Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7148
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Article Citation - WoS: 6Citation - Scopus: 5Envelope Soliton Resonances and Broer-Kaup Non-Madelung Fluids(Pleiades Publishing, 2012) Pashaev, Oktay; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyWe derive an extended nonlinear dispersion for envelope soliton equations and also find generalized equations of the nonlinear Schrödinger (NLS) type associated with this dispersion. We show that space dilatations imply hyperbolic rotation of the pair of dual equations, the NLS and resonant NLS (RNLS) equations. For the RNLS equation, in addition to the Madelung fluid representation, we find an alternative non-Madelung fluid system in the form of a Broer-Kaup system. Using the bilinear form for the RNLS equation, we construct the soliton resonances for the Broer-Kaup system and find the corresponding integrals of motion and existence conditions for the soliton resonance and also a geometric interpretation in terms of a pseudo-Riemannian surface of constant curvature. This approach can be extended to construct a resonance version and the corresponding Broer-Kaup-type representation for any envelope soliton equation. As an example, we derive a new modified Broer-Kaup system from the modified NLS equation.Conference Object Citation - WoS: 8Citation - Scopus: 7Soliton Resonances for the Mkp-Ii(Pleiades Publishing, 2005) Lee, Jiunhung; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyUsing the second flow (derivative reaction-diffusion system) and the third one of the dissipative SL(2, ℝ) Kaup-Newell hierarchy, we show that the product of two functions satisfying those systems is a solution of the modified Kadomtsev-Petviashvili equation in 2+1 dimensions with negative dispersion (MKP-II). We construct Hirota's bilinear representations for both flows and combine them as the bilinear system for the MKP-II. Using this bilinear form, we find one- and two-soliton solutions for the MKP-II. For special values of the parameters, our solution shows resonance behavior with the creation of four virtual solitons. Our approach allows interpreting the resonance soliton as a composite object of two dissipative solitons in 1+1 dimensions.Conference Object Citation - WoS: 16Citation - Scopus: 14Degenerate Four-Virtual Resonance for the Kp-Ii(Pleiades Publishing, 2005) Pashaev, Oktay; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyWe propose a method for solving the (2+1)-dimensional Kadomtsev- Petviashvili equation with negative dispersion (KP-II) using the second and third members of the disipative version of the AKNS hierarchy. We show that dissipative solitons (dissipatons) of those members yield the planar solitons of the KP-II. From the Hirota bilinear form of the SL(2,ℝ) AKNS flows, we formulate a new bilinear representation for the KP-II, by which we construct one- and two-soliton solutions and study the resonance character of their mutual interactions. Using our bilinear form, for the first time, we create a four-virtual-soliton resonance solution of the KP-II, and we show that it can be obtained as a reduction of a four-soliton solution in the Hirota-Satsuma bilinear form for the KP-II