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We study two members of the multi-component AKNS hierarchy. These are multi-NLS and multi-MKdV systems. We derive the Hirota bilinear forms of these equations and obtain soliton solutions. We find all possible local and nonlocal reductions of these systems of equations and give a prescription to obtain their soliton solutions. We derive also $(2+1)$-dimensional extensions of the multi-component AKNS systems., 19 pages

Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same $N$-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in $(1+1)$-dimensions., 7 pages

We first construct a ( 2 + 1 ) -dimensional negative AKNS hierarchy and then we give all possible local and (discrete) nonlocal reductions of these equations. We find Hirota bilinear forms of the negative AKNS hierarchy and give one- and two-soliton solutions. By using the soliton solutions of the negative AKNS hierarchy we find one-soliton solutions of the local and nonlocal reduced equations.

We find one- and two-soliton solutions of shifted nonlocal NLS and MKdV equations. We discuss the singular structures of these soliton solutions and present some of the graphs of them., 13 pages, 3 figures

We study Kerr-Schild-Kundt class of metrics in generic gravity theories with Maxwell's field. We prove that these metrics linearize and simplify the field equations of generic gravity theories with Maxwell's field., Comment: 15 pages, no figures, matches the published version

No! We show that the field equations of Einstein-Gauss-Bonnet theory defined in generic $D>4$ dimensions split into two parts one of which always remains higher dimensional, and hence the theory does not have a non-trivial limit to $D=4$. Therefore, the recently introduced four-dimensional, novel, Einstein-Gauss-Bonnet theory does not admit an intrinsically four-dimensional definition as such it does not exist in four dimensions. The solutions (the spacetime, the metric) always remain $D>4$ dimensional. As there is no canonical choice of 4 spacetime dimensions out of $D$ dimensions for generic metrics, the theory is not well defined in four dimensions., 8 pages, typos corrected, discussions expanded

Writing the Hirota-Satsuma (HS) system of equations in a symmetrical form we find its local and new nonlocal reductions. It turns out that all reductions of the HS system are Korteweg-de Vries (KdV), complex KdV, and new nonlocal KdV equations. We obtain one-soliton solutions of these KdV equations by using the method of Hirota bilinearization., Comment: 15 pages, 7 figures

We summarize our proof that the "Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime" introduced in Phys. Rev. Lett. 124, 081301 (2020) does not have consistent field equations, as such the theory does not exist. The proof is given in both the metric and the first order formalisms.

We show that (1) the Einstein field equations with a perfect fluid source admit a nonlinear superposition of two distinct homogenous Friedman-Lemaitre-Robertson-Walker (FLRW) metrics as a solution, (2) the superposed solution is an inhomogeneous geometry in general, (3) it reduces to a homogeneous one in the two asymptotes which are the early and the late stages of the universe as described by two different FLRW metrics, (4) the solution possesses a scale factor inversion symmetry and (5) the solution implies two kinds of topology changes: one during the time evolution of the superposed universe and the other occurring in the asymptotic region of space., 13 pages, 4 figures, minor changes

We study some physical properties of black holes in Null Aether Theory (NAT)--a vector-tensor theory of gravity. We first review the black hole solutions in NAT and then derive the first law of black hole thermodynamics. The temperature of the black holes depends on both the mass and the NAT \textquotedblleft charge" of the black holes. The extreme cases where the temperature vanishes resemble the extreme Reissner-Nordstr\"{o}m black holes. We also discuss the contribution of the NAT charge to the geodesics of massive and massless particles around the NAT black holes., Comment: 24 pages, 1 figure, 1 table

The Ba\~nados-Teitelboim-Zanelli (BTZ) black hole metric solves the three-dimensional Einstein's theory with a negative cosmological constant as well as all the generic higher derivative gravity theories based on the metric; as such it is a universal solution. Here, we find, in all generic higher derivative gravity theories, new universal non-Einsteinian solutions obtained as Kerr-Schild type deformations of the BTZ black hole. Among these, the deformed non-extremal BTZ black hole loses its event horizon while the deformed extremal one remains intact as a black hole in any generic gravity theory., Comment: 5 pages; v2: Clarifications added. Version to appear in PRD

We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and integrable. They admit Lax representations and hence possess infinitely many conserved quantities., 19 Pages

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 \textquotedblleft aether." In this paper, we put forward the idea of a \textit{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\"{o}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 $D\geq3$. 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 \textquotedblleft mass" determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory., Comment: 32 pages, no figures, matches the published version

