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The nonlocal symmetries of the nonlinear Dirac system are constructed on the basis of the known Lax pair. With the help of suitable auxiliary variable, the nonlocal symmetries are transformed into Lie point symmetries, while the nonlinear Dirac system is enlarged to a prolonged system that contains auxiliary variable. The soliton periodic wave interaction solution and periodic wave solution for the Dirac system are obtained using the method of similarity reduction on the prolonged system. Furthermore, the conservation laws of the Dirac system are derived. [ABSTRACT FROM AUTHOR]

Electron–positron–ion plasmas emerge in active galactic nuclei, the primordial Universe, peripheries of neutron stars, and surroundings of black holes. Thus, this article showcases the analytical examination of a multi-dimensional fifth-order generalized Zakharov–Kuznetsov model with dual power-law nonlinearities in an electron–positron–ion magnetoplasma. This interesting electron–positron–ion plasma model, with enough nonlinear mathematics and astrophysics/cosmology considerations, is observed to possess various copious real-world scenarios, especially in cosmic plasmas. In essence, a thorough investigation of the model is carried out with a view to see the application of its results in various science and engineering disciplines. Abundant soliton solutions to the models are to be generated, and various wave structures of interest are to be simulated numerically. In the wake of the robust Lie group theory, a comprehensive Lie group analysis of this equation with power-law nonlinearities is further performed. This consequently leads to the emergence of diverse invariants and solutions associated with the model. In addition, the equation is reduced to diverse ordinary differential equations using its point symmetries, and consequently, diverse closed-form solutions of interest are achieved for some particular cases of n. These outcomes are obtained in the form of complex and non-complex dark solitons, topological solitons, as well as various algebraic solutions with arbitrary functions. Moreover, by utilizing the power series method, one derives some series solutions of the understudy models for some specific cases of some of the consequential difficult nonlinear ordinary differential equations. A deep understanding of the found solutions is aided by simulating some of the solutions. Consequently, various soliton collisions ensued, thus giving rise to diverse structures of psychedelic bump waves, parabolic waves, concentric wave shapes with strata, as well as other wave forms of interest which are discussed. The real-world applications of the various achieved wave dynamics are presented in detail to bring the pertinence of the research results home. Thereafter, strict self-adjointness as well as formal Lagrangian formulation, leading to various conservation laws via Ibragimov’s theorem, are entrenched. Consequently, conservation of energy, momentum, and angular momentum is achieved, which has a very wide spectrum of pertinence and significance, especially in the fields of physics and mathematics. [ABSTRACT FROM AUTHOR]

This paper investigates the invariance properties of the fractional Kersten–Krasil’shchik coupled KdV–mKdV system. By extending Lie symmetry analysis method to a fractional system, two vector fields are derived. The similarity transformation elegantly reduces the fractional system to a set of ordinary differential equations. Furthermore, the explicit power series solutions for the system are derived by employing the power series theory. Lastly, conservation laws for the system are established utilizing Ibragimov’s method. [ABSTRACT FROM AUTHOR]

For a fractional potential-KdV equation, Lie point symmetries are studied and their infinitesimal generators are investigated to construct conservation laws. Furthermore, the invariants of the obtained generators are used to reduce the studied equation to a nonlinear fractional ordinary differential equation (NFODE) and with Erdélyi–Kober fractional operator the reduced equation is solved explicitly by using the power series approach. [ABSTRACT FROM AUTHOR]

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]

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether's theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent's constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain. [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]

In this paper, we propose a quaternion form of equations describing electromagnetic field in a homogeneous isotropic medium without dispersion. It is shown that by renormalizing the values of field inductions and sources, one can transform the asymmetrical Maxwell equations to a highly symmetric form. This provides a possibility to introduce the scalar and vector field potentials and represent the generalized equation for electromagnetic field in the form of a single second-order quaternionic wave equation. We demonstrate that this equation reduces to the system of ordinary hyperbolic wave equations for the field potentials and on the other hand, the same equation is equivalent to the system of Maxwell equations for renormalized field intensities. The symmetry of the renormalized Maxwell equations allows one to obtain the second-order relations for energy and momentum, as well as for Lorentz invariants of the renormalized fields, which formally have the same form as for the fields in a vacuum. In addition, the generalization of renormalized equations to the case of magnetic sources corresponding to the models of Dirac magnetic monopoles and Schwinger dyons is discussed. [ABSTRACT FROM AUTHOR]

