Iterative solvers for soliton solutions of nonlinear Schrödinger-type equations

Funded by the Deutsche Forschungsgemeinschaft(DFG) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD(EXC 2122, Project ID 390833453 German Research Foundation Affiliations: (RotisSans, blue, 36 pt) O. Melchert, A. Demircan Open-source solvers for soliton solutions of nonlinear Schrödinger-type equations Computational problem solved by SWtools Usage examples thoroughly documented under Ref. [1] One-dimensional (1D) NSE Software integration with py-fmas Two-dimensional (2D) NSE References Further features Leibniz Universität Hannover, IQO, Welfengarten 1, 30167 Hannover Leibniz Universität Hannover, PxD, Welfengarten 1A, 30167 Hannover *[email protected] We present open-source Python tools for the numerical calculation of soliton solutions for nonlinear Schrödinger-type equations in nonlinear optics and quantum mechanics. Two variants of the corresponding nonlinear eigenvalue problem (NEVP) are considered: a bare NEVP, where a solution with prescribed eigenvalue is computed, and a constrained NEVP, where a solution with prescribed norm is computed. Linear stability of the solutions is assessed in the framework of soliton internal modes. ■ Integrates well with py-fmas [2] and GNLStools.py [3] ■ Pictorial outline of SWtools [1] ■ 1D Gross-Pitaevskii equation ■ Higher-order NSE [5] ■ 2D nonlinear Schrödinger equation ■ NEVP with normalization constraint [1] ■ Usage example [1,4] ■ Usage example [1,7] ■ Extendible software framework ■ Tools for linear stability analysis ■ Library of solitary wave examples ■ Generalized nonlinear Schrödinger equation (generalized NSE) ■ Solitary wave ansatz ■ Nonlinear eigenvalue problem (NEVP) ■ Conservation laws required by the implemented algorithms ■ Aim: obtain solitary wave U by suitable iterative procedure propagation coordinate linear differential operator transverse coordinate nonlinear functional complex-valued envelope Funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). Code: Compute capsule: Poster: [1] O. Melchert, A. Demircan, CPC 317 (2025) 109851 [2] O. Melchert, A. Demircan, CPC 273 (2022) 108257 [3] O. Melchert, A. Demircan, SFX 20 (2022) 101232 [4] W. Bao, Q. Du, SIAM 25 (2004) 1674 [5] O. Melchert, A. Demircan, PRA 110 (2004) 043518 [7] J. Yang, Z. Musslimani, OL 28 (2003) 2094 [6] O. Sinkin et al., JLT 21 (2003) 61 • trapping potential • BEC wavefunction • potential • NEVP • 2D transverse vector • iterative solution method: nonlinear successive overrelaxation (NSOM) • initial trial function • iterative solution method: spectral renormalization method (SRM) • easy-to-use code interface • (a) examples of solitary wave solutions • (b) evolution of the accuracy upon iteration • interaction parameter here: ■ Software integration [1,2] • SWtools ``bare'' NEVP solver [1] • pulse propagation using py-fmas [2] (adaptive stepsize local error method [6]) iterate until: (accuracy)