUTHOR]

For the Burgers' equation, the entropy solution becomes instantly BV with only L∞ initial data. For conservation laws with genuinely nonlinear discontinuous flux, it is well known that the BV regularity of entropy solutions is lost. Recently, this regularity has been proved to be fractional with s = 1/2. Moreover, for less nonlinear flux, the solution still has a fractional regularity 0< s ≤ 1/2. The resulting general rule is that the regularity of entropy solutions for a discontinuous flux is less than for a smooth flux. In this paper, an optimal geometric condition on the discontinuous flux is used to recover the same regularity as for the smooth flux with the same kind of nonlinearity. [ABSTRACT FROM AUTHOR]

We consider the energy-minimizing configuration of the Sadowsky-type functional for narrow rectifying strips. We show that the functional is proportional to the p -Willmore functional using classical analysis techniques and the geometry of developable surfaces. We introduce hyperelastic strips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variational approach. We show a naturally expected relationship between the planar stationary points of the Sadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, the internal force and torque, using Euclidean motions and obtain the first and second conservation laws for hyperelastic strips. [ABSTRACT FROM AUTHOR]

A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional $(\alpha +2)$(α+2)-dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem. [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]

Summary: The paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first‐order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first‐order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex‐based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi‐discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation. [ABSTRACT FROM AUTHOR]

This article utilizes the Lie symmetry method to analyze the (n + 1) -dimensional modified Zakharov–Kuznetsov (mZK) equation, which characterizes weakly nonlinear traveling waves in plasma with a constant magnetic field, comprising cold ions and hot isothermal electrons. The model is also applicable to dusty and magnetized plasma. Lie point symmetries and associated group invariant solutions are computed using Lie group theory, with the underlying equation. The improved tan ( Φ (ξ) 2) -expansion method is then employed to derive soliton solutions, including hyperbolic, trigonometric, rational, and exponential forms. Graphic interpretations of specific solutions are provided. Nonlinear self-adjointness is used to compute non-local conservation laws in lower dimensions, and conservation laws are developed based on the equation's formal Lagrangian structure, applying Ibragimov's theorem. These findings highlight the novelty and reliability of the methodology employed. [ABSTRACT FROM AUTHOR]

For the disciplinary training of the undergraduate student in Physics at the Francisco José de Caldas District University, it is essential to understand and analyze the concepts of collision and disintegration of relativistic particles and the conservation laws that govern them, these are immersed within the area of modern physics and quantum mechanics, mandatory core subjects belonging to the 7th and 8th semester curriculum, the aforementioned contents are challenging due to their abstract and complex nature, therefore the need emerges to provide tools that energize and strengthen the interpretation, analysis and therefore the memory fixation of said concepts, which with visual and interactive aids such as Virtual Learning Objects (OVA) provide greater flexibility in teaching and learning. For this, a virtual learning object is designed using the ADDIE instructional model: (Analysis, Design, Development, Implementation and Evaluation) which has a constructivist model that shows the interactions of collisions and decay between particles constituted by the first generations of the model. standard and that give rise to the SU(3) symmetries, the hadrons; Taking bubble chamber experiments as a starting point as a tool that helps solve this abstract nature, this work contemplates the first two stages of the ADDIE model. [ABSTRACT FROM AUTHOR]

In this research, the (2+1)‐dimensional normal biological population model, incorporating the Verhulst law for population growth, is employed to explore species population dynamics. Employing Lie symmetry analysis, we address a nonlinear degenerate parabolic partial differential equation, yielding much‐improved results. This analysis includes computing one‐dimensional optimal subalgebras, reduced ordinary differential equations, and obtaining invariant solutions with a visual depiction of the physical behavior of the Verhulst biological population model through symmetry group transformations. Additionally, the multiplier method leads to novel conservation laws and potential systems not locally connected to the governing partial differential equation (PDE). These findings have significant implications for understanding and controlling biological populations, offering insights for applications in ecology and the environment. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities is considered. A series of direct transformations is performed for the traveling wave reduction of the original equation. The implicit function method is used for nonlinear ordinary differential equation with some constraints on the parameters. Some analytical solutions are found in the form of periodic and solitary waves. Three conservation laws, corresponding to the generalized nonlinear Schrödinger equation are obtained and conserved quantities corresponding to the solution in implicit form are found [ABSTRACT FROM AUTHOR]

We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0 . This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities. [ABSTRACT FROM AUTHOR]

