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BORLUK, Handan

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Handan

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    Orbital stability of periodic standing waves for the cubic fractional nonlinear Schrödinger equation
    (Elsevier, 2022-12-25) Bittencourt Moraes, G. E.; Borluk, Handan; de Loreno, G.; Muslu, G. M.; Natali, F.; Natural and Mathematical Sciences; BORLUK, Handan
    In this paper, the existence and orbital stability of the periodic standing wave solutions for the nonlinear fractional Schrödinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding the oscillation theorem for fractional Hill operators and the Vakhitov-Kolokolov condition, well known for Schrödinger equations. We then perform a numerical approach to generate the periodic standing wave solutions of the fNLS equation by using the Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative.
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    On the stability of solitary wave solutions for a generalized fractional Benjamin–Bona–Mahony equation
    (IOP Publishing, 2022-03-03) Oruc, G.; Natali, F.; Borluk, Handan; Muslu, G. M.; Natural and Mathematical Sciences; BORLUK, Handan
    In this paper we establish a rigorous spectral stability analysis for solitary waves associated to a generalized fractional Benjamin-Bona-Mahony type equation. Besides the well known smooth and positive solitary wave with large wave speed, we present the existence of smooth negative solitary waves having small wave speed. The spectral stability is then determined by analysing the behaviour of the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum at the solitary wave. Since the analytical solution is not known, we generate the negative solitary waves numerically by using Petviashvili method. We also present some numerical experiments to observe the stability properties of solitary waves for various values of the order of nonlinearity and fractional derivative. Some remarks concerning the orbital stability are also celebrated.
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    Lump solutions of the fractional Kadomtsev–Petviashvili equation
    (Springer, 2024-01) Borluk, Handan; Bruell, G.; Nilsson, D.; Natural and Mathematical Sciences; BORLUK, Handan
    Of concern is the fractional Kadomtsev–Petviashvili (fKP) equation and its lump solution. As in the classical Kadomtsev–Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case α>45 by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation [9] nor for the fKP-I when α≤45 [26]. Furthermore, we show that for any α>45 lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.
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    The generalized fractional Benjamin–Bona–Mahony equation: Analytical and numerical results
    (Elsevier, 2020-08) Oruc, G.; Borluk, Handan; Muslu, G. M.; Natural and Mathematical Sciences; BORLUK, Handan
    The generalized fractional Benjamin-Bona-Mahony (gfBBM) equation models the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The equation involves two fractional terms unlike the well-known fBBM equation. In this paper, we prove local existence and uniqueness of the solutions for the Cauchy problem by using energy method. The sufficient conditions for the existence of solitary wave solutions are obtained. The Petviashvili method is proposed for the generation of the solitary wave solutions and their evolution in time is investigated numerically by Fourier spectral method. The efficiency of the numerical methods is tested and the relation between nonlinearity and fractional dispersion is observed by various numerical experiments.
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    On the existence, uniqueness, and stability of periodic waves for the fractional Benjamin–Bona–Mahony equation
    (Wiley, 2022-01) Amaral, S.; Borluk, Handan; Muslu, G. M.; Natali, F.; Oruc, G.; Natural and Mathematical Sciences; BORLUK, Handan
    The existence, uniqueness, and stability of periodic traveling waves for the fractional Benjamin–Bona–Mahony equation is considered. In our approach, we give sufficient conditions to prove a uniqueness result for the single-lobe solution obtained by a constrained minimization problem. The spectral stability is then shown by determining that the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum and mass at the periodic wave has no negative eigenvalues. We propose the Petviashvili's method to investigate the spectral stability of the periodic waves for the fractional Benjamin–Bona–Mahony equation, numerically. Some remarks concerning the orbital stability of periodic traveling waves are also presented.
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    Traveling waves and transverse instability for the fractional Kadomtsev–Petviashvili equation
    (Wiley, 2022-07) Borluk, Handan; Bruell, G.; Nilsson, D.; Natural and Mathematical Sciences; BORLUK, Handan
    Of concern are traveling wave solutions for the fractional Kadomtsev–Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension-breaking bifurcation. Moreover, the line solitary wave solutions and their transverse (in)stability are discussed. Analogous to the classical Kadmomtsev–Petviashvili (KP) equation, the fKP equation comes in two versions: fKP-I and fKP-II. We show that the line solitary waves of fKP-I equation are transversely linearly instable. We also perform numerical experiments to observe the (in)stability dynamics of line solitary waves for both fKP-I and fKP-II equations.