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High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities. [ABSTRACT FROM AUTHOR]

The rational solitons and other wave solutions demonstrate the existence of wave packets propagating through optical Kerr media, exhibiting non-diffractive and non-dispersive spatiotemporal localization. This paper focuses on constructing rational solitons, multi-wave solutions, and other solutions for a dimensionless time-dependent paraxial wave model (dt-pPWM). These solutions are derived through the application of symbolic computation and the ansatz function method, using both traveling wave and logarithmic transformations. Our approach encompasses three distinct methods: the positive quadratic function approach, the three-wave approach, and the double exponential approach. These wave results find practical applications and contribute to a deeper understanding of the physical phenomena described by this model. We have computed conservative quantities corresponding to the power, momentum, and energy of solitons. To assess the stability of the wave equation, we conducted modulational instability analysis, confirming the stability and accuracy of all soliton solutions. The structure of the solution provides a physical interpretation. By selecting appropriate parametric values, we generated 3D graphics illustrating various multi-wave solutions, including lump waves, rogue waves, and multipeak solitons. Additionally, we observed intriguing phenomena arising from the interaction of multi-waves, which have applications in telecommunications, quantum optics and other related fields. [ABSTRACT FROM AUTHOR]

Potential vorticity (PV) is one of the most important quantities in atmospheric science. In the absence of dissipative processes, the PV of each fluid parcel is known to be conserved, for a dry atmosphere. However, a parcel's PV is not conserved if clouds or phase changes of water occur. Recently, PV conservation laws were derived for a cloudy atmosphere, where each parcel's PV is not conserved but parcel‐integrated PV is conserved, for integrals over certain volumes that move with the flow. Hence a variety of different statements are now possible for moist PV conservation and non‐conservation, and in comparison to the case of a dry atmosphere, the situation for moist PV is more complex. Here, in light of this complexity, several different definitions of moist PV are compared for a cloudy atmosphere. Numerical simulations are shown for a rising thermal, both before and after the formation of a cloud. These simulations include the first computational illustration of the parcel‐integrated, moist PV conservation laws. The comparisons, both theoretical and numerical, serve to clarify and highlight the different statements of conservation and non‐conservation that arise for different definitions of moist PV. [ABSTRACT FROM AUTHOR]

Nonlinear waves are pivotal in analyzing the propagation of electromagnetic waves and the dynamics of oceanic systems. These waves are indispensable for modeling the long-term impacts of wave energy in coastal regions, addressing critical issues such as climate change adaptation, erosion and coastal flooding. The Bogoyavlensky–Konoplechenko system describes the interaction of the Riemann wave and the long wave propagating in two dimensions. The present work aims to elaborate symmetry reductions and derive invariant solutions of the proposed system. Meanwhile, the infinitesimal generators under one-parameter transformation are constructed, which render the system invariant. Therefore, a repeated process of reductions results in an equivalent system of ordinary differential equations and hence, leads to exact solutions. The solutions have rich physical significance and are efficient in defining several phenomena due to existing arbitrary functions and constants. To examine the physical nature of these solutions, numerical simulation is performed and thus, wave structures like bright and dark lumps, multisoliton, line multisoliton, periodic and annihilation profiles are analyzed. The bifurcation theory has been used to investigate the stability of dynamical system and examine corresponding phase portraits. Furthermore, the conserved vectors with underlying symmetries are constructed using Noether’s theorem. [ABSTRACT FROM AUTHOR]

Analytical solutions of the generalized nonlinear Schrödinger with four powers of nonlinearity for description of propagating pulses in optical fiber are presented. Optical solitons corresponding to the mathematical model are given. Conservation laws of the generalized model for propagation pulses with four powers of nonlinearity are written. To the best of our knowledge, the conservation laws obtained have not yet been presented in literature. The equation investigated generalizes several well-known models, which allows us to evaluate the influence of various processes on pulse propagation. Conservative quantities for the bright optical soliton, corresponding to its power, momentum and energy, are calculated. The analytical expressions for conservative quantities obtained can be applied to check whether numerical schemes for the explored equation are conservative. [ABSTRACT FROM AUTHOR]

