Aslı Pekcan

Hacettepe University · Department of Mathematics

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 ge...

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)...

We study the 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 shifted nonlocal reductions by the Hirota bilinear method. We present particular e...

We study the local and shifted nonlocal reductions of the integrable coupled Kundu system. We then consider particular cases of the Kundu 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...

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.

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.

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...

In our previous work (Gürses 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 diffi...

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 Schr\"{o}dinger, Fokas, and long wave resonance sy...

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.

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.

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.

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 Hir...

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.

We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.

We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.

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.

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...

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 whic...

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...

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...

We present some nonlocal integrable systems by using the Ablowitz–Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schrödinger (NLS) and modified Korteweg–de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to non...

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...

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...

We study three types of nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (AKNS equations) by using Ablowitz-Musslimani type nonlocal reductions. 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 soli...

We present some nonlocal integrable systems by using the Ablowitz-Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schr\"{o}dinger (NLS) and modified Korteweg-de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to...

We study the nonlocal modified Korteweg-de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz-Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz-Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of local...

We study the $(3+1)$-dimensional eight-order nonlinear wave equation associated with the principal representation of the exceptional affine Lie algebra $E_6^{(1)}$, which was constructed by Kac and Wakimoto and stated that $N$-soliton solution of the equation can be formulated. We show that the equation is not Hirota integrable since it does not ha...

Traveling wave solutions of degenerate coupled ℓ-KdV equations are studied. Due to symmetry reduction these equations reduce to one ordinary differential equation (ODE), i.e., ( f ′)2 = Pn( f ) where Pn( f ) is a polynomial function of f of degree n = ℓ + 2, where ℓ ≥ 3 in this work. Here ℓ is the number of coupled fields. There is no known method...

a new version of a known method to find novel solutions of the KdV and coupled KdV equations

A new approach to double-sub equation method is introduced to construct novel solutions for the nonlinear partial differential equations. It is applied to the Korteweg-de Vries (KdV) equation and yields new complexiton solutions of both the KdV and coupled KdV equations. The graphs of the solutions are also illustrated.

Traveling wave solutions of degenerate three-coupled and four-coupled KdV equa- tions are studied. Due to symmetry reduction these equations reduce to one ODE, (f0)2 = Pn(f) where Pn(f) is a polynomial function of f of degree n = ` + 2, where ` � 3 in this work. Here ` is the number of coupled �elds. There is no known method to solve such ordinary...

We give a detailed study of the traveling wave solutions of (� = 2) Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutio...

The Kadomtsev-Petviashvili and Boussinesq equations (u xxx - 6uu x)x - ut x ± uyy = 0, (u xxx - 6uu x)x + u xx ± u tt = 0, are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. I...

We give a detailed study of the traveling wave solutions of $(\ell=2)$ Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solu...

We present the complete classification of equations of the form uxy = f (u, ux, uy) and the Klein-Gordon equations vxy = F (v) connected with one another by differential substitutions v = Φ (u, ux, uy) such that Φ ux Φuy ≠ 0 over the ring of complex-valued variables.

We present the complete classification of equations of the form $u_{xy}=f(u,u_x,u_y)$ and the Klein-Gordon equations $v_{xy}=F(v)$ connected with one another by differential substitutions $v=\varphi(u,u_x,u_y)$ such that $\varphi_{u_x}\varphi_{u_y}\neq 0$ over the ring of complex-valued variables.

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 solution structures. The only difference in these solutions are the dispersion relations. We also show that they possess the Painlev´e property. C�2011 American Insti...

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 solution structures. The only difference in these solutions are the dispersion relations. We also show that they possess the Painlev´e property.

The Kadomtsev-Petviashvili and Boussinesq equations (u xxx − 6uu x) x − u tx ± u yy = 0, (u xxx − 6uu x) x + u xx ± u tt = 0, are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory...

We study differential-difference equation (d/dx)t(n+1,x) = f(t(n,x),t(n+1,x),(d/dx)t(n,x)) with unknown t(n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n+k,x)}k = −∞∞, {(dk/dxk)t(n,x)}k = 1∞, such that DxF =...

We show that we can apply the Hirota direct method to some non-integrable equations. For this purpose, we consider the extended Kadomtsev-Petviashvili-Boussinesq equation with M variable which is (u xxx -6uu x )+a 11 u xx +2∑ k=2 M a 1k u xx k +∑ i,j=2 M a ij u x i x j =0, where a ij =a ji are constants and x i =(x,t,y,z,⋯,x M ). We give the result...

We study differential-difference equation of the form $t_{x}(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$ (called an $n$-integral), both of a finite number of variables $x$, $t(n)$, $t(n\pm 1)$, $t(n\pm 2)$, $...$,...

Nonlinear semi-discrete equations of the form t_x(n+1)=f(t(n), t(n+1), t_x(n)) are studied. An adequate algebraic formulation of the Darboux integrability is discussed and the attempt to adopt this notion to the classification of Darboux integrable chains has been undertaken.

We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains.

We show that we can also apply the Hirota method to some non-integrable equations. For this purpose, we consider the extensions of the Kadomtsev-Petviashvili (KP) and the Boussinesq (Bo) equations. We present several solutions of these equations.

ABSTRACT THE HIROTA DIRECT METHOD Aslı Pekcan M.S. in Mathematics Supervisor: Prof. Dr. Metin G¨urses July, 2005 The search for integrability of nonlinear partial dierential,and dierence,equa-