Exact Solutions of the φ6 + φ5 Model Using Truncated Method

In this letter, the φ6 φ5 model in (D 1) dimensions can be solved by a truncated series method. A series of solitary solutions of the φ6 φ5 model in (D 1) dimensions have be obtained.     

Radiating black hole solutions in arbitrary dimensions

We prove a theorem that characterizes a large family of non-static solutions to Einstein equations in N-dimensional space-time, representing, in general, spherically symmetric Type II fluid. It is shown that the best known Vaidya-based (radiating) black hole solutions to Einstein equations, in both four dimensions (4D) and higher dimensions (HD), are particular cases from this family. The spherically symmetric static black hole solutions for Type I fluid can also be retrieved. A brief discussion on the energy conditions, singularities and horizons is provided.     

THE EXACT SOLUTIONS OF THE BURGERS EQUATION AND HIGHER-ORDER BURGERS EQUATION IN (2+1) DIMENSIONS

Some exact solutions of the Burgers equation and higher-order Burgers equation in (2+1) dimensions are obtained by using the extended homogeneous balance method. In these solutions there are solitary wave solutions, close formal solutions for the initial value problems of the Burgers equation and higher-order Burgers equation, and also infinitely many rational function solutions. All of the solutions contain some arbitrary functions that may be related to the symmetry properties of the Burgers equation and the higher-order Burgers equation in (2+1) dimensions.     

Composite S-Brane Solutions on the Product of Ricci-Flat Spaces

A family of generalized S-brane solutions with orthogonal intersection rules and n Ricci-flat factor spaces in the theory with several scalar fields and antisymmetric forms is considered. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. These subclasses contain sub-families of solutions with accelerated expansion of certain factor spaces. The solutions depend on charge densities of branes, their dimensions and intersections, dilatonic couplings and the number of dilatonic fields.     

A Moving Pseudo-Boundary MFS for Three-Dimensional Void Detection

We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a three-dimensional void (rigid inclusion or cavity) within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary. The algorithm for imaging the interior of the medium also makes use of radial spherical parametrization of the unknown star-shaped void and its centre in three dimensions. We also include the contraction and dilation factors in selecting the fictitious surfaces where the MFS sources are to be positioned in the set of unknowns in the resulting regularized nonlinear least-squares minimization. The feasibility of this new method is illustrated in several numerical examples.     

Modified Kadomtsev-Petviashvili Equation in (3+1) Dimensions:Multiple Front-Wave Solutions

A modified Kadomtsev-Petviashvili(mKP) equation in(3+1) dimensions is presented.We reveal multiple front-waves solutions for this higher-dimensional developed equation,and multiple singular front-wave solutions as well.The constraints on the coefficients of the spatial variables,that assure the existence of these multiple front-wave solutions are investigated.We also show that this equation fails the Pamlevi test,and we conclude that it is not integrable in the sense of possessing the Pain/eve property,although it gives multiple front-wave solutions.     

Properties of Solutions in 2 + 1 Dimensions

We solve the Einstein equations for the 2 + 1 dimensions with and without scalar fields. We calculate the entropy, Hawking temperature and the emission probabilities for these cases. We also compute the Newman-Penrose coefficients for different solutions and compare them.     

Exact Solutions of the φ^6 ＋ φ^5 Model Using Truncated Method

In this letter, the φ^6 ＋ φ^5 model in （D ＋ 1） dimensions can be solved by a truncated series method. A series of solitary solutions of the φ^6 ＋ φ^5 model in （D ＋ 1） dimensions have be obtained.     

Using Gaussian Eigenfunctions to Solve Boundary Value Problems

Kernel-based methods are popular in computer graphics, machine learning, and statistics, among other fields; because they do not require meshing of the domain under consideration, higher dimensions and complicated domains can be managed with reasonable effort. Traditionally, the high order of accuracy associated with these methods has been tempered by ill-conditioning, which arises when highly smooth kernels are used to conduct the approximation. Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems. This paper will extend these techniques to the solution of boundary value problems using collocation. The method of particular solutions will also be considered for elliptic problems, using Gaussian eigenfunctions to stably produce an approximate particular solution.     

Some solutions of the Gauss–Bonnet gravity with scalar field in four dimensions

We give all exact solutions of the Einstein–Gauss–Bonnet Field Equations coupled with a scalar field in four dimensions under certain assumptions. The main assumption we make in this work is to take the second covariant derivative of the coupling function proportional to the spacetime metric tensor. Although this assumption simplifies the field equations considerably, to obtain exact solutions we assume also that the spacetime metric is conformally flat. Then we obtain a class of exact solutions.     

