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Metin Gürses

Publications and source records attributed to Metin Gürses.

At least 19 recordsLinked to original sources
arXiv2026

Multi-place shifted nonlocal reductions of a multi-component AKNS system

Starting from a multi-component AKNS system, we obtain new shifted nonlocal nonlinear Schrödinger equations. We find 13 different shifted nonlocal nonlinear Schrödinger equations with two-place nonlocalities and 10 shifted nonlocal nonlinear Schrödinger equations with four-place nonlocalities. We first obtain one-soliton solutions of the multi-component AKNS system by the Hirota method. Applying the shifted nonlocal reduction formulas to this solution, we obtain one-soliton solutions for the shifted nonlocal nonlinear Schrödinger equations. In cases yielding nontrivial solutions, we discuss the singularity structures of the solutions and show that the one-soliton solutions we obtain are nonsingular for certain values of the parameters. We plot representative nonsingular solutions obtained for admissible parameter values.

nlin.SI↗
arXiv2026

FLRW-Cosmology in Scalar-Vector-Tensor Theories of Gravity

We generalize our previous theorem for FLRW spacetimes within the framework of generic metric gravity theories. In earlier work, we proved that, in the absence of matter fields, the field equations of any metric gravity theory constructed from the curvature tensor and its covariant derivatives reduce in FLRW spacetime to the Einstein equations with an effective perfect-fluid source. In the present work, we extend this result to a broad class of scalar-vector-tensor theories in which the gravitational action contains arbitrary scalar and vector fields together with their covariant derivatives at any order. We prove that, under the symmetry conditions imposed by FLRW geometry, the metric field equations necessarily take the Einstein form with an effective perfect-fluid source, supplemented by the corresponding scalar and vector field equations. This result shows that FLRW metrics belong to the class of universal metrics: the tensorial structure of the gravitational field equations is solely fixed by the symmetry of the FLRW spacetime and is independent of the specific form of the gravitation theory, while the resulting cosmological dynamics remains theory dependent. We illustrate our theorem using recently proposed Einstein-scalar and Einstein-Proca theories.

gr-qc↗
arXiv2025

Higher order Hirota bilinear forms

In this paper we study Hirota bilinear forms of the type $P(D) \{f\cdot f\}=0$. We prove that for $P(D)=D_x^mD_y^rD_t^n$ the equations have three-soliton solutions if only if two of nonzero $m,n,p$ are odd and the other one even. We explicitly derive the nonlinear partial differential equations corresponding to this form for $m+n+p=4$ and $m+n+p=6$. We show that the equations for $P(D)=D_x(D_x^3+α_1 D_t+α_2 D_y)^{2k+1}$ possess three-soliton solutions for any constants $(α_1,α_2)\neq (0,0)$ and $k\in \mathbb{N}$. We conjecture that these equations have four-soliton solution only for $k=0$. Finally, we consider the equations for $P(D)=D_x^{m_1}D_y^{m_2}D_t^{m_3}D_z^{m_4}$. We prove that these equations have three-soliton solutions if only if one of $m_i=1$, and all the other $m_i$'s are odd for $i=1,2,3,4$. We observe that the monomials $D_x^mD_y^rD_t^n$ and $D_x^{m_1}D_y^{m_2}D_t^{m_3}D_z^{m_4}$ do not result genuine four-soliton solutions. In addition, we obtain three-soliton, lump, and hybrid solutions of these three type of equations for particular powers of the Hirota $D$-operators.

nlin.SI↗
arXiv2025

Generalized Vaidya Spacetime in Cotton and Conformal Killing Theories

We demonstrate that the non-vacuum field equations of Cotton gravity and Conformal Killing gravity admit a generalized class of Vaidya-type solutions. In particular, beyond the standard induced term associated with the matter source, the generalized metric incorporates two additional correction terms of purely geometric origin, arising from the unique structure of these theories. This extended solution generalizes the classical Vaidya spacetime in General Relativity and offers new insights into the dynamics of radiating spacetimes within the framework of these third-rank gravity theories.

gr-qc↗
arXiv2025

Method of ${\cal M}_{n}$-Extension via Frobenius Companion Matrices

Frobenius companion matrices arise when we write an $n$-th order linear ordinary differential equation as a system of first order differential equations. These matrices and their transpose have very nice properties. By using the powers of these matrices we form a closed algebra under the matrix multiplication. Structure constants of this commuting algebra are the components of companion matrix. We use these matrices in our method of ${\cal M}_{n}$-extension of scalar integrable equations to produce new systems of integrable equations with recursion operators.

nlin.SI↗
arXiv2025

The Method of ${\cal M}_{n}$-Extension: The KdV Equation

In this work we generalize ${\cal M}_{2}$-extension that has been introduced recently. For illustration we use the KdV equation. We present five different ${\cal M}_{3}$-extensions of the KdV equation and their recursion operators. We give a compact form of ${\cal M}_{n}$-extension of the KdV equation and recursion operator of the coupled KdV system. The method of ${\cal M}_{n}$-extension can be applied to any integrable scalar equation to obtain integrable multi-field system of equations. We also present unshifted and shifted nonlocal reductions of an example of ${\cal M}_{3}$-extension of KdV.

