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1. Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

   https://doi.org/10.1111/sapm.12580  

   Caudrelier, Vincent; Gkogkou, Aikaterini; Prinari, Barbara (May 2023, Studies in Applied Mathematics)

   Abstract The complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process. 

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2. Observation of dense collisional soliton complexes in a two-component Bose-Einstein condensate

   https://doi.org/10.1038/s42005-024-01659-w  

   Mossman, Sean_M; Katsimiga, Garyfallia_C; Mistakidis, Simeon_I; Romero-Ros, Alejandro; Bersano, Thomas_M; Schmelcher, Peter; Kevrekidis, Panayotis_G; Engels, Peter (May 2024, Communications Physics)

   Abstract Solitons are nonlinear solitary waves which maintain their shape over time and through collisions, occurring in a variety of nonlinear media from plasmas to optics. We present an experimental and theoretical study of hydrodynamic phenomena in a two-component atomic Bose-Einstein condensate where a soliton array emerges from the imprinting of a periodic spin pattern by a microwave pulse-based winding technique. We observe the ensuing dynamics which include shape deformations, the emergence of dark-antidark solitons, apparent spatial frequency tripling, and decay and revival of contrast related to soliton collisions. For the densest arrays, we obtain soliton complexes where solitons undergo continued collisions for long evolution times providing an avenue towards the investigation of soliton gases in atomic condensates. 

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3. Quantum energies of solitons with different topological charges

   https://doi.org/10.1103/PhysRevD.111.085031  

   Graham, N; Weigel, H (April 2025, Physical Review D)

   The vacuum polarization energy is the leading quantum correction to the classical energy of a soliton. We study this energy for two-component solitons in one space dimension as a function of the soliton’s topological charge. We find that both the classical and the vacuum polarization energies are linear functions of the topological charge with a small offset. Because the combination of the classical and quantum offsets determines the binding energies, either all higher charge solitons are energetically bound or they are all unbound, depending on model parameters. This linearity persists even when the field configurations are very different from those of isolated solitons and would not be apparent from an analysis of their bound state spectra alone. Published by the American Physical Society2025 

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4. Polarization-diverse soliton transitions and deterministic switching dynamics in strongly-coupled and self-stabilized microresonator frequency combs

   https://doi.org/10.1038/s42005-024-01773-9  

   Wang, Wenting; Aldhafeeri, Alwaleed; Zhou, Heng; Melton, Tristan; Jiang, Xinghe; Vinod, Abhinav_Kumar; Yu, Mingbin; Lo, Guo-Qiang; Kwong, Dim-Lee; Wong, Chee_Wei (August 2024, Communications Physics)

   Abstract Dissipative Kerr soliton microcombs in microresonators have enabled fundamental advances in chip-scale precision metrology, communication, spectroscopy, and parallel signal processing. Here we demonstrate polarization-diverse soliton transitions and deterministic switching dynamics of a self-stabilized microcomb in a strongly-coupled dispersion-managed microresonator driven with a single pump laser. The switching dynamics are induced by the differential thermorefractivity between coupled transverse-magnetic and transverse-electric supermodes during the forward-backward pump detunings. The achieved large soliton existence range and deterministic transitions benefit from the switching dynamics, leading to the cross-polarized soliton microcomb formation when driven in the transverse-magnetic supermode of the single resonator. Secondly, we demonstrate two distinct polarization-diverse soliton formation routes – arising from chaotic or periodically-modulated waveforms via pump power selection. Thirdly, to observe the cross-polarized supermode transition dynamics, we develop a parametric temporal magnifier with picosecond resolution, MHz frame rate and sub-ns temporal windows. We construct picosecond temporal transition portraits in 100-ns recording length of the strongly-coupled solitons, mapping the transitions from multiple soliton molecular states to singlet solitons. This study underpins polarization-diverse soliton microcombs for chip-scale ultrashort pulse generation, supporting applications in frequency and precision metrology, communications, spectroscopy and information processing. 

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5. Inverse scattering transform for the complex coupled short‐pulse equation

   https://doi.org/10.1111/sapm.12463  

   Gkogkou, Aikaterini; Prinari, Barbara; Feng, Bao‐Feng; Trubatch, A. David (October 2021, Studies in Applied Mathematics)

   Abstract In this paper, we develop the Riemann–Hilbert approach to the inverse scattering transform (IST) for the complex coupled short‐pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real short‐pulse equation (SPE) and complex short‐pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one‐soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather. 

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