Conservation laws with two-parameter hysteretic fluxes are introduced. The total variation of the flux of a viscous solution does not increase as time progresses. The maximum principle for the flux is also established. Viscous shock waves are identified, and solutions to the Riemann problem are obtained. The one with the least total variation of flux is preferred. A definition of weak solutions for the hysteretic flow model, a non-conservative hyperbolic system, is provided. Approximate solutions of the system are constructed using the front tracking method with randomly adjusted discretization locations. These approximate solutions have uniformly bounded total variation, demonstrating their compactness. [ABSTRACT FROM AUTHOR]

We investigate variational, Lie and Killing symmetries of the Lagrangian of an essential class of four-dimensional (pseudo-)Riemannian manifolds, i.e., non-reductive homogeneous spaces. Moreover, a Lie algebra analysis is shown. With the help of a result of the Noether's theorem, we have presented the expressions for conservation laws corresponding to all variational symmetries. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional time-fractional chiral non-linear Schrödinger equation in physical media is considered in this paper. At the outset, the Lie group analysis is applied to build a set of infinitesimal generators for this equation with the aid of the Riemann–Liouville fractional derivatives. Consequently, the reduction for the considered equation into an ordinary differential equation of fractional order is obtained by using these generators and the ErdLélyi–Kober fractional operator. As a result of this reduction, we use power series analysis to get an analytical solution provided by a convergence analysis of the obtained solution. Furthermore, we construct a numerical solution based on hyperbolic functions using the fractional reduced differential transform method in the sense of Caputo fractional derivatives. Also, we detect absolute errors by performing a comparison between the exact and numerical solutions of the equation under study, while investigating the effect of fractional order α on the numerical solution. Finally, conservation laws are derived using the the formal Lagrangian and new conservation theorem. [ABSTRACT FROM AUTHOR]

A two-layer rod with a box section twisted by angle under the action of shear stresses is considered. It is assumed that the rod deformations are elastic and its lateral surface is stress-free. The layers have different elastic properties and different thicknesses. The contact line between the layers is assumed to be rigid, i.e., the stresses on it are the same. An exact solution describing the stress state of this structure is constructed using conservation laws. The stress state is determined at each point of the cross section using outer contour integrals. [ABSTRACT FROM AUTHOR]

For a certain natural generalization of the Infeld—Rowlands equation we prove nonexistence of nontrivial local Hamiltonian structures and nontrivial local symplectic structures of any order, as well as of nontrivial local Noether and nontrivial local inverse Noether operators of any order, and exhaustively characterize all cases when the equation in question admits nontrivial local conservation laws of any order; the method of establishing the above nonexistence results can be readily applied to many other PDEs. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

Following the solution formula method given in Dong et al. (High order discontinuities decomposition entropy condition schemes for Euler equations. CFD J. 2002;10(4): 448–457), this article studies a type of one‐step fully‐discrete scheme, and constructs a third‐order scheme which is written into a compact form via a new limiter. The highlights of this study and advantages of new third‐order scheme are as follows: ① We proposed a very simple new methodology of constructing one‐step, consistent high‐order and non‐oscillation schemes that do not rely on Runge–Kutta method; ② We systematically studied new scheme's theoretical problems about entropy conditions, error analysis, and non‐oscillation conditions; ③ The new scheme achieves exact solution in linear cases and performing better in nonlinear cases when CFL → 1; ④ The new scheme is third order but high resolution with excellent shock‐capturing capacity which is comparable to fifth order WENO scheme; ⑤ CPU time of new scheme is only a quarter of WENO5 + RK3 under same computing condition; ⑥ For engineering applications, the new scheme is extended to multi‐dimensional Euler equations under curvilinear coordinates. Numerical experiments contain 1D scalar equation, 1D,2D,3D Euler equations. Accuracy tests are carried out using 1D linear scalar equation, 1D Burgers equation and 2D Euler equations and two sonic point tests are carried out to show the effect of entropy condition linearization. All tests are compared with results of WENO5 and finally indicate EC3 is cheaper in computational expense. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Riemann–Liouville (R-L) time-fractional (2 + 1) dimensional extended quantum Zakharov–Kuznetsov (EQZK) equation by using the Lie symmetry method which arises in hydrodynamic that describes the nonlinear propagation of the quantum ion-acoustic waves. By using its symmetry, we convert the equation under consideration to a fractional order non-linear ordinary differential equation (ODE). In this reduced ODE, we use a special type of derivative which is known as Erdélyi–Kober (EK) derivative. This enables us to obtain explicit solutions with convergence analysis of the considered problem. By using Ibragimov's conservation laws theorem, we compute the conservation laws of the problem under investigation. Moreover, by employing the two potent methods explicit power series and ( 1 G ′ )-expansion technique, we get the explicit solutions to the problem under discussion. This analysis leads to the derivation of various key findings, including the identification of symmetries, the establishment of similarity reductions involving the EK fractional differential operator, the determination of exact solutions, and the formulation of conservation laws for the considered equation. We have confidence that these remarkable findings can provide valuable insights and contribute to the exploration of additional evolutionary mechanisms associated with the studied equation. [ABSTRACT FROM AUTHOR]

