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Aslı Pekcan

Publications and source records attributed to Aslı Pekcan.

At least 19 recordsLinked to original sources
arXiv2026

Comment on "Space-time shifted solitons and solution interactions for a generalized nonlocal nonlinear Schrödinger equation"

Zhou and Yu [Appl. Math. Lett. 178 (2026) 109937] studied the space-time shifted nonlocal nonlinear Schrödinger (NLS) equation $iq_t(x,t)=q_{xx}(x,t)-2σq^2(x,t) \bar{q}(x_0-x,t_0+λt),\,\, λ=\pm1$, and claimed that for both choices of $λ$ the equation is integrable. Here we examine this claim from the perspective of reductions of the standard integrable NLS system. We find that the compatibility of the equation with the NLS system is true only when there is no time reversal in the argument. In other words, for this equation it requires $λ=1$ and $t_0=0$. The case with $λ=-1$ is a valid nonlocal equation, but cannot be derived from the standard NLS system by the corresponding complex conjugate shifted nonlocal reduction, and its integrability therefore does not follow from that integrable NLS system. We also discuss the equivalence between shifted (with real shifts) and unshifted nonlocal reductions and give the adjusted form of the Hirota bilinear equation.

nlin.SI↗
arXiv2026

Loop-type geometric folding of exact solutions of shifted nonlocal NLS and MKdV equations

Based on the notion of foldon, we introduce a geometric framework for constructing folded parametric wave representations of exact solutions of some shifted nonlocal nonlinear Schrödinger and modified Korteweg-de Vries equations. Unlike the method of constructing loops in $(2+1)$-dimensional integrable models based on universal variable separation approach or hodograph transformation, we consider a simplified geometric approach of constructing loop-type folded profiles via non-monotonic parametrization of the spatial coordinate associated with the exact solution of the $(1+1)$-dimensional shifted nonlocal equations. A sufficient condition under which folding takes place is provided in the form of sign change of the derivative of folding map. Applying one- and two-soliton solutions of various shifted nonlocal nonlinear Schrödinger and modified Korteweg-de Vries equations found earlier, we show how different folding maps generate different loop-type folded profiles. In particular, we analyze the influence of deformation parameters and solution parameters on the geometry of folded waves. We show that the effect of the folding leads only to the modification of the spatial parametrization and generates various geometric structures like regular loop-type, oscillating-type, and singular-type folded profiles for certain values of parameters.

nlin.SI↗
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↗
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

Shifted nonlocal reductions of 5-component Maccari system

In this work, we prove that shifted nonlocal reductions of integrable $(2+1)$-dimensional $5$-component Maccari system are particular cases of shifted scale transformations. We present all shifted nonlocal reductions of this system and obtain new two-place and four-place integrable systems and equations. In addition to that we use the Hirota direct method and obtain one-soliton solution of the $5$-component Maccari system. By using the reduction formulas with the solution of the Maccari system we also derive soliton solutions of the shifted nonlocal reduced Maccari systems and equations. We give some particular examples of solutions with their graphs.

nlin.SI↗
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

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↗
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↗
arXiv2021

Shifted nonlocal Kundu type equations: Soliton solutions

We study the local and shifted nonlocal reductions of the integrable coupled Kundu type system. We then consider particular cases of this system; namely Chen-Lee-Liu, Gerdjikov-Ivanov, and Kaup-Newell systems. We obtain one- and two-soliton solutions of these systems and their local and shifted nonlocal reductions by the Hirota bilinear method. We present particular examples for one- and two-soliton solutions of the reduced local and shifted nonlocal Chen-Lee-Liu, Gerdjikov-Ivanov, and Kaup-Newell equations.

nlin.SI↗
arXiv2020

Local and nonlocal $(2+1)$-dimensional Maccari systems and their soliton solutions

In this work, by using the Hirota bilinear method, we obtain one- and two-soliton solutions of integrable $(2+1)$-dimensional $3$-component Maccari system which is used as a model describing isolated waves localized in a very small part of space and related to very well-known systems like nonlinear Schrödinger, Fokas, and long wave resonance systems. We represent all local and Ablowitz-Musslimani type nonlocal reductions of this system and obtain new integrable systems. By the help of reduction formulas and soliton solutions of the $3$-component Maccari system, we obtain one- and two-soliton solutions of these new integrable local and nonlocal reduced $2$-component Maccari systems. We also illustrate our solutions by plotting their graphs for particular values of the parameters.

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

Nonlocal Coupled HI-MKdV Systems

We first study coupled Hirota-Iwao modified KdV (HI-mKdV) systems and give all possible local and nonlocal reductions of these systems. We then present Hirota bilinear forms of these systems and give one-soliton solutions of them with the help of pfaffians. By using the soliton solutions of the coupled HI-mKdV systems for $N=2,3,$ and $N=4$ we find one-soliton solutions of the local and nonlocal reduced equations.

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↗