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In this paper, we discuss a new approach to balancing (known as dynamic–quadratic balancing) and model reduction for affine nonlinear system. We give a fresh look to balancing in terms of the dynamics of the system, rather than simply a structural property. With this perspective, we also develop a new approach to obtaining the balancing transformation in one step, instead of a three‐step process as proposed in earlier methods. Further, we explore the relationship between quadratic balancing and input–output stability. In addition, we also develop a computational approach for obtaining the balancing transformation by solving a coupled system of PDEs or inequalities (Lyapunov/Hamilton–Jacobi type). After that, model reduction can be carried out in the conventional way using Hankel singular‐value functions or using a new criterion. Finally, we present some examples to clarify the results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the Hamilton–Jacobi equation with phase constraints arising in molecular biology. The problem has no classical solutions, and the Hamiltonian does not satisfy the conditions guaranteeing the existence of minimax or viscous generalized solutions. A new continuous generalized solution is obtained. We present sufficient conditions for the existence of a global solution preserving the structure given by the initial manifold. The behavior of such solutions is examined for large values of time. [ABSTRACT FROM AUTHOR]

This paper studies a maximal L q -regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a L p duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold q = d (γ − 1) / γ , d being the dimension of the ambient space and γ > 1 the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range q > 1. [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 the setting of Carnot groups, we prove the q -logarithmic Sobolev inequality for probability measures as a function of the Carnot–Carathéodory distance. As an application, we use the Hamilton–Jacobi equation in the setting of Carnot groups to prove the p -Talagrand inequality and hypercontractivity. [ABSTRACT FROM AUTHOR]

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices. [ABSTRACT FROM AUTHOR]

• The thermal process of NEPE propellant under transient condition is simulated. • The reaction performance of NEPE propellant is firstly developed. • The simulated microscale flame structure evolution is captured. • The non-planar moving surface is used to represent the burning process. A sophisticated numerical framework has been proposed as a powerful tool to comprehend the thermal behavior of NEPE propellant during rapid pressure decay. This framework consists of two crucial components: a 2D propellant pack generation module based on the sequential algorithm and a refined combustion model code. To track the regression of the propellant surface, we employ a straightforward single-value equation of the Hamilton-Jacobi type. This equation enables us to model the movement and evolution of the propellant surface as it undergoes combustion. To validate the credibility of the simulations, comparisons between the simulated burning rate and predicted average surface temperature are made with experimental studies. Encouragingly, reasonable agreement has been achieved between the numerical predictions and experimental data, further affirming the reliability and effectiveness of the proposed numerical framework. Simulation results suggest that the flame structure transforms from a complex diffusion flame in the initial stage of this transient burning to a premixed flame with a heat core altering with time in the final stage of fast depressurization. The surface cores of coarse AP and HMX particles regress most slowly, while the interface between AP and HMX powders moves most fast. Regarding the regression of the propellant surface, it is observed that the surface core of coarse AP and HMX particles exhibits a slower regression rate, while the interface between AP and HMX powders experiences a faster movement. This discrepancy in regression rates underscores the heterogeneity of NEPE propellant combustion, where different components contribute differently to the overall regression process. Furthermore, the entire surface morphology undergoes significant changes during rapid depressurization and combustion in the atmospheric environment. This suggests that pressure changes alone are not the sole factors influencing the shape of the propellant surface. The presence of oxidizing particles within the propellant pack emerges as an important factor that influences the regression mechanism of the propellant surface. This investigation provides valuable insights into the dynamic behavior of NEPE propellant, shedding light on the underlying physical phenomena at play. [ABSTRACT FROM AUTHOR]

On (pseudo)Riemannian manifolds of two and three dimensions, we list all metrics that admit a complete separation of variables in the Hamilton–Jacobi equation for geodesics. There are three different classes of separable metrics on two-dimensional surfaces. Three-dimensional manifolds admit six classes of separable metrics. Within each class, metrics are related by canonical transformations and a nondegenerate transformation of parameters that does not depend on coordinates. [ABSTRACT FROM AUTHOR]