In our previous work (Gurses and Pekcan, 2019, [40]) we started to investigate negative AKNS( − N ) hierarchy in ( 2 + 1 ) -dimensions. We were able to obtain only the first three, N = 0 , 1 , 2 , members of this hierarchy. The main difficulty was the nonexistence of the Hirota formulation of the AKNS( N ) hierarchy for N ≥ 3 . Here in this work we overcome this difficulty for N = 3 , 4 and obtain Hirota bilinear forms of ( 2 + 1 ) -dimensional AKNS( − N ) equations for these members. We study the local and nonlocal reductions of these systems of equations and obtain several new integrable local and nonlocal equations in ( 2 + 1 ) -dimensions. We also give one-, two-, and three-soliton solutions of the reduced equations.

We present two classes of inhomogeneous, spherically symmetric solutions of the Einstein-Maxwell-Perfect Fluid field equations with cosmological constant generalizing the Vaidya-Shah solution. Some special limits of our solution reduce to the known inhomogeneous charged perfect fluid solutions of the Einstein field equations and under some other limits we obtain new charged and uncharged solutions with cosmological constant. Uncharged solutions in particular represent cosmological models where the universe may undergo a topology change and in between is a mixture of two different Friedmann-Robertson-Walker universes with different spatial curvatures. We show that there exist some spacelike surfaces where the Ricci scalar and pressure of the fluid diverge but the mass density of the fluid distribution remains finite. Such spacelike surfaces are known as (sudden) cosmological singularities. We study the behavior of our new solutions in their general form as the radial distance goes to zero and infinity. Finally, we briefly address the null geodesics and apparent horizons associated to the obtained solutions., Comment: 35 pages one figure and two tables

We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations., Comment: 12 pages

We show that the torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\mathcal F}_{n}$ ($n=1,2,3, \ldots$) defined over compact surfaces (closed membranes) in ${\mathbb R}^3$. When the Lagrange function ${\mathcal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the torus, the radii ($a,r$) of the torus are constrained to satisfy $\frac{a^2}{r^2}=\frac{n^2-n}{n^2-n-1},~~ n \ge 2$. A simple generalization of torus in ${\mathbb R}^3$ is a tube of radius $r$ along a curve ${\bf \alpha}$ which we call it toroidal surface (TS). We show that toroidal surfaces with non-circular curve ${\bf \alpha}$ do not provide minimal energy surfaces of the functionals ${\mathcal F}_{n}$ ($n=2,3$) on closed surfaces. We discuss possible applications of the functionals discussed in this work on cell membranes.

We construct surfaces from modified Korteweg-de Vries (mKdV) and sine-Gordon (SG) soliton solutions by the use of parametric deformations. For each case there are two types of deformations. The first one gives surfaces on spheres and the second one gives highly complicated surfaces in three dimensional Euclidean space (${\mathbb R}^3$). The SG surfaces that we obtained are not the critical points of functional where the Lagrange function is a polynomial function of the Gaussian ($K$) and mean ($H$) curvatures of the surfaces. We also give the graph of interesting mKdV and SG surfaces arise from parametric deformations.

The classical double copy idea relates some solutions of Einstein's theory with those of gauge and scalar field theories. We study the Kerr-Schild-Kundt (KSK) class of metrics in $d$-dimensions in the context of possible new examples of this idea. We first show that it is possible to solve the Einstein-Yang-Mills system exactly using the solutions of a Klein-Gordon type scalar equation when the metric is the $pp$-wave metric which is the simplest member of the KSK class. In the more general KSK class, the solutions of a scalar equation also solve the Yang-Mills, Maxwell and Einstein-Yang-Mills-Maxwell equations exactly albeit with a null fluid source. Hence in the general KSK class, the double copy correspondence is not as clean-cut as in the case of the $pp$-wave. In our treatment all the gauge fields couple to dynamical gravity, and are not treated as test fields. We also briefly study G\"{o}del type metrics along the same lines., Comment: 9 pages, published version, in v3 small typos are corrected

We study standard and nonlocal nonlinear Schr\"{o}dinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function $|q(t,x)|^2$ for the standard and nonlocal NLS equations., Comment: 22 pages, 10 figures(typo corrections)

We define (non-Einsteinian) universal metrics as the metrics that solve the source-free covariant field equations of generic gravity theories. Here, extending the rather scarce family of universal metrics known in the literature, we show that the Kerr-Schild--Kundt class of metrics are universal. Besides being interesting on their own, these metrics can provide consistent backgrounds for quantum field theory at extremely high energies., Comment: 31 pages, To appear in the Journal of Classical and Quantum Gravity