Multi-bump, self-similar, blow-up solutions of the Ginzburg-Landau equation

For the Ginzburg-Landau equation (GL), we establish the existence and local uniqueness of two classes of multi-bump, self-similar, blow-up solutions for all dimensions 2<d<4 (under certain conditions on the coefficients in the equation). In numerical simulation and via asymptotic analysis, one class of solutions was already found; the second class of multi-bump solutions is new.In the analysis, we treat the GL as a small perturbation of the cubic nonlinear Schrödinger equation (NLS). The existence result given here is a major extension of results established previously for the NLS, since for the NLS the construction only holds for d close to the critical dimension d=2.The behaviour of the self-similar solutions is described by a nonlinear, non-autonomous ordinary differential equation (ODE). After linearisation, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. We call the region where the type of behaviour changes the mid-range. All of the bumps of the solutions that we construct lie in the mid-range.For the construction, we track a manifold of solutions of the ODE that satisfy the condition at the origin forward, and a manifold of solutions that satisfy the asymptotic conditions backward, to a common point in the mid-range. Then, we show that these manifolds intersect transversely. We study the dynamics in the mid-range by using geometric singular perturbation theory, adiabatic Melnikov theory, and the Exchange Lemma.     

A Vaidya-type radiating solution in Einstein-Gauss-Bonnet gravity and its application to braneworld

We consider a Vaidya-type radiating spacetime in Einstein gravity with the Gauss-Bonnet combination of quadratic curvature terms. Simply generalizing the known static black hole solutions in Einstein-Gauss-Bonnet gravity, we present an exact solution in arbitrary dimensions with the energy-momentum tensor given by a null fluid form. As an application, we derive an evolution equation for the “dark radiation” in the Gauss-Bonnet braneworld.     

A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem

In this paper, a meshless regularization method of fundamental solutions is proposed for a two-dimensional, two-phase linear inverse Stefan problem. The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions. Furthermore, the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution. Therefore, regularization is necessary in order to obtain a stable solution. Numerical results for several benchmark test examples are presented and discussed.     

Non-Static Local String in Higher-Dimensional Gravity

We analyze the space-time structure of local gauge string with a phenomenological energy–momentum tensor, as prescribed by Vilenkin, in an arbitrary number of space-time dimensions with a non-zero cosmological constant Λ. A set of solutions of the full non-linear Einstein's equations for the interior region of such a string is presented.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

Wave Functions for Time-Dependent Dirac Equation under GUP

In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle(GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In(1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials.     

Heterotic string from a higher dimensional perspective

The (abelian bosonic) heterotic string effective action, equations of motion and Bianchi identity at order α^′ in ten dimensions, are shown to be equivalent to a higher dimensional action, its derived equations of motion and Bianchi identity. The two actions are the same up to the gauge fields: the latter are absorbed in the higher dimensional fields and geometry. This construction is inspired by heterotic T-duality, which becomes natural in this higher dimensional theory.We also prove the equivalence of the heterotic string supersymmetry conditions with higher dimensional geometric conditions. Finally, some known Kähler and non-Kähler heterotic solutions are shown to be trivially related from this higher dimensional perspective, via a simple exchange of directions. This exchange can be encoded in a heterotic T-duality, and it may also lead to new solutions.     

A Modified Gravity Theory: Null Aether

General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the "aether". In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory(NAT) — a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordstr?m-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT:We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions — AdS-plane waves and pp-waves — in this theory in any dimension D ≥ 3. Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a "mass" determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.     

On the decay of infinite energy solutions to the Navier-Stokes equations in the plane

Infinite energy solutions to the Navier-Stokes equations in R² may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon (2002) [2] as well as solutions constructed from a “radial energy decomposition”. Our proof uses the Fourier Splitting technique of M.E. Schonbek.     

Analytical Solutions of the Kratzer-Fues Potential in an Arbitrary Number of Dimensions

No Heading Some aspects of the N dimensional Kratzer-Fues potential are discussed, which is an extension of the combined Coulomb-like potential with inverse quadratic potential in N dimensions. The analytical solutions obtained (eigenfunctions and eigenvalues) are dimensionally dependent, so also, the solutions depend on the value of the coefficient of the inverse quadratic term. The expectation values for < r^–2 >, < r^–1 > and the virial theorem for this potential are obtained and the values are also dimensions and parameter dependent.