This paper discusses the multi-player non-cooperative game of nonlinear stochastic time-varying systems described by Itô-type differential equations in a finite time interval. Multi-player non-cooperative game problems are represented by multi-objective Pareto (MOP) control problems to describe the fact that each player has their own goals. By applying Hamilton–Jacobi inequalities (HJIs), the criterion of upper bounds of the MOP boundary is obtained for nonlinear stochastic systems, and the corresponding strategies are designed for such games, so the MOP problem is transformed into a HJI-constrained MOP problem. In order to overcome the difficulty of solving HJIs, a global linearization method is proposed to approximate the nonlinear systems. By the proposed global linearization method, multi-player non-cooperative game problems are transformed into Riccati equation-constrained MOP problems, and the approximate solutions of HJI-constrained MOP problems are obtained. Finally, a practical example is given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a finite-horizon two-person zero-sum differential game in which the system dynamics is described by a linear differential equation with a Caputo fractional derivative and the goals of the players' control are to minimize and maximize a quadratic terminal-integral cost function, respectively. We present conditions for the existence of a game value and obtain formulas for players' optimal feedback control strategies with memory of motion history. The results are based on the construction of a solution to an appropriate Hamilton–Jacobi equation with fractional coinvariant derivatives under a natural right-end boundary condition. [ABSTRACT FROM AUTHOR]

A review of the basic Hansen's ideal frame algorithms for accurate numerical integration of perturbed elliptic motion is carried out. The fundamental approaches rely on the use of nonsingular variables and differ in the ways in which the ellipse in the orbital plane is determined. It is well known that the accuracy of the propagation of the orbit geometry is notably increased when using time-regularization techniques to transform the independent variable. However, this is at the expense of adding a differential equation to compute the time, which gathers the Lyapunov-type instabilities that are removed from the coordinates. The asynchronism resulting from errors in the numerical integration of the time may be palliated with the use of time elements, to which end a constant and a linear nonsingular time element are presented, which are new to our knowledge. [ABSTRACT FROM AUTHOR]

This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels. [ABSTRACT FROM AUTHOR]

In this paper, we consider the full velocity difference model for traffic flow and we study the existence and uniqueness of traveling wave solutions. First, using the monotony of the car’s interdistance, we derive necessary conditions for the existence of such solutions. Then, in the framework of viscosity solutions, we construct a traveling wave solution by considering an approximate non-local Hamilton–Jacobi equation on a bounded domain. This traveling wave solution can be interpreted as a phase transition between a congested state and a free-flow one. [ABSTRACT FROM AUTHOR]

The method of characteristics is extended to set-valued Hamilton-Jacobi equations. This problem arises from a calculus of variations’ problem with a multicriteria Lagrangian function: through an embedding into a set-valued framework, a set-valued Hamilton-Jacobi equation is derived, where the Hamiltonian function is the Fenchel conjugate of the Lagrangian function. In this paper a method of characteristics is described and some results are given for the Fenchel conjugate. [ABSTRACT FROM AUTHOR]

The notion of strict singular characteristics is important in the wellposedness issue of singular dynamics on the cut locus of the viscosity solutions. We provide an intuitive and rigorous proof of the existence of the strict singular characteristics of Hamilton–Jacobi equation H (x , D u (x) , u (x)) = 0 in two dimensional case. We also proved if x is a strict singular characteristic, then we really have the right-differentiability of x and the right-continuity of x ˙ + (t) for every t. Such a strict singular characteristic must give a selection p (t) ∈ D + u (x (t)) such that p (t) = arg min p ∈ D + u (x (t)) H (x (t) , p , u (x (t))) . [ABSTRACT FROM AUTHOR]

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem, we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves. [ABSTRACT FROM AUTHOR]

We study a reaction-diffusion problem formulated with a higher- order operator, a non-linear advection, and a Fisher-KPP reaction term de- pending on the spatial variable. The higher-order operator induces solutions to oscillate in the proximity of an equilibrium condition. Given this oscillatory character, solutions are studied in a set of bounded domains. We introduce a new extension operator, that allows us to study the solutions in the open domain RN, but departing from a sequence of bounded domains. The analysis about regularity of solutions is built based on semigroup theory. In this approach, the solutions are interpreted as an abstract evolution given by a bounded continuous operator. Afterward, asymptotic profiles of solutions are studied based on a Hamilton-Jacobi equation that is obtained with a single point exponential scaling. Finally, a numerical assessment, with the function bvp4c in Matlab, is introduced to discuss on the validity of the hypothesis. [ABSTRACT FROM AUTHOR]