This study deals with exact solutions and soliton dynamics of Burgers equation. The Lie symmetry method under one parameter transformation is employed to establish symmetry condition, infinitesimals and their commutative relations. In consequence, the similarity variables are derived and cause to first symmetry reduction. A twice employment of method allows further symmetry reductions of test equations and results to systems of ODEs. Then, the ODEs have been integrated under parametric constraints and evolve desired exact solutions. These solutions consist of all the functions f 1 (t) , f 2 (t) , g 1 (y) , g 2 (y) existed in infinitesimals, more arbitrary functions F(X), H(X) and several arbitrary constants that make obtained results generalize than previous findings. To analyze the nature of physical phenomena associated with Burgers equation, these solutions have been supplemented with numerical simulation. Consequently, single soliton, doubly soliton, multisoliton, asymptotic nature, soliton fusion and fission nature are discussed systematically. Moreover, the Lagrangian formulation and conserved vectors associated with Lie symmetries are obtained. [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]

This paper presents a comprehensive analysis of the (1 + 1) -dimensional Klein-Gordan equation which plays a significant role in various areas of theoretical and applied physics. The main focus of this research centers on several key areas. First, infinitesimal generators of symmetries were found by Lie symmetry invariance analysis. Then, using the adjoint representation, an ideal system was created based on the found Lie vectors. Secondly, by utilizing the analytical approach, namely the modified Sardar sub equation method, we systematically derive various novel soliton solution in the form of dark, bright, periodic, singular, combo, hyperbolic as well as mixed trigonometric. Finally, bifurcation analysis is performed at the system's fixed points, revealing chaotic behavior when an external force is introduced into the dynamic system. To identify the chaotic characteristics, a range of tools, such as 3D and 2D phase plots, time series, Lyapunov exponents, and multistability analysis, are utilized. Additionally, the sensitivity analysis of the model is examined under different initial conditions. These results enhance the understanding of nonlinear wave phenomena in mathematical physics and hold potential applications across numerous scientific disciplines. [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]

The entropic regularization of the conservative stable discontinuous Galerkin method (DGM) in conservative variables for three-dimensional Euler equations is constructed through the use of a special slope limiter. This limiter ensures the fulfillment of the three-dimensional analogs of the monotonicity conditions and a discrete analog of the entropic inequality. The developed method is tested on a three-dimensional model problem of a Taylor–Green vortex. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the study of Lie symmetries and conservation laws for fractional partial differential equations featuring mixed derivatives of fractional and integer orders. Firstly, we explore the Lie symmetries permitted by the (2+1)-dimensional time-fractional dissipative long-wave system. Subsequently, leveraging these symmetries, we reduce the original (2+1)-dimensional partial differential equations to (1+1)-dimensional counterparts. We then proceed to develop explicit power series solutions along with several exact solutions. Lastly, employing the concept of nonlinear self-adjointness in the fractional context, we establish the conservation properties of the system, resulting in the derivation of novel non-trivial conservation law. [ABSTRACT FROM AUTHOR]

A detailed Lie symmetry analysis of the nonlinear damped Klein-Gordon Fock equation: $u_{tt}\,+\alpha (u)\,u_{t}\,=\,u_{xx}\,+\,f(u)$ is addressed in this paper. Applying the Lie symmetry method, a comprehensive Lie group classification is performed for the arbitrary smooth functions $\alpha (u)$ and $f(u)$ present in the equation, leading to two distinct cases. Additionally, for each case an optimal system of one-dimensional subalgebras is derived, which is a minimal set of all the linearly independent symmetry generators without redundant symmetries. Using the similarity transformation method, the above-mentioned partial differential equation is reduced into a set of ordinary differential equations. In certain cases, several exact invariant solutions encompassing the travelling wave solutions and soliton waves are obtained. The graphs of the soliton solutions and traveling wave solutions are also presented. Finally, the conservation laws are identified via the partial Noether approach, leading to distinct cases with several subcases. The derived conservation laws provide valuable tools for examining the dynamics and stability of physical systems, making this research suitable to a range of scientific studies.

