Stabilizing the Self-Localized Solitons of the Kundu-Eckhaus Equation by Dissipation


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Yurtbak H., Bayındır C.

ICAME 2021, Balıkesir, Turkey, 1 - 03 September 2021, no.25, pp.1-6

  • Publication Type: Conference Paper / Full Text
  • City: Balıkesir
  • Country: Turkey
  • Page Numbers: pp.1-6

Abstract

The Kundu-Eckhaus equation (KEE) is a nonlinear partial differential equation in the nonlinear Schrödinger equation (NLSE) class. This equation was introduced to the scientific literature independently by Kundu [1] and Eckhaus [2]. It is well-known that KEE admits many different analytical solutions like the NLSE. Those solutions of the KEE are widely used in fields such as nonlinear optics, fiber optical waveforms, water waves mechanics, and hydraulics, just to name a few. In this study, the effect of loss/gain on the soliton solutions of the KEE has been investigated. With this aim, we study the dissipative Kundu-Eckhaus equation (dKEE) [4-5]. We analyze the effects of dissipation in the form of a loss term on the self-localized solitons of the dKEE. For this purpose, we propose a Petviashvili’s method (PM) for the numerical construction of the soliton solution of the dKEE [6]. Using PM, we first numerically compute the soliton solutions of the dKEE and discuss their properties. Then, we analyze the effects of dissipation on the dynamics and stabilities of those soliton using a split-step Fourier method (SSFM) implemented for time-stepping purposes. We show that the dKEE equation admits one and two soliton solutions for zero potential and for photorefractive potential (V = (x)) cases. Since the solitons under the photorefractive potentials turned out to be unstable during temporal evolution, we introduce and discuss the effects of dissipation on the dynamics and stabilization of those solitons. The effects of dissipation on soliton characteristics and power are also discussed.The Kundu-Eckhaus equation (KEE) is a nonlinear partial differential equation in the nonlinear Schrödinger equation (NLSE) class. This equation was introduced to the scientific literature independently by Kundu [1] and Eckhaus [2]. It is well-known that KEE admits many different analytical solutions like the NLSE. Those solutions of the KEE are widely used in fields such as nonlinear optics, fiber optical waveforms, water waves mechanics, and hydraulics, just to name a few. In this study, the effect of loss/gain on the soliton solutions of the KEE has been investigated. With this aim, we study the dissipative Kundu-Eckhaus equation (dKEE) [4-5]. We analyze the effects of dissipation in the form of a loss term on the self-localized solitons of the dKEE. For this purpose, we propose a Petviashvili’s method (PM) for the numerical construction of the soliton solution of the dKEE [6]. Using PM, we first numerically compute the soliton solutions of the dKEE and discuss their properties. Then, we analyze the effects of dissipation on the dynamics and stabilities of those soliton using a split-step Fourier method (SSFM) implemented for time-stepping purposes. We show that the dKEE equation admits one and two soliton solutions for zero potential and for photorefractive potential (V =Iocos2(x)) cases. Since the solitons under the photorefractive potentials turned out to be unstable during temporal evolution, we introduce and discuss the effects of dissipation on the dynamics and stabilization of those solitons. The effects of dissipation on soliton characteristics and power are also discussed.