A class of exact (non-perturbative) models of strong gravitational waves based on Shapovalov type III spacetimes and Einstein's vacuum equations is obtained. Exact solutions are found for the trajectories of particles and radiation in a gravitational wave in privileged coordinate systems. Exact solutions are obtained for the equations of geodesic deviation and tidal acceleration of particles in a gravitational wave in privileged coordinate systems. An explicit analytical law of transition from a privileged coordinate system to a synchronous reference system associated with a freely falling observer with an explicit selection of time and spatial coordinates is obtained. An explicit form of the metric of a gravitational wave in a synchronous frame of reference is obtained. For a synchronous frame of reference, the trajectories of particles and radiation, the deviation of geodesics, and tidal accelerations in a gravitational wave are obtained. The presented methods and approaches are applicable both to Einstein's general theory of relativity and to modified theories of gravity. [ABSTRACT FROM AUTHOR]

We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. The proof is based on a theorem by Marie-Claude Arnaud ([4]). We also gave a new proof of the theorem in time-dependent case. [ABSTRACT FROM AUTHOR]

This paper proposes a novel strategy to design a suboptimal attitude tracking control law for a near-space hypersonic vehicle (NSHV) based on the Koopman operator and stable manifold theory. The nonlinear vector field of the NSHV attitude model is locally Lipschitz continuous and can be approximated by a high-dimensional linear system over a compact set. Linear and nonlinear parts of the attitude dynamics are determined based on this system. Subsequently, the stable manifold theory is applied to determine the unconstrained approximated optimal control law that is used to further consider the control input constraints of the NSHV attitude model. The suboptimality of the control law is analyzed, and the local exponential stability of the closed-loop system with input constraints is proven. Furthermore, the eigenvalues for the closed-loop nonlinear attitude error dynamics are analyzed. After the control input saturation, the nonlinear closed-loop error dynamics of the NSHV can be approximated by a high-dimensional linear system with a minimal dimension. The eigenvalues of this linear system indicate the stability and time response characteristics of the attitude error dynamics of the NSHV. The numerical simulation results demonstrate the effectiveness and suboptimality of the proposed attitude tracking control law and workflow of the eigenvalue analysis. [ABSTRACT FROM AUTHOR]

Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation U t + H (U x) = 0 and signed Radon measure valued entropy solutions of the conservation law u t + [ H (u) ] x = 0 . After having proved a precise statement of the formal relation U x = u , we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton–Jacobi equation and signed singular measures in case of the conservation law. [ABSTRACT FROM AUTHOR]

Optimal control problem is typically solved by first finding the value function through the Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value function, we propose a new formulation for the gradient of the value function (value-gradient) as a decoupled system of partial differential equations in the context of a continuous-time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the fact that they share the same characteristic curves as the HJE for the value function. For the theoretical part, we prove that this iterative scheme converges linearly in L²α sense for some suitable exponent α in a weight function. For the numerical method, we combine a characteristic line method with machine learning techniques. Specifically, we generate multiple characteristic curves at each policy iteration from an ensemble of initial states and compute both the value function and its gradient simultaneously on each curve as the labeled data. Then supervised machine learning is applied to minimize the weighted squared loss for both the value function and its gradients. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates, particularly with less characteristics data or fewer training steps. [ABSTRACT FROM AUTHOR]

This paper discusses the multi-player non-cooperative game of nonlinear stochastic time-varying systems described by Itô-type differential equations in a finite time interval. Multi-player non-cooperative game problems are represented by multi-objective Pareto (MOP) control problems to describe the fact that each player has their own goals. By applying Hamilton–Jacobi inequalities (HJIs), the criterion of upper bounds of the MOP boundary is obtained for nonlinear stochastic systems, and the corresponding strategies are designed for such games, so the MOP problem is transformed into a HJI-constrained MOP problem. In order to overcome the difficulty of solving HJIs, a global linearization method is proposed to approximate the nonlinear systems. By the proposed global linearization method, multi-player non-cooperative game problems are transformed into Riccati equation-constrained MOP problems, and the approximate solutions of HJI-constrained MOP problems are obtained. Finally, a practical example is given to illustrate the effectiveness of the proposed method.