We present nonlocal integrable reductions of the Fordy-Kulish system of nonlinear Schrodinger equations and the Fordy system of derivative nonlinear Schrodinger equations on Hermitian symmetric spaces. Examples are given on the symmetric space $\frac{SU(4)}{SU(2) \times SU(2)}$., 10 pages, to be published in Physics Letters A

It was previously proved that the G\"{o}del-type metrics with flat three-dimensional background metric solve exactly the field equations of the Einstein-Aether theory in four dimensions. We generalize this result by showing that the stationary G\"{o}del-type metrics with nonflat background in $D$ dimensions solve exactly the field equations of the Einstein-Aether theory. The reduced field equations are the $(D-1)$-dimensional Euclidean Ricci-flat and the $(D-1)$-dimensional source-free Maxwell equations, and the parameters of the theory are left free except $c_{1}-c_{3}=1$. We give a method to produce exact solutions of the Einstein-Aether theory from the G\"{o}del-type metrics in $D$ dimensions. By using this method, we present explicit exact solutions to the theory by considering the particular cases: ($D-1$)-dimensional Euclidean flat, conformally flat, and Tangherlini backgrounds., Comment: 18 pages, no figures, matches the published version

The surfaces in three dimensional Euclidean space ${\mathbb R}^3$ obtained through the use of the soliton techniques are called integrable surfaces. Integrable equations and their Lax equations possess certain symmetries. Infinitesimal versions of these symmetries are deformations which are responsible in constructing the integrable surfaces. There are four different types of deformations. The spectral parameter, the gauge, the generalized symmetries and integration parameters deformations. We shall present here how these deformations generate surfaces in ${\mathbb R}^3$ and also in three-dimensional Minkowski ${\mathbb M}_{3}$ space. The key point here is to start with an integrable equation and its Lax representation. In this work we assume that the Lax equations of integrable equations are given in terms of a group $G$ and its algebra ${\mathfrak g}$ valued functions. The surfaces in ${\mathbb R}^3$ are also represented via respect ${\mathfrak g}$ valued functions. In constructing integrable surfaces we need the solutions of both the integrable equations and their corresponding Lax equations. In this work we use the one soliton solutions of the integrable equations. We solve the Lax equations for one soliton solutions of the integrable equations. Then choosing a deformation one can construct several types of surfaces. After obtaining these surfaces the next is to search for their properties. Most of these surfaces are Weingarten surfaces, Willmore-like surfaces and surfaces which are derivable from a variational principle. We give sketches of the interesting surfaces of Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV) and Nonlinear Schrodinger (NLS), sine Gordon (SG) equations.

We consider Einstein Born-Infeld theory with a null fluid in Kerr-Schild Geometry. We find accelerated charge solutions of this theory. Our solutions reduce to the Plebanski solution when the acceleration vanishes and to the Bonnor-Vaidya solution as the Born-Infeld parameter b goes to infinity. We also give the explicit form of the energy flux formula due to the acceleration of the charged sources., Latex file (12 pp)

Traveling wave solutions of degenerate coupled $\ell$-KdV equations are studied. Due to symmetry reduction these equations reduce to one ODE, $(f')^2=P_n(f)$ where $P_n(f)$ is a polynomial function of $f$ of degree $n=\ell+2$, where $\ell \geq 3$ in this work. Here $\ell$ is the number of coupled fields. There is no known method to solve such ordinary differential equations when $\ell \geq 3$. For this purpose, we introduce two different type of methods to solve the reduced equation and apply these methods to degenerate three-coupled KdV equation. One of the methods uses the Chebyshev's Theorem. In this case we find several solutions some of which may correspond to solitary waves. The second method is a kind of factorizing the polynomial $P_n(f)$ as a product of lower degree polynomials. Each part of this product is assumed to satisfy different ODEs., Comment: 25 pages, 14 figures. arXiv admin note: text overlap with arXiv:1308.5649

We find the explicit forms of the anti-de Sitter plane, anti-de Sitter spherical, and pp waves that solve both the linearized and exact field equations of the most general higher derivative gravity theory in three dimensions. As a sub-class, we work out the six derivative theory and the critical version of it where the masses of the two spin-2 excitations vanish and the spin-0 excitations decouple., Comment: 14 pages, matches the published version

Non-autonomous Svinolupov-Jordan systems are considered. The integrability criteria of such systems are associated with the existence of recursion operators. A new non-autonomous KdV system is obtained and its recursion operator is given for all $N$. The examples for N=2 and N=3 are studied in detail. Some possible transformations are also discussed which map some systems to autonomous cases., Comment: Latex file (amssymb), 10 pages

We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge–Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge–Ampère equations. Local as well nonlocal conserved densities are obtained.

