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Dressing the boundary: On soliton solutions of the nonlinear Schr?dinger equation on the half-line
摘要： Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schr?dinger equation on the half-line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux-dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self-modulated solitons on a constant background. Half-line solitons in both cases are explicitly computed. In particular, the boundary-bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.
关键词： the NLS equation on the half-line,dressing transformations,integrable boundary conditions,inverse scattering transform,soliton solutions
更新于2025-09-23 15:22:29
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Spectral characteristics of periodic tunnelling of chirped soliton in nonautonomous system with external potentials
摘要： We present an exact chirped soliton solution for a nonautonomous system with external potentials and periodic nonlinear barriers/wells (NBs/NWs). Based on the soliton solution, we disclose the interesting spectral features of the chirped solitons through periodic NBs and NWs. In addition to the spectrum being broadened through NBs and being compressed through NWs, we find that distinct spectral characteristics of the chirped soliton appear at the centre, leading and trailing edges of NBs and NWs, which are related to the chirp of the solitons, the period of NBs/NWs and external potentials. When the period of NBs/NWs decreases and the soliton chirp increases, the spectrum appears additional broadening near the leading and trailing edges of both NBs and NWs, but the spectral widths at the centre of both NBs and NWs remain unchanged. Moreover, the linear potential affects the trajectory and the frequency shift of the chirped soliton.
关键词： chirped soliton,nonautonomous NLS equation,Periodic tunnelling,external potentials
更新于2025-09-23 15:22:29
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[IEEE 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Munich, Germany (2019.6.23-2019.6.27)] 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Theoretical and Experimental Studies of Breather Wave Molecules
摘要： The formation of localized wave structures has attracted a significant research interest in a variety of conservative and dissipative systems, including photonics, plasma physics, Bose-Einstein condensates, fluid mechanics, biology and chemistry. Besides their formation, their mutual interactions such as collisions and formation of stable bound states have been particularly investigated in laser physics [1-3]. Such dissipative multi-soliton complexes, also called photonic molecules, result from self-organization phenomena, showing dynamics similar to matter molecules, such as synthesis and vibration. The build-up and transient regimes of dissipative localized structures in mode-locked fiber lasers are currently the subject of intense studies by means of real-time ultrafast measurements [2-3]. By contrast to the soliton molecules in dissipative cavities, which are described by Ginzburg-Landau equation, here we study conservative fiber systems driven by the integrable NLS equation [4]. More specifically, we present the theoretical description of the breather molecule based on the NLS equation. We find that the breather molecule exhibits quasi-periodic or exact periodic oscillatory dynamics according to the characteristic frequencies of elementary breathers. Figures 1(a) and 1(b) show the evolution of the intensity profile of two (a) and three (b) breathers based on the N-breather solution of standard NLS equation, where each breather can be accurately described by four key parameters [5-7]. These co-propagative breathers can exhibit both dissociated state (a1-b1) and bound state (a2-b2) behaviours, depending on the temporal matching between them. Next, we present our experimental results based on the propagation of arbitrarily shaped light waves and using a single-mode optical fiber workbench (a similar setup can be found in ref. [7]). As a general case, when temporal matching is satisfied, two individual breathers will be tightly bounded as a chain during propagation, [see Fig. 1(c1), where the key parameters of the breathers are listed in the caption]. We select three nodes of this breather molecule and present the snapshot of the measured amplitude profile in Fig. 1(c3) – each node exhibits the typical X-shape signature in the plane (ξ, τ), which is predicted by the theory in Fig. 1(c2).
关键词： soliton molecules,localized wave structures,photonic molecules,breather wave molecules,NLS equation
更新于2025-09-19 17:13:59
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Optical solitons and stability analysis with coupled nonlinear schrodinger’s equations having double external potentials
摘要： We consider coupled nonlinear Schrodinger equation (CNLSE) of the Gross-Pitaevskii-type, with linear mixing and nonlinear cross-phase modulation. Motivated by the study of matter waves in Bose-Einstein condensates and multicomponent (vectorial) nonlinear optical systems, we investigate the eigenvalue problem of the CNLSE with double external potentials in a self-defocusig Kerr medium. For this system, we obtain different kinds of wave structures induced by two injected beams, of physical relevance in nonlinear optics and Bose-Einstein condensation. Exact solutions are found by the extended unified method. The linear stability of these solutions is analyzed through the formulation of an eigenvalue problem. The spectral problem is constructed by perturbing the frequency of stationary solutions and by linearizing the resulting equations near the stationary (or steady) states. Our study may simulate experimental work on multiple injected laser beams in a medium with Kerr-type nonlinearity.
关键词： Coupled NLS equation,double external potentials,stability,the eigenvalue problem
更新于2025-09-16 10:30:52
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Nonlinear Fiber Optics || Self-phase modulation
摘要： The ?rst nonlinear effect that we focus on is the self-phase modulation (SPM), a phenomenon that leads to spectral broadening of optical pulses [1–9]. SPM is the temporal analog of self-focusing of CW beams occurring inside any nonlinear medium with n2 > 0. It was ?rst observed in 1967 in the context of transient self-focusing of optical pulses propagating through a CS2-?lled cell [1]. By 1970, SPM had been observed in solids and glasses by using picosecond pulses. The earliest observation of SPM in optical ?bers was made with a ?ber whose core was ?lled with CS2 liquid [7]. This work led by 1978 to a systematic study of SPM in a silica-core ?ber [9]. This chapter considers SPM as a simple example of the nonlinear effects that can occur inside optical ?bers. Section 4.1 is devoted to the case of pure SPM as it neglects the GVD effects and focuses on spectral changes induced by SPM. The combined effects of GVD and SPM are discussed in Section 4.2 with emphasis on the SPM-induced frequency chirp. Section 4.3 presents two analytic techniques and uses them to solve the NLS equation approximately. Section 4.4 extends the analysis to include the higher-order nonlinear effects such as self-steepening.
关键词： GVD,self-steepening,spectral broadening,optical pulses,frequency chirp,SPM,nonlinear effects,NLS equation,self-phase modulation
更新于2025-09-12 10:27:22