We recall the question of geometric integrators in the context of Poisson geometry and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and compared to traditional methods. [ABSTRACT FROM AUTHOR]

Consider a Conservation Law and a Hamilton-Jacobi equation with a flux/Hamiltonian depending also on the space variable. We characterize first the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed profile. An explicit example then shows the deep differences between the cases of x-independent and x-dependent fluxes/Hamiltonians. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to compute exact solutions for the generalized (3+1)-dimensional KadomtsevPetviashvili Benjamin-Bona–Mahony (gnKP-BBM) equation by invoking an effective method, namely the Lie symmetry technique. Firstly, we derive the infinitesimals and write down the Lie symmetries. Using these symmetries the gnKP-BBM equation is reduced to various nonlinear ordinary differential equations (NLODEs). Thereafter, solutions of the NLODEs are derived by using the Jacobi elliptic cosine method, the (G′ /G)-expansion method, the simplest equation technique and Kudryashov’s method. Conclusively, we derive conservation laws of the gnKP-BBM equation by using the Ibragimov’s method. [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]

It is shown that the Boussinesq–Peregrine system, which describes long waves of small amplitude at the surface of an inviscid fluid with variable depth, admits a number of approximate conservation equations. Notably, this paper provides accurate estimations for the approximate conservation of the mechanical balance laws associated with mass, momentum, and energy. These precise estimates offer valuable insights into the behavior and dynamics of the system, shedding light on the conservation principles governing the wave motion. [ABSTRACT FROM AUTHOR]

All low‐order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time‐dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time‐dependent self‐interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well‐known variational symmetries admitted by nonlinear undamped wave equations in one dimension. [ABSTRACT FROM AUTHOR]

In this paper, we study a three‐component modified nonlinear Schrödinger equation and main results include: (1) Lax pair and infinitely‐many conservation laws, (2) modulation instability of continuous waves, (3) semi‐rational solutions obtained through the Darboux transformation, and (4) four new types of mixed semi‐rational solutions provided for the three components q1$$ {q}_1 $$, q2$$ {q}_2 $$ and q3$$ {q}_3 $$. [ABSTRACT FROM AUTHOR]

In this study, we take into account the (2 + 1)-dimensional Boussinesq equation, a nonlinear evolution partial differential equation that describes how gravity waves move across the surface of the ocean. The symmetry reductions and group invariant precise solutions are systematically determined using the Lie symmetry analysis. We derive the precise multiple wave solutions using the multiple exp-function method, and then, using the multiplier method, we give the conservation laws. The dynamics of complicated waves and their interplay are faithfully recreated by the findings. [ABSTRACT FROM AUTHOR]

This study addresses the integrable Akbota equation which is a Heisenberg ferromagnet-type equation and has a significant study of the curve and surface geometry. The conservation laws of the model are computed using the multipliers approach. The Jacobi elliptic function solutions for the model are derived by utilizing the new extended auxiliary equation method. The Jacobi elliptic functions solutions are degenerated to dark, bright, singular and singular periodic solitons in their limit conditions. Further, the modulation instability analysis of the equation is studied. Physical interpretations of the obtained results are represented graphically. To the best of our knowledge, the obtained results in this article are novel and have not been reported before. [ABSTRACT FROM AUTHOR]

The present paper intends to thoroughly study an evolutionary model called the Zakharov–Kuznetsov modified equal-width (ZK–MEW) equation. More precisely, Lie symmetries as well as invariant solutions to the ZK–MEW equation describing shallow and stratified waves in nonlinear LC circuits are first derived, and then a general theorem established by Ibragimov is adopted to retrieve its conservation laws. Additionally, by applying the qualitative theory of dynamical systems, the bifurcation analysis of the dynamical system is carried out and several Jacobi elliptic solutions to the ZK–MEW equation are formally constructed. In some case studies, the impact of the nonlinear coefficient on the physical features of bright and kink solitary waves as well as periodic continuous waves is examined in detail. [ABSTRACT FROM AUTHOR]

