PHYSICAL REVIEW A 101, 043822 (2020) Editors’ Suggestion Generalized dispersion Kerr solitons Kevin K. K. Tam * and Tristram J. Alexander Institute of Photonics and Optical Science, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia Andrea Blanco-Redondo Nokia Bell Labs, 791 Holmdel Road, Holmdel, New Jersey 07733, USA C. Martijn de Sterke † Institute of Photonics and Optical Science, School of Physics, and University of Sydney Nano Institute, University of Sydney, Sydney, New South Wales 2006, Australia (Received 20 October 2019; accepted 6 February 2020; published 17 April 2020) We report a continuum of pulselike soliton solutions to the generalized nonlinear Schrödinger equation with both quadratic and quartic dispersion and a Kerr nonlinearity. We show that the well-known nonlinear Schrödinger solitons, which occur in the presence of only negative (anomalous) quadratic dispersion, and pure-quartic solitons, which occur in the presence of only negative quartic dispersion, are members of a large superfamily, encompassing both. The members of this family, none of which are unstable, have exponentially decaying tails, which can exhibit oscillations. We find analytic solutions for positive quadratic dispersion and negative quartic dispersion and investigate the soliton dynamics. We also find evidence that a combination of the quadratic and quartic dispersion, rather than exclusively quadratic or quartic dispersion, is likely to improve the performance of soliton lasers. DOI: 10.1103/PhysRevA.101.043822 I. INTRODUCTION Nonlinear Schrödinger (NLS) solitons, solitons that are solutions to the nonlinear Schrödinger equation, have been widely studied and have enabled a plethora of applications. They occur in a great variety of fields, including water waves [1], Bose-Einstein condensates [2,3], and plasmas [4]. In an optics context, they are characterized by quadratic dispersion and a Kerr nonlinear medium, i.e., a medium in which the refractive index depends linearly on intensity [5]. Many generalizations have been studied over the past decades, particularly higher-order nonlinearities [6], and more complicated geometries [7] with coupled modes involving different waveguides, polarizations, frequencies, propagation directions, or combinations of these. In comparison to this, deviations from perfectly quadratic dispersion have not been widely studied and have generally been treated as a perturba- tion of NLS solitons. Recently we studied Kerr nonlinear me- dia at a frequency where the dispersion is purely quartic and demonstrated experimentally and theoretically that in such media pure quartic solitons (PQSs) can arise [8,9]. Though the experiments were carried out in a photonic crystal waveguide, PQSs should similarly occur in optical fibers [10] and in microresonators [11]. In practice it is difficult to achieve purely quadratic or purely quartic dispersion. We therefore consider here Kerr nonlinear media in the presence of both quadratic and quartic * ktam6495@uni.sydney.edu.au † martijn.desterke@sydney.edu.au dispersion, without treating either as a perturbation. We take the quartic dispersion coefficient to be negative (β 4 < 0), whereas the quadratic dispersion coefficient (β 2 ) can have either sign. Nonlinear pulse propagation in the presence of negative β 2 and negative β 4 has been considered earlier [12–19]. Karlsson and Höök [12] found pulselike analytic solutions in the form of a squared hyperbolic secant, whereas Akhmediev et al. [13] found that, depending on the pa- rameters, the exponentially decaying tails of these solutions can have additional oscillations. Akhmediev and Buryak [14] studied the interactions between the solitons and also showed [15] that the solutions with the oscillating tails may form bound states. The properties of these bound states depend on the relative alignment of the oscillations. Piché et al. [16] rederived the solutions found by Karlsson and Höök, and also considered the effect of nonzero β 3 . More recently, Roy and Biancalana [17] considered the propagation of high-intensity pulses in specially designed slot waveguides in numerical ex- periments. By considering a geometry that minimizes Raman scattering, they found that it is possible to generate a large number of solitons, the spectral interference of which leads to a continuum. The work of Bansal et al. [18] and Biswas et al. [19] concentrates on finding analytic solutions in the presence of quartic and cubic dispersion. In this paper, we demonstrate that NLS solitons and PQSs are in fact part of a single continuous soliton superfamily, which we refer to as generalized dispersion Kerr solitons (GDKSs), that also includes the set of analytic solutions for β 2 < 0 and β 4 < 0 reported by Karlsson and Höök [12]. By considering the tails of the solutions we can, based on analytic arguments, divide the parameter space into three distinct 2469-9926/2020/101(4)/043822(11) 043822-1 ©2020 American Physical Society