The goal of this article is to compute conservation law, Lagrangian and μ-conservation law of the generalized Rosenau-type equation using the homotopy operator, the μ-symmetry method and the variational problem method. The generalized Rosenau-type equation includes the generalized Rosenau equation, the generalized Rosenau-RLW equation and the generalized Rosenau-KdV equation, which admits the third-order Lagrangian. The article also compares the conservation law and the μ-conservation law of these three equation. [ABSTRACT FROM AUTHOR]

We prove that rough Follow-the-Leader (FtL) approximations are compact in the strong $ L^1 $–topology, with a compensated compactness argument. We then apply this result to prove the existence of a weak solution of the one-dimensional Hughes model. Here, we consider general initial data $ \rho_0 \in L^\infty $ and general costs $ c \in W^{1, \infty} $. We also investigate the ill-posedness of the FtL–Hughes model, and how the ill-posedness issues affect numerical simulations. [ABSTRACT FROM AUTHOR]

The goal of this article is to compute conservation law, Lagrangian and $\mu$-conservation law of the generalized Rosenau-type equation using the homotopy operator, the $\mu$-symmetry method and the variational problem method. The generalized Rosenau-type equation includes the generalized Rosenau equation, the generalized Rosenau-RLW equation and the generalized Rosenau-KdV equation, which admits the third-order Lagrangian. The article also compares the conservation law and the $\mu$-conservation law of these three equation.

In this article, we mainly prove low regularity conservation laws for the Fokas-Lenells equation in Besov spaces with small initial data both on the line and on the circle. We develop a new technique in Fourier analysis and complex analysis to obtain the a priori estimates. It is based on the perturbation determinant associated with the Lax pair introduced by Killip, Vişan, and Zhang for completely integrable dispersive partial differential equations. Additionally, we also utilize the perturbation determinant to derive the global a priori estimates for the Schwartz solutions to the Camassa-Holm (CH) equation in H1{H}^{1}. Even though the energy conservation law of the CH equation is a fact known to all, the perturbation determinant method indicates that we cannot get any conserved quantities for the CH equation in Hk{H}^{k} except k=1k=1.

We examine the vanishing adsorption limit of solutions of Riemann problems for the Glimm–Isaacson model of chemical flooding of a petroleum reservoir. A contact discontinuity is deemed admissible if it is the limit of traveling waves or rarefaction waves for an augmented system that accounts for weak chemical adsorption onto the rock. We prove that this criterion justifies the admissibility criteria adopted previously by Keyfitz–Kranzer, Isaacson–Temple and de Souza–Marchesin, provided that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence. [ABSTRACT FROM AUTHOR]

In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation. [ABSTRACT FROM AUTHOR]

In this paper, we define the ruled surface in terms of the Darboux vector field of an isotropic curve in a complex 3-space, and we study the ruled surface as an elastic strip along the isotropic curve. First of all, we show that elastic strips along isotropic curves that serve critical points of the modified Sadowsky functional are characterized by three Euler-Lagrange equations. As a result, we give two conservation laws to characterize elastic strips of isotropic curves in complex 3-space and explain an isotropic helix in terms of the force vector. Finally, we give some examples to illustrate elastic strips with isotropic curves in a complex 3-space. [ABSTRACT FROM AUTHOR]

Partial differential equations are widely used to describe complex phenomena in various branches of science, including: physics, mechanics, etc. Therefore, obtaining high-precision target solutions and, if possible, finding exact analytical solutions of these equations play an important role in these sciences. In this paper, we first examine Lie symmetry and invariant transformations. Next, we obtain the equation solutions with the help of symmetries. We compute invariants and similarity solutions. Spectral method was used to numerically solve the equation. Then, we introduce two approaches, the Homotopy method and the direct method, which are used to construct conservation laws. Then we construct three conservation laws for the Degasperis-Procesi equation by using the direct method. Numerical solution and analysis of the Degasperis-Procesi equation by the methods proposed in this research have not been done in previous studies. Also, in Iran, this is the first time that this research has been done, which is a combination of mathematics, physics and computer science, and is considered interdisciplinary and applied. [ABSTRACT FROM AUTHOR]

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma. [ABSTRACT FROM AUTHOR]