The primary goal of this work is to derive newly constructed invariant solutions, conservation laws, and modulation instability in the context of the (2+1)-dimensional cubic nonlinear Schrödinger's equation (cNLSE), which explains the phenomenon of soliton propagation along optical fibers. The nonlinear Schrödinger equations and their various formulations hold significant importance across a wide spectrum of scientific disciplines, especially in nonlinear optics, optical fiber, quantum electronics, and plasma physics. In this context, we utilize Lie symmetry analysis to determine the vector fields and assess the optimality of the governing equation. Upon establishing the optimal system, we derive similarity reduction equations, thereby converting the system of partial differential equations into a set of ordinary differential equations. Through the simplification of these ordinary differential equations, we are able to construct several optically invariant solutions for the governing equation. Furthermore, through the utilization of the generalized exponential rational function (GERF) approach, we have derived additional intriguing closed-form solutions. Conservation laws are derived for the governing equation by utilizing the resulting symmetries introduced by the Ibragimov scheme. Furthermore, modulation instability and gain spectrum are derived for this equation to understand the correlation between nonlinearity and dispersive effects. To provide a comprehensive and insightful portrayal of our findings, we have created three-dimensional (3D) visualizations of these solutions, which reveal the periodic waves and the solitary wave structures. The form of cubic nonlinear Schrödinger's equation discussed in this article and the optical solutions obtained have never been studied before. Also, these attained solutions can be beneficial to study analytically the identical models arising in fluid dynamics, birefringent fibers, plasma physics, and other optical areas. [ABSTRACT FROM AUTHOR]

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem. [ABSTRACT FROM AUTHOR]

Riemann problem for a class of systems of conservation laws of Keyfitz–Kranzer type is solved. The Riemann solutions contain three kinds of interesting structures, two of which contain vacuum and the other includes delta‐shock. The generalized Rankine–Hugoniot relation and improved entropy condition are proposed to solve the delta‐shock. Besides, with the use of the vanishing viscosity method, all of the existence, uniqueness, and stability of the solutions involving the delta‐shock are proved. [ABSTRACT FROM AUTHOR]

We propose a new data-driven method to learn the dynamics of an unknown hyperbolic system of conservation laws using deep neural networks. Inspired by classical methods in numerical conservation laws, we develop a new conservative form network (CFN) in which the network learns to approximate the numerical flux function of the unknown system. Our numerical examples demonstrate that the CFN yields significantly better prediction accuracy than what is obtained using a standard nonconservative form network, even when it is enhanced with constraints to promote conservation. In particular, solutions obtained using the CFN consistently capture the correct shock propagation speed without introducing nonphysical oscillations into the solution. They are furthermore robust to noisy and sparse observation environments. [ABSTRACT FROM AUTHOR]

Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax--Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are used to prove an extension of the Lax--Wendroff theorem incorporating pseudotime iterations with explicit Runge--Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of conservation laws, where each equation is modified by a particular scalar factor multiplying the spatial flux terms. Local conservation is further established for Krylov subspace methods with and without restarts, and for Newton's method under certain assumptions on the discretization. It is thus shown that Newton--Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2 dimensional compressible Euler equations corroborate the theoretical results. A simple technique for enforcing flux consistency of Newton--Krylov methods is presented. Experiments indicate that its efficacy is case dependent, and diminishes as the number of iterations grow. [ABSTRACT FROM AUTHOR]

The current study covers the traveling wave structures of a fourth-order nonlinear symmetric regularized long-wave (SRLW) equation which emerges in a few actual applications incorporating particle sound waves in plasma using the new extended direct algebraic method. This model depicts nonlinear particle acoustic and space-charge waves. This model admits the two-dimensional Lie algebra. We have computed the optimal system and reduced our given model corresponding to each element of the optimal system. After that, by using a proposed method, we obtained the solitary wave profiles of the model under investigation. By using the concept of nonlinear self-adjointness, we have shown that our given model is not self-adjoint. Moreover, we have computed the conserved quantities corresponding to each element of the Lie algebra. The obtained solutions are in different classes of solitary wave solutions with dark, combined dark-bright, singular, periodic-singular, combined dark-singular, combined singular, dark-singular combo solitons, and rational solutions. Furthermore, existing constraints for the resultant solutions are reported. Also, a graphical interpretation of some particular solutions is shown by considering the specific values of the parameters. In the end, the sensitive visualization of the model is done by considering different initial states. We are confident that these outstanding results will offer insightful information and further the investigation of other evolutionary mechanisms connected to the equation under study. [ABSTRACT FROM AUTHOR]