In this work we develop a general procedure for constructing the recursion operators fro non-linear integrable equations admitting Lax representation. Svereal new examples are given. In particular we find the recursion operators for some KdV-type of integrable equations., Latex File (AMS format), 23 pages, to be published in Journal of Mathematical Physics

Correspondence is established between sigma models, minimal surfaces and the Monge–Ampere equation. The Lax pairs of the minimality condition of the minimal surfaces and the Monge–Ampere equations are given. Existence of infinitely many nonlocal conservation laws is shown and some Backlund transformations are also given.

We consider 2-surfaces arising from the Korteweg-de Vries (KdV) hierarchy and the KdV equation. The surfaces corresponding to the KdV equation are in a three-dimensional Minkowski (M-3) space. They contain a family of quadratic Weingarten and Willmore-like surfaces. We show that some KdV surfaces can be obtained from a variational principle where the Lagrange function is a polynomial function of the Gaussian and mean curvatures. We also give a method for constructing the surfaces explicitly, i.e., finding their parameterizations or finding their position vectors. (C) 2013 Elsevier Ltd. All rights reserved.

We construct the AdS-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives in analogy with the pp-wave solutions. In constructing the wave solutions of the generic theory, we show that the most general two tensor built from the Riemann tensor and its derivatives can be written in terms of the traceless-Ricci tensor. Quadratic gravity theory plays a major role; therefore, we revisit the wave solutions in this theory. As examples to our general formalism, we work out the six-dimensional conformal gravity and its nonconformal deformation as well as the tricritical gravity, the Lanczos-Lovelock theory, and string-generated cubic curvature theory., Comment: 34 pages, no figures, references added, discussions amplified, version to appear in Phys. Rev. D

Cataloged from PDF version of article. We show that the recently found anti-de Sitter (AdS)-plane and AdS-spherical wave solutions of quadratic curvature gravity also solve the most general higher derivative theory in D dimensions. More generally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild spacetimes having type-N Weyl and type-N traceless Ricci tensors. © 2013 American Physical Society.

Collision of plane waves in dilaton gravity theories and low energy limit of string theory is considered. The formulation of the the problem and some exact solutions are presented., Comment: 23 pages, Latex, to appear in Phys. Rev.D

We find constant scalar curvature Type-N and Type-D solutions in all higher curvature gravity theories with actions of the form $f({R}_{\ensuremath{\mu}\ensuremath{\nu}})$ that are built on the Ricci tensor, but not on its derivatives. In our construction, these higher derivative theories inherit some of the previously studied solutions of the cosmological topologically massive gravity and the new massive gravity field equations, once the parameters of the theories are adjusted. Besides the generic higher curvature theory, we have considered in some detail the examples of the quadratic curvature theory, the cubic curvature theory, and the Born-Infeld extension of the new massive gravity.

It is a known fact that the Kerr-Schild type solutions in general relativity satisfy both exact and linearized Einstein field equations. We show that this property remains valid also for a special class of the Kerr-Schild metrics in arbitrary dimensions in generic quadratic curvature theory. In addition to the AdS-wave (or Siklos) metric which represents plane waves in an AdS background, we present here a new exact solution, in this class, to the quadratic gravity in D-dimensions which represents a spherical wave in an AdS background. The solution is a special case of the Kundt metrics belonging to spacetimes with constant curvature invariants., 20 pages, references added, typos corrected, Appendix B improved, version to appear in Phys. Rev. D

We present some nonlinear partial differential equations in 2+1-dimensions derived from the KdV Equation and its symmetries. We show that all these equations have the same 3-soliton structures. The only difference in these solutions are the dispersion relations. We also showed that they pass the Painlev\'e test., Comment: 15 pages

We give an exact solution of the quadratic gravity in D dimensions. The solution is a plane fronted wave metric with a cosmological constant. This metric solves not only the full quadratic gravity field equations but also the linearized ones which include the linearized equations of the recently found critical gravity. A subset of the solutions change the asymptotic structure of the anti-de Sitter space due to their logarithmic behavior., 11 pages, references added, version to appear in Phys. Rev. D