Abstract Potential vorticity (PV) is one of the most important quantities in atmospheric science. In the absence of dissipative processes, the PV of each fluid parcel is known to be conserved, for a dry atmosphere. However, a parcel's PV is not conserved if clouds or phase changes of water occur. Recently, PV conservation laws were derived for a cloudy atmosphere, where each parcel's PV is not conserved but parcel‐integrated PV is conserved, for integrals over certain volumes that move with the flow. Hence a variety of different statements are now possible for moist PV conservation and non‐conservation, and in comparison to the case of a dry atmosphere, the situation for moist PV is more complex. Here, in light of this complexity, several different definitions of moist PV are compared for a cloudy atmosphere. Numerical simulations are shown for a rising thermal, both before and after the formation of a cloud. These simulations include the first computational illustration of the parcel‐integrated, moist PV conservation laws. The comparisons, both theoretical and numerical, serve to clarify and highlight the different statements of conservation and non‐conservation that arise for different definitions of moist PV.

Abstract Weighted compact nonlinear schemes (WCNS) are a family of nonlinear shock capturing schemes that are suitable for solving problems with discontinuous solutions. The schemes are based on grids staggered by flux points and solution points, resulting in algorithms with the nonlinear interpolation step independent of the difference step. Thus, only linear difference operators are needed, such that geometric conservation law can be preserved easily, resulting in the preservation of freestream condition. In recent years, these schemes have attracted a lot of attention in the community of computational fluid dynamics. This paper intends to give a brief review of the basic algorithms of these schemes and present some related recent developments.

Predicting future discontinuous phenomena that are unobservable from training data sets has long been a challenging problem in scientific machine learning. We introduce a novel paradigm to predict the emergence and evolution of various discontinuities of hyperbolic partial differential equations (PDEs) based on given training data over a short window with limited discontinuity information. Our method is inspired by the classical Roe solver [P. L. Roe, J Comput Phys., vol. 43, 1981], a basic tool for simulating various hyperbolic PDEs in computational physics. By carefully designing the computing primitives, the data flow, and the novel pseudoinverse processing module, we enable our data‐driven predictor to satisfy all the essential mathematical criteria of a Roe solver and hence deliver accurate predictions of hyperbolic PDEs. We demonstrate through various examples that our data‐driven Roe predictor outperforms original human‐designed Roe solver and deep neural networks with weak priors in terms of accuracy and robustness. [ABSTRACT FROM AUTHOR]

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates. [ABSTRACT FROM AUTHOR]

In this work, a field representation of the conservation law of linearmomentum is derived fromthe atomistic, using the theory of distributions as the mathematical tool, and expressed in terms of temperature field by defining temperature as a derived quantity as that in molecular kinetic theory and atomistic simulations. The formulation leads to a unified atomistic and continuum description of temperature and a new linear momentum equation that, supplemented by an interatomic potential, completely governs thermal and mechanical processes across scales from the atomic to the continuum. The conservation equation can be used to solve atomistic trajectories for systems at finite temperatures, as well as the evolution of field quantities in space and time, with atomic or multiscale resolution. Four sets of numerical examples are presented to demonstrate the efficacy of the formulation in capturing the effect of temperature and thermal fluctuations, including phonon density of states, thermally activated dislocation motion, dislocation formation during epitaxial processes, and attenuation of longitudinal acoustic waves as a result of their interaction with thermal phonons. [ABSTRACT FROM AUTHOR]

A simple cell population growth model is proposed, where cells are assumed to have a physiological structure (e.g., a model describing cancer cell maturation, where cells are structured by maturation stage, size, or mass). The main question is whether we can guarantee, using the death rate as a control mechanism, that the total number of cells or the total cell biomass has prescribed dynamics, which may be applied to modeling the effect of chemotherapeutic agents on malignant cells. Such types of models are usually described by partial differential equations (PDE). The population dynamics are modeled by an inverse problem for PDE in our paper. The main idea is to reduce this model to a simplified integral equation that can be more easily studied by various analytical and numerical methods. Our results were obtained using the characteristics method. [ABSTRACT FROM AUTHOR]

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

Weighted compact nonlinear schemes (WCNS) are a family of nonlinear shock capturing schemes that are suitable for solving problems with discontinuous solutions. The schemes are based on grids staggered by flux points and solution points, resulting in algorithms with the nonlinear interpolation step independent of the difference step. Thus, only linear difference operators are needed, such that geometric conservation law can be preserved easily, resulting in the preservation of freestream condition. In recent years, these schemes have attracted a lot of attention in the community of computational fluid dynamics. This paper intends to give a brief review of the basic algorithms of these schemes and present some related recent developments. [ABSTRACT FROM AUTHOR]