National parks often serve as hotspots for environmental crime such asillegal deforestation and animal poaching. Previous attempts to modelenvironmental crime were either discrete and network-based or required veryrestrictive assumptions on the geometry of the protected region and made heavyuse of radial symmetry. We formulate a level set method to track criminalsinside a protected region which uses real elevation data to determine speed oftravel, does not require any assumptions of symmetry, and can be applied toregions of arbitrary shape. In doing so, we design a Hamilton-Jacobi equationto describe movement of criminals while also incorporating the effects ofpatrollers who attempt to deter the crime. We discuss the numerical schemesthat we use to solve this Hamilton-Jacobi equation. Finally, we apply ourmethod to Yosemite National Park and Kangaroo Island, Australia and designpractical patrol strategies with the goal of minimizing the area that isaffected by criminal activity.

We extend the invariant subspace method (ISM) to a class of Hamilton–Jacobi equations (HJEs) and a family of third-order time-fractional dispersive PDEs with the Caputo fractional derivative in this letter. More precisely, the complete classification is presented for such HJEs that admit invariant subspaces governed by solutions of the second-order and third-order linear ordinary differential equations (ODEs). Meanwhile, some concrete equations are derived for the construction of new exact solutions u(x,t)=∑i=1nCi(t)fi(x). Then a set of invariant subspaces of the considered third-order time-fractional non-linear dispersive equations are obtained. Based on the Laplace transform method (LTM) and applying several properties of the well known Mitta-Leffer (ML) function, the different types of explicit solutions of a family of third-order time-fractional dispersive PDEs are finally derived.

In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated. [ABSTRACT FROM AUTHOR]

We study a reaction-diffusion problem formulated with a higherorder operator, a non-linear advection, and a Fisher-KPP reaction term depending on the spatial variable. The higher-order operator induces solutions to oscillate in the proximity of an equilibrium condition. Given this oscillatory character, solutions are studied in a set of bounded domains. We introduce a new extension operator, that allows us to study the solutions in the open domain RN, but departing from a sequence of bounded domains. The analysis about regularity of solutions is built based on semigroup theory. In this approach, the solutions are interpreted as an abstract evolution given by a bounded continuous operator. Afterward, asymptotic profiles of solutions are studied based on a Hamilton-Jacobi equation that is obtained with a single point exponential scaling. Finally, a numerical assessment, with the function bvp4c in Matlab, is introduced to discuss on the validity of the hypothesis. [ABSTRACT FROM AUTHOR]

Deflection of light due to massive objects was predicted by Einstein in his general theory of relativity. This deflection of light has been calculated by many researchers in the past, for spherically symmetric objects. But, in reality, most of these gravitating objects are not spherical; instead they are ellipsoidal (oblate) in shape. The objective of the present work is to study theoretically the effect of this ellipticity on the trajectory of a light ray. In the present work, we obtain a converging series expression for the deflection of a light ray due to an ellipsoidal gravitating object, characterised by an ellipticity parameter. As a boundary condition, by setting the ellipticity parameter to be equal to zero, we get back the same expression for deflection as due to Schwarzschild black hole. It is also found that the additional contribution in deflection angle due to this ellipticity though small, but could be typically higher than the similar contribution caused by the rotation of a celestial object. Therefore for a precise estimate of the deflection due to a celestial object, the calculations presented here would be useful. [ABSTRACT FROM AUTHOR]

In this paper, we consider a steady state multi-anticipative traffic model and we provide necessarily and sufficient conditions for the existence of traveling solutions. In our work, the word "traveling" means that the distance between two consecutive vehicles travels continuously between two different states. As application to our result, we show that taking a strictly concave optimal velocity, we can construct a traveling solution such that the distance between two vehicles decreases. The existence, uniqueness and the study of the asymptotic behavior of such solutions is done at the level of the Hamilton-Jacobi equation. [ABSTRACT FROM AUTHOR]