nlin.SI↗
arXiv2024

Wave Metrics in the Cotton and Conformal Killing Gravity Theories

We study wave metrics in the context of Cotton Gravity and Conformal Killing Gravity. First, we consider pp-wave metrics with flat and non-flat wave surfaces and show that they are exact solutions to the field equations of these theories. More explicitly, the field equations reduce to an inhomogeneous Laplace and Helmholtz differential equations, depending on the curvature of the two-dimensional geometry of the wave surfaces. An interesting point here is that the ones with non-flat wave surfaces are not present in classical GR, which manifests a crucial distinction between these theories and GR. Moreover, we investigate Kerr-Schild-Kundt metrics in the context of these theories and show that, from among these metrics, only the AdS wave metrics solve the field equations of these theories. However, AdS spherical and dS hyperbolic wave metrics do not solve the field equations of these theories, which is in contrast to the classical GR. In the case of AdS wave metrics, the field equations of these theories reduce to an inhomogeneous Klein-Gordon equation. We give all the necessary and sufficient conditions for the metric function $V$ to solve these field equations.

gr-qc↗
arXiv2024

Geometric Perfect Fluids and Dark Side of the Universe

Recently we showed that in FLRW cosmology, the contribution from higher curvature terms in any generic metric gravity theory to the energy-momentum tensor is of the perfect fluid form. Such a geometric perfect fluid can be interpreted as a fluid remaining from the beginning of the universe where the string theory is thought to be effective. Just a short time after the beginning of the Universe, it is known that the Einstein-Hilbert action is assumed to be modified by adding all possible curvature invariants. We propose that the observed late-time accelerating expansion of the Universe can be solely driven by this geometric fluid. To support our claim, we specifically study the quadratic gravity field equations in $D$-dimensions. We show that the field equations of this theory for the FLRW metric possess a geometric perfect fluid source containing two critical parameters $σ_1$ and $σ_2$. To analyze this theory concerning its parameter space $(σ_1, σ_2)$, we obtain the general second-order nonlinear differential equation governing the late-time dynamics of the deceleration parameter $q$. Hence using some present-day cosmological data as our initial conditions, our findings for the $σ_2=0$ case are as follows: $ (i)$ In order to have a positive energy density for the geometric fluid $ρ_g$, the parameter $σ_1$ must be negative for all dimensions up to $D = 11$, $(ii)$ For a suitable choice of $σ_1$, the deceleration parameter experiences signature changes in the past and future, and in the meantime it lies within a negative range which means that the current observed accelerated expansion phase of the Universe can be driven solely by the curvature of the spacetime, $(iii)$ $q$ experiences a signature change and as the dimension $D$ of spacetime increases, this signature change happens at earlier and later times, in the past and future, respectively.

gr-qc↗
arXiv2024

On SK and KK Integrable Systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods which are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the ${\cal M}_{2}$-extension of integrable scalar equations. For illustration we give Korteweg-de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations we obtain some new integrable systems with recursion operators. We give also the soliton solutions of the system equations and integrable standard nonlocal and shifted nonlocal reductions of these systems.

nlin.SI↗
arXiv2024

Moving null curves and integrability

We study the null curves and their motion in a $3$-dimensional flat space-time $M_{3}$. We show that when the motion of null curves forms two surfaces in $M_{3}$ the integrability conditions lead to the well-known AKNS hierarchy. In this case we obtain all the geometrical quantities of the surfaces arising from the whole hierarchy but we particulary focus on the surfaces of the MKdV and KdV equations. We obtain one- and two-soliton surfaces associated to the MKdV equation and show that the Gauss and mean curvatures of these surfaces develop singularities in finite time. We show that the tetrad vectors on the curves satisfy the spin vector equation in the ferromagnetism model of Heisenberg.

nlin.SI↗
arXiv2023

The Method of Hirota Bilinearization

Bilinearization of a given nonlinear partial differential equation is very important not only to find soliton solutions but also to obtain other solutions such as the complexitons, positons, negatons, and lump solutions. In this work we study the bilinearization of nonlinear partial differential equations in $(2+1)$-dimensions. We write the most general sixth order Hirota bilinear form in $(2+1)$-dimensions and give the associated nonlinear partial differential equations for each monomial of the product of the Hirota operators $D_{x}$, $D_{y}$, and $D_{t}$. The nonlinear partial differential equations corresponding to the sixth order Hirota bilinear equations are in general nonlocal. Among all these we give the most general sixth order Hirota bilinear equation whose nonlinear partial differential equation is local which contains 12 arbitrary constants. Some special cases of this equation are the KdV, KP, KP-fifth order KdV, and Ma-Hua equations. We also obtain a nonlocal nonlinear partial differential equation whose Hirota form contains all possible triple products of $D_{x}$, $D_{y}$, and $D_{t}$. We give one- and two-soliton solutions, lump solutions with one, two, and three functions, and hybrid solutions of local and nonlocal $(2+1)$-dimensional equations. We proposed also solutions of these equations depending on dynamical variables.

nlin.SI↗
arXiv2023

Multi-component AKNS systems

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.

nlin.SI↗
arXiv2020

Nonlocal KdV Equations

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.

nlin.SI↗
arXiv2020

Hirota bilinear forms of the AKNS($N$) systems

We study the AKNS($N$) hierarchy for $N=3,4,5,6$. We give the Hirota bilinear forms of these systems and present local and nonlocal reductions of them. We give the Hirota bilinear forms of the reduced equations. The compatibility of the commutativity diagrams of the application of the recursion operator, reductions of the AKNS($N$) systems, and Hirota bilinearization is also studied.

nlin.SI↗
arXiv2019

$(2+1)$-dimensional AKNS($-N$) Systems: $ N=3,4$

In this work we continue to study negative AKNS($N$) that is AKNS($-N$) system for $N=3,4$. We obtain all possible local and nonlocal reductions of these equations. We construct the Hirota bilinear forms of these equations and find one-soliton solutions. From the reduction formulas we obtain also one-soliton solutions of all reduced equations.

nlin.SI↗
arXiv2019

Superposition of the Coupled NLS and MKdV Systems

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.

nlin.SI↗