In this paper, we present a finite difference scheme for the triple-coupled Schrödinger equations (T-CNLS) in optics. The T-CNLS is approximated by Crank-Nicolson scheme in time and finite difference method in space. Some mathematical characters are investigated, such as structure-preserving properties, unique solvability, convergence in L ∞ norm. Some numerical examples are reported to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Several definitions of the energy–momentum tensor in classical field models on differential manifolds are examined in this work. We specifically talk about the connection between - Hilbert's symmetric and Noether's asymmetric energy–momentum tensors. We consider a matter field Lagrangian on a manifold endowed with a teleparallel (coframe) structure. The associated 3-form of coframe conserved current can be viewed as a link between Noether's and Hilbert's energy–momentum tensors. In these circumstances, we are able to show the complete equivalence of Hilbert's and Noether's currents for the matter field. [ABSTRACT FROM AUTHOR]

The central objective of this study is to examine how the Lie symmetry technique and conservation law theories can be employed to analyze fractional partial differential equations with a mixed derivative ∂ t 2 β (u xx) , specifically considering the Riemann–Liouville time-fractional and integer-order x-derivatives. In the beginning, we shed light on the prolongation formula that encompasses the infinitesimal generator, particularly in the presence of mixed derivatives. Following that, through the identification of Lie symmetries, we introduce a comprehensive formula for deriving conservation laws, incorporating the notion of nonlinear self-adjointness. Utilizing the Lie symmetries and an alternative reduction technique, we achieve a novel reduction that eliminates the need for the Erd e ´ lyi-Kober operator. Additionally, family of wave solutions have been obtained corresponding to the longitudinal wave equation in magneto-electro-elastic structures and bifurcation analysis and chaotic analysis has been done for the governing model via Galilean transformation and phase plane portraits have been used to depict the changes in the behavior of the model by changing the values of the potential parameter. [ABSTRACT FROM AUTHOR]

We study the initial-value problem for semilinear hyperbolic systems in one space dimension. When the power of nonlinearity is even (n = 2 k) , we have global existence of solution which is related with the repulsive Liouville equation. When n = 2k−1, we may have a finite time blow-up solution which is related with the attractive Liouville equation. [ABSTRACT FROM AUTHOR]

The slender circle submarine pipeline possesses both shell and beam characteristics, which are widely used in practical engineering. Unfortunately, for some reason, there will be some geometric defects in the cross-section of the pipeline (such as machining errors and seawater corrosion, etc.), resulting in the stiffness of the circular submarine pipeline being different. The cracked variable stiffness pipeline is actually a typical three-dimensional complex crack problem. In this paper, taking the outer circle and inner ellipse submarine pipeline as an example, a method based on elementary mechanics and the concept of the conservation law is proposed to determine the stress intensity factors (SIFs) of variable stiffness pipeline, and a series of closed-form expressions of SIF are derived for the cracked pipeline. The results from the present method are shown to agree well with available solutions and FEM. In particular, the present method can also be easily applied to other similar variable stiffness submarine pipelines. [ABSTRACT FROM AUTHOR]

The paper analyzes one of the models of equations of magnetohydrodynamics (MHD) derived earlier. The model was obtained as a result of group classification of the MHD equations in mass Lagrangian coordinates, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. The use of Lagrangian coordinates made it possible to solve four equations, which led to the form of reduced equations containing four arbitrary functions: entropy and a three‐dimensional vector associated with the magnetic field. The objective of this work is to develop conservation laws and exact solutions for the model. Conservation laws are obtained using Noether's theorem, while exact solutions are obtained either explicitly or by solving a system of ordinary or partial differential equations with two independent variables. Numerical methods are employed for the latter solutions. [ABSTRACT FROM AUTHOR]

The multi-component nonlinear Schrödinger equations (MNLS) are derived by extending the single-component nonlinear Schrödinger equation to multiple interacting fields. These equations often describe the dynamics of wave packets in quantum mechanics or nonlinear optics. In this paper, we investigate MNLS equations via the Lie symmetry method. The Lie infinitesimal symmetries of the MNLS equations are derived by solving recursive determining equations, and the symmetry reductions of the equations are given by using symmetry variables. Moreover, some interesting explicit solutions for the equations are constructed. Finally, the conservation laws of the MNLS equations are obtained utilizing Ibragimov's method with detailed derivation. [ABSTRACT FROM AUTHOR]

