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In this paper, we present a comprehensive examination of the Lie symmetry analysis, optimal system, exact solutions, and conservation laws pertaining to the variable-coefficient higher-order Boussinesq–Burgers system. The Lie symmetry analysis method is initially employed to derive the vector field, followed by the utilization of Olver’s method to determine the optimal system of one-dimensional subalgebras for the system. By utilizing the optimal system, the Boussinesq–Burgers system is similarly reduced. Subsequently, the $(\frac{G^{\prime }}{G})$-expansion method is applied to this reduced system, yielding kink soliton solution, dark soliton solution, trigonometric-function solution and rational solution for the variable-coefficient higher-order Boussinesq–Burgers system. By assigning various functions to the coefficient functions, different types of soliton solutions are obtained, and these solutions’ structural features are visualized through the use of three-dimensional figures. Last, an examination of the conservation laws governing the system is conducted.

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.

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.

The Korteweg–de Vries (KdV) equation is an essential model to characterize shallow water waves in fluid mechanics. Here, we investigated the generalized time and time-space fractional KdV equation with fractional derivative of Riemann–Liouville. At the beginning of, we applied the fractional Lie symmetry scheme to derive their symmetry, respectively. We found that the vector fields of these considered equations decrease as the independent variables fractionalize. Subsequently, the one-parameter Lie transformation groups of these concerned models were yielded. At the same time, they can be reduced into fractional order ordinary differential equations with the Erdélyi–Kober fractional operators. Finally, by obtaining the nonlinear self-adjointness, conservation laws of the generalized time-space fractional KdV equation were also found. These good results provide a basis for us to further understand the phenomenon of shallow water waves.

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.

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 $\alpha$, 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 $\alpha$ impact the precision of the model.

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.

In this paper, we study symmetries, Newtonoid vector fields, conservation laws, Noether's theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher–Nijenhuis formalism on the space of k¹-velocities of the configuration manifold.

For the case k = 1, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's theorem and its converse. For the case k > 1, we provide a new proof for Noether's theorem, which shows that, in the k-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

Under investigation in this paper is a more general discrete $2\times 2$ 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.

In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We prove that if the middle components of the non-stationary solutions of the Maxwell equations verify a Morawetz-type estimate supported on a compact region in space around the trapped surface, then the components of the Maxwell fields decay uniformly in the entire exterior of the Schwarzschild black hole, including the event horizon. This is shown by making only use of Sobolev inequalities combined with energy estimates using the Maxwell equations directly. The proof does not pass through the scalar wave equation on the Schwarzschild black hole, does not need to decouple the middle components for the Maxwell fields, and would be in particular useful for the non-abelian case of the Yang–Mills equations where the decoupling of the middle components cannot occur. In fact, the estimates for the hereby argument are still valid for the Yang–Mills fields except for the Lie derivatives of the fields that are involved in the proof.

We investigate second-order quasilinear equations of the form f_iju_xixj = 0, where u is a function of n independent variables x₁, …, x_n, and the coefficients f_ij depend on the first-order derivatives p¹ = u_x1, …, pⁿ = u_xn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pⁿ with coordinates p¹, …, pⁿ. The coefficient matrix f_ij defines on Pⁿ a conformal structure f_ij(p)dpⁱdp^j. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure f_ij(p) dpⁱdp^j is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.

In the setting of a product of distributions which is not defined by approximation processes, we are able to consider a Riemann problem for the system u_t + [ϕ(u)]_x = 0, v_t + [ψ(u)v]_x = 0, with the unknown states u, v in convenient spaces of distributions and ϕ, ψ : ℝ → ℝ continuous. A consistent extension of the classical solution concept will show the possible arising of a δ-shock wave solution. This procedure affords a simpler and more general framework to construct singular solutions and can surely be applied to other equations or systems.

In this paper, we mainly investigate the time fractional nonlinear wave equation which can be usually used to express nonlinear phenomena appearing in shallow water waves by using group analysis scheme. First, the symmetry can be obtained make uses of the group analysis to the time fractional nonlinear wave equation. Based on the above-found symmetry, this equation was able to reduce into an ordinary differential equation of fractional order. As a result, some new invariant solutions were also constructed for this considered equation. Second, the scaling transformation was also obtained by introducing new independent and dependent variables. Finally, the conservation laws were also found to satisfy the time fractional nonlinear wave equation with the help of the Ibragimov theorem. These novel results show unique nonlinear phenomena.

Numerical modelling of detonation using shock fitting is described in detail. A complete set of jump conditions that hold across the detonation front is presented in a simple form. Validity and accuracy of the model has been established by comparison with published results and results of another model utilizing the method of integral relations. A brief description of the later model is given to highlight its validity and limitations.

Simulation of critical relaxation of the magnetization in a microcanonical Ising model agrees with the expected exponent z ≃2 after very long times on the square lattice, while on the simple cubic lattice z≃3.3 seems different.

A covariant symplectic structure can be identified, in general, starting from conservation laws connected to generic Hamiltonian invariants constructed from covariant vectors, bivectors and tensors. This feature leads to both holonomic and anholonomic formulations of Hamilton equations and Poisson brackets and it seems a deep link common to all interactions including gravity. A key role in the approach is played by affine connections instead of metric.

All the Doubly Special Relativity (DSR) models studied in literature so far involve a deformation of the energy conservation rule that forces us to release the hypothesis of the additivity of the energy for composite systems. In view of the importance of the issue for a consistent formulation of a DSR statistical mechanics and a DSR thermodynamics, we show that DSR models preserving the usual (i.e. additive) energy conservation rule can be found. These models allow the construction of a DSR-covariant extensive energy. The implications of the analysis for the dynamics of DSR-covariant multiparticle systems are also briefly discussed.

For describing wave propagation in an inhomogeneous erbium-doped nonlinear fiber with higher-order dispersion and self-steepening, an inhomogeneous coupled Hirota–Maxwell–Bloch system is considered with the aid of symbolic computation. Through Painlevé singularity structure analysis, the integrable condition of such a system is analyzed. Via the Painlevé-integrable condition, the Lax pair is explicitly constructed based on the Ablowitz–Kaup–Newell–Segur scheme. Furthermore, the analytic soliton-like solutions are obtained via the Darboux transformation which makes it exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some obtained analytic one- and two-soliton-like solutions, our concerns are mainly on the envelope soliton excitation, the propagation features of optical solitons and their interaction behaviors in actual fiber communication. Finally, the conservation laws for the system are also presented.

The aim of this paper is to classify Bianchi type II spacetimes according to their teleparallel Killing vector fields using the direct integration technique. Studying teleparallel Killing vector fields in the above spacetimes, it turns out that the dimensions of the teleparallel Killing vector fields are 4, 5 or 7. A brief comparison between teleparallel and general relativity Killing vector fields are given. It is shown that for the above spacetimes in the presence of torsion we get more conservation laws which are different from the theory of general relativity.

This paper discusses the approximate Noether symmetries of the action of plane symmetric spacetime. A time-dependent conformal factor is introduced in the plane symmetric metric in such a way so as to obtain the same number of Noether symmetries as for the exact plane symmetric spacetime. The corresponding approximate conservation is obtained to discuss the conservation laws of the energy and momentum during the formation of gravitational wave.