We consider the continuous dependence on the initial and flux functions for entropy solutions to the Cauchy problem of non-homogeneous conservation laws. Under only the assumption that the flux function and the non-homogeneous term's function are locally Lipschitz continuous, we prove the continuous dependence on the initial and flux functions for the entropy solutions. [ABSTRACT FROM AUTHOR]

In this paper, the Lie symmetry analysis method is applied to the high dimensional fractional 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 derivatives are obtained and can be solved by different methods in the literatures herein. [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]

This article presents an analytical investigation performed on a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science. To start with, achieving diverse solitary wave solutions to the generalized power‐law model involves using wave transformation, which reduces the model to a nonlinear ordinary differential equation. A direct integration approach is adopted to construct solutions in the beginning. This brings the emergence of interesting solutions like non‐topological solitons, trigonometric functions, exponential functions, elliptic functions, and Weierstrass functions in general structures. Besides, in a bid to secure more general exact solutions to the model, one adopts the extended Jacobi elliptic function expansion technique (for some specific cases of n$$ n $$). Thus, various cnoidal, snoidal, and dnoidal wave solutions to the understudied model are attained. The copolar trio explicated in a tabular form reveals that these solutions can be relapsed to various hyperbolic and trigonometric functions under certain criteria. Additionally, diverse graphical exhibitions of the dynamical attributes of the gained results are presented in a bid to have a sound understanding of the physical phenomena of the underlying model. Later, one gives the conserved vectors of the aforementioned equation by employing the standard multiplier approach. [ABSTRACT FROM AUTHOR]

We systematically classify Lie symmetries of a class of (2 + 1)-dimensional nonlinear wave equations. Our methodology proposes a symmetry classification for Lie generators applicable to four distinct cases inherent within this equation. For each identified category, we comprehensively analyze symmetry reduction and delineate the invariant solutions. Furthermore, we extend our Lie symmetry analysis to encompass reduced 1 + 1 partial differential equations (PDEs). Through our investigations, we establish local conservation laws corresponding to each conserved vector, employing the formal Lagrangian approach. Significantly, this classification constitutes a novel contribution to the scientific discourse, as it remains absent from extant literature to date. [ABSTRACT FROM AUTHOR]

We give a complete description of inequivalent nontrivial local conservation laws of all orders for a natural generalization of the dissipative Westervelt equation and, in particular, show that the equation under study admits an infinite number of inequivalent nontrivial local conservation laws for the case of more than two independent variables. [ABSTRACT FROM AUTHOR]

The Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation is often used in dealing with long-wave propagation interactions, and is widely used in mathematics, physics, and engineering. This paper proposes a new extended (3+1)-dimensional KdV-CBS equation, and it’s never been studied. Additionally, we verify the integrability of the equation based on the Painlevé test. By employing Hirota’s method, a bilinear auto-Bäcklund transformation, the multiple-soliton solutions, and the soliton molecules of the equation are derived. New exact solutions of the equation are constructed utilizing the power series expansion method and (G ′ / G) -expansion method. These exact solutions are also presented graphically. Finally, the conservation laws of the equation are obtained. Our results are helpful for understanding nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983. https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023. https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023. https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022. https://doi.org/10.1007/s00205-021-01736-2). For L t q L x r suitable Leray–Hopf solutions of the d - dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure P s , which gives s = d - 2 as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity. [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]

This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end. [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]

In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim is to construct closed-form solutions of the equation using a powerful technique, namely the Lie group analysis method. Firstly, we derive the Lie point symmetries of the equation. Thereafter, the equation is reduced to non-linear ordinary differential equations using symmetry reductions. Furthermore, the solutions of the equation are derived using the extended Jacobi elliptic function technique, the simplest equation method, and the power series method. In conclusion, we construct conservation laws for the equation using Noether's theorem and the multiplier approach, which plays a crucial role in understanding the behavior of non-linear equations, especially in physics and engineering, and these laws are derived from fundamental principles such as the conservation of mass, energy, momentum, and angular momentum. [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]