Based on classical Lie symmetry method, the one-dimensional non-linear wave equation is investigated. By using four-dimensional subalgebras of the equation, the invariant groups and commutator table are constructed. Furthermore, optimal system of the equation is obtained, and the exact solutions can be gained by solving reduced equations. Finally, a complete derivation of the conservation law is given by using conservation multipliers. [ABSTRACT FROM AUTHOR]

This study investigates via Lie symmetry analysis the Hunter–Saxton equation, an equation relevant to the theoretical analysis of nematic liquid crystals. We employ the multiplier method to obtain conservation laws of the equation that arise from first-order multipliers. Conservation laws of the equation, combined with the admitted Lie point symmetries, enable us to perform symmetry reductions by employing the double reduction method. The method exploits the relationship between symmetries and conservation laws to reduce both the number of variables and the order of the equation. Five nontrivial conservation laws of the Hunter–Saxton equation are derived, four of which are found to have associated Lie point symmetries. Applying the double reduction method to the equation results in a set of first-order ordinary differential equations, the solutions of which represent invariant solutions for the equation. While the double reduction method may be more complex to implement than the classical method, since it involves finding Lie point symmetries and deriving conservation laws, it has some advantages over the classical method of reducing PDEs. Firstly, it is more efficient in that it can reduce the number of variables and order of the equation in a single step. Secondly, by incorporating conservation laws, physically meaningful solutions that satisfy important physical constraints can be obtained. [ABSTRACT FROM AUTHOR]

This article investigates the potential Kadomtsev–Petviashvili (pKP) equation, which describes the evolution of small-amplitude nonlinear long waves with slow transverse coordinate dependence. For the first time, we employ Lie symmetry methods to calculate the Lie point symmetries of the equation, which are then utilized to derive exact solutions through symmetry reductions and with the help of Kudryashov’s method. The solutions obtained include exponential, hyperbolic, elliptic, and rational functions. Furthermore, we provide one-parameter group of transformations for the pKP equation. To gain a better understanding of the nature of each solution, we present 3D, 2D, and density plots. These obtained solutions, along with their associated physical characteristics, offer valuable insights into the propagation of small yet finite amplitude waves in shallow water.In addition, the pKP equation conserved vectors are derived by utilizing the multiplier method and the theorems by Noether and Ibragimov.

In nonlinear conservation laws, there is a possibility of intersecting characteristic lines or even regions that are not traversed by characteristic lines. In regions where there are two or more characteristic lines, a gradien catastrophic phenomenon can occur, resulting in the emergence of a breaking time, thus a weak solution of the nonlinear conservation law is considered. Such a weak solution is capable of producing shock wave solutions at the breaking time. Additionally, there are rarefaction wave solutions for regions that do not have characteristics. The construction of shock wave and rarefaction wave solutions can be observed in a case study in traffic flow, specifically before and after a traffic light turns red. Shock waves are observed in the flow of vehicles before the red light, while rarefaction waves occur when the red light changes to green

Split-step quintic B-spline collocation (SS5BC) methods are constructed for nonlinear Schrödinger equations in one, two and three dimensions in this paper. For high dimensions, new notations are introduced, which make the schemes more concise and achievable. The solvability, conservation and linear stability are discussed for the proposed methods. Numerical tests are carried out, and the present schemes are numerically verified to be convergent with second-order in time and fourth-order in space. The conserved quantity is also computed which agrees with the exact one. And solitary waves in one, two and three dimensions are simulated numerically which coincide with the exact ones. The SS5BC scheme is compared with the split-step cubic B-spline collocation (SS3BC) method in the numerical tests, and the former scheme is more efficient than the later one. Finally, the SS5BC scheme is also applied to compute Bose-Einstein condensates.

