Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 4Citation - Scopus: 6Dissipative Hierarchies and Resonance Solitons for Kp-Ii and Mkp-Ii(Elsevier Ltd., 2007) Francisco, Meltem L. Y.; Lee, Jyh Hao; Pashaev, OktayWe show that dissipative solitons (dissipatons) of the second and the third members of SL(2,R) AKNS hierarchy give rise to the real solitons of KP-II, while for SL(2,R) Kaup-Newell hierarchy they give solitons of MKP-II. By the Hirota bilinear form for both flows, we find new bilinear system for these equations, and one- and two-soliton solutions. For special values of parameters our solutions show resonance behaviour with creation of four virtual solitons. We first time created four virtual soliton resonance solution for KP-II and established relations of it with degenerate four-soliton solution in the Hirota-Satsuma bilinear form for KP-II. Our approach allows one to interpret the resonance soliton as a composite object of two dissipative solitons in 1 + 1 dimensions.Conference Object Citation - WoS: 24Citation - Scopus: 24Solitons of the Resonant Nonlinear Schrödinger Equation With Nontrivial Boundary Conditions: Hirota Bilinear Method(Pleiades Publishing, 2007) Lee, Jyh Hao; Pashaev, OktayWe use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.Conference Object Citation - WoS: 8Citation - Scopus: 7Soliton Resonances for the Mkp-Ii(Pleiades Publishing, 2005) Lee, Jiunhung; Pashaev, OktayUsing 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; Francisco, Meltem L. Y.We 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