In this work, a laboratory scaled industrial interconnected nonlinear Multi-Input | Multi-Output (MIMO) three-tank system, was modelled to control the liquid levels. Ensuing the tradition in the process industry to apply linear controller to most control processes, a linear control scheme was developed for this system. However, since linear schemes are proximate to actual process models, they may not be adequate, especially for highly nonlinear systems. Therefore, a nonlinear control scheme was also developed and compared with the linear scheme. Specifically, optimal linear and nonlinear controllers were designed. In summary, the results of the two control schemes showed adequate performance. However, the linear controller had more robust control and required lesser computational demand compared to the nonlinear scheme. To enhance the computational demand of the nonlinear scheme, a third-party MATLAB toolbox, Automatic Control and Dynamic Optimization (ACADO) toolbox, that interfaces MATLAB with C++ to speed up computations was also utilised, and its results compared, and tentatively validate the earlier solved nonlinear control scheme. • A Multi-Input | Multi-Output (MIMO) nonlinear three-tank system with experimentally verified valve constants is considered. • Discretised linear and nonlinear optimal controllers are designed for the system. • The linear optimal controller was designed based on the Jacobian linearisation of the nonlinear system. • The two designed controllers were used to control the simulated system and their results were compared. • Automatic Control and Dynamic Optimization (ACADO) was also used to compare and validate the nonlinear optimal controllers. [ABSTRACT FROM AUTHOR]

For the quadratic theory of gravity with a scalar field, exact solutions are found for gravitational-wave models in Shapovalov I-type spacetimes, which do not arise in models of the general theory of relativity. The theory of gravity under consideration can effectively describe the early stages of the universe. Type I Shapovalov spaces are the most general forms of gravitational-wave Shapovalov spacetimes, whose metrics in privileged coordinate systems depend on three variables, including the wave variable. For Einstein vacuum spacetimes, these wave models degenerate into simpler types. The exact models of gravitational waves in the quadratic theory of gravity can be used to test the realism of such theories of gravity. [ABSTRACT FROM AUTHOR]

Exact solutions for Shapovalov wave spacetimes of type I in Brans–Dicke's scalar-tensor theory of gravity are constructed. Shapovalov wave spacetimes describe gravitational wave models that allow for the the separation of wave variables in privileged coordinate systems. In contrast to general relativity, the vacuum field equations of the Brans–Dicke scalar-tensor theory of gravity lead to exact solutions for type I Shapovalov spaces, allowing for the the construction of observational tests to detect such wave disturbances. Furthermore, the equations for the trajectories of the test particles are obtained for the models considered. [ABSTRACT FROM AUTHOR]

This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels.

We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equations and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable x / t and in which the time and space derivatives are coupled together. We first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s = x / t. In terms of the standard Fisher-KPP equation of reaction-diffusion type, we give explicit formulas of the spreading speed when the environment has one or two shifting speeds. As a byproduct, we also introduce a novel class of "asymptotically homogeneous" environments which share the same spreading speed with the corresponding homogeneous environments. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantum tunneling of scalar particles near the event horizon of Kerr–Newman black hole under Lorentz violation theory. First, the modified Hamilton–Jacobi equation under Lorentz violation theory in the curved spacetime is reviewed. The modified Hawking temperatures, heat capacities and entropies of Kerr–Newman black hole in frame dragging coordinates and Eddington coordinate are derived. The modified Hawking temperature, heat capacity and change in Bekenstein–Hawking entropy of Kerr–Newman black hole are found to be modified due to Lorentz violation theory and they depend not only on Lorentz violation parameter λ but also on ether-like vector u α . [ABSTRACT FROM AUTHOR]

We discuss various kinds of representation formulas for the viscosity solutions of the contact type Hamilton-Jacobi equations by using the Herglotz' variational principle. [ABSTRACT FROM AUTHOR]

A class of exact (non-perturbative) models of strong gravitational waves based on Shapovalov type III spacetimes and Einstein’s vacuum equations is obtained. Exact solutions are found for the trajectories of particles and radiation in a gravitational wave in privileged coordinate systems. Exact solutions are obtained for the equations of geodesic deviation and tidal acceleration of particles in a gravitational wave in privileged coordinate systems. An explicit analytical law of transition from a privileged coordinate system to a synchronous reference system associated with a freely falling observer with an explicit selection of time and spatial coordinates is obtained. An explicit form of the metric of a gravitational wave in a synchronous frame of reference is obtained. For a synchronous frame of reference, the trajectories of particles and radiation, the deviation of geodesics, and tidal accelerations in a gravitational wave are obtained. The presented methods and approaches are applicable both to Einstein’s general theory of relativity and to modified theories of gravity.