In this paper, we study the a -contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient a can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for 2 × 2 conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for 2 × 2 systems of hyperbolic conservation laws are proved. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity properties of bounded entropy solutions to the isentropic Euler equations with γ = 3. First, we use a blow-up technique to obtain a new trace theorem for all such solutions. Second, we use a modified De Giorgi type iteration on the kinetic formulation to show a new partial regularity result on the Riemann invariants. We are able to conclude that in fact for any bounded entropy solution u, the density p is almost everywhere upper semicontinuous away from vacuum. To our knowledge, this is the first example of a nonlinear hyperbolic system, which fails to be Temple class, but has the property that generic L∞ initial data give rise to bounded entropy solutions with a form of near classical regularity. This provides one example that 2 × 2 hyperbolic systems can possess some of the more striking regularizing effects known to hold generically in the genuinely nonlinear, multidimensional scalar setting. While we are not able to use our regularity results to show unconditional uniqueness, the results substantially lower the likelihood that current methods of convex integration can be used in this setting. [ABSTRACT FROM AUTHOR]

We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry. This gives exact conservation laws for several theories, e.g., lattice electrodynamics and gauge theory. In particular, we construct a conserved discrete energy–momentum tensor, approximating the continuum one at least for free fields. The theory is stated in topological terms, such as coboundary and products of cochains. [ABSTRACT FROM AUTHOR]

We consider the differential equations that describe the Darcy–Brinkman flow. We provide the Lie symmetry classification of this system, construct conservation laws and study the system that describes the traveling wave solutions. We show that the integration of the last system is reducible to the Abel ordinary differential equation and indicate a case when this ordinary differential equation (ODE) is integrable in quadratures. [ABSTRACT FROM AUTHOR]

Split-step quintic B-spline collocation (SS5BC) methods are constructed for nonlinear Schrödinger equations in one, two and three dimensions in this paper. For high dimensions, new notations are introduced, which make the schemes more concise and achievable. The solvability, conservation and linear stability are discussed for the proposed methods. Numerical tests are carried out, and the present schemes are numerically verified to be convergent with second-order in time and fourth-order in space. The conserved quantity is also computed which agrees with the exact one. And solitary waves in one, two and three dimensions are simulated numerically which coincide with the exact ones. The SS5BC scheme is compared with the split-step cubic B-spline collocation (SS3BC) method in the numerical tests, and the former scheme is more efficient than the later one. Finally, the SS5BC scheme is also applied to compute Bose-Einstein condensates. [ABSTRACT FROM AUTHOR]

The solution of the integrable Hirota equation has attracted considerable attention in the applications of nonlinear optics, electromagnetics, and many other natural sciences. In this paper, we propose an improved physics-informed neural network (IPINN) method to study numerical solutions of the Hirota equation, which embeds energy conservation laws into a traditional neural network through the Lax pair formulation. Our simulation results show that the proposed method can predict the solutions and parameters of the Hirota equation more accurately than the traditional physics-informed neural network method. In addition, the influence on the rogue wave solution for the Hirota equation of the three factors of the IPINN method that are, number of network layers and hidden layer neurons, sampling points, and noises, is also analyzed in detail. In our study, it is worth noting that the presented method can achieve good prediction with fewer training data and iterations. [ABSTRACT FROM AUTHOR]

In this paper, the Lie group of point symmetries for a kind of Benjamin-Bona-Mahoney (BBM) equation is obtained by applying the classical Lie symmetry method. An optimal system of sub-algebras of dimension one for the BBM equation is deduced by classifying the adjoint representation orbits of the Lie symmetry group. Then, for any infinitesimal symmetry generators of the Lie group, the related similarity reductions are generated and new group invariant solutions for the BBM equation are obtained. Also, new conservation laws for this equation are constructed by the method of scaling. The conservation laws densities are calculated by using the concept of variables weight, scaling symmetry and Euler operator and their fluxes are computed by applying the homotopy operator. [ABSTRACT FROM AUTHOR]

We prove that the two-component peakon solutions are orbitally stable in the energy space. The system concerned here is a two-component Novikov system, which is an integrable multi-component extension of the integrable Novikov equation. We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures. Moreover, we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method. [ABSTRACT FROM AUTHOR]

This work is intended to compute the $\mu$-symmetry and $\mu$-conservation laws for the Cammasa-Holm (CH) equation and the Hunter-Saxton (HS) equation. In other words, $\mu$-symmetry, $\mu$-symmetry reduction, variational problem, and $\mu$-conservation laws for the CH equation and the HS equation are provided. Since the CH equation and the HS equation are of odd order, they do not admit a variational problem. First we obtain $\mu$-conservation laws for both of them in potential form because they admit a variational problem and then using them, we obtain $\mu$-conservation laws for the CH equation and the HS equation.