Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity

It is well known that a symmetric soliton in coupled nonlinear Schroedinger (NLS) equations with the cubic nonlinearity loses its stability with the increase of its energy, featuring a transition into an asymmetric soliton via a subcritical bifurcation. A similar phenomenon was found in a dual-core system with quadratic nonlinearity, and in linearly coupled fiber Bragg gratings, with a difference that the symmetry-breaking bifurcation is supercritical in those cases. We aim to study transitions between symmetric and asymmetric solitons in dual-core systems with saturable nonlinearity. We demonstrate that a basic model of this type, viz., a pair of linearly coupled NLS equations with the intra-core cubic-quintic (CQ) nonlinearity, features a bifurcation loop: a symmetric soliton loses its stability via a supercritical bifurcation, which is followed, at a larger value of the energy, by a reverse bifurcation that restores the stability of the symmetric soliton. If the linear-coupling constant is small enough, the second bifurcation is subcritical, and there is a broad interval of energies in which the system is bistable, with coexisting stable symmetric and asymmetric solitons. At larger values of the coupling constant, the reverse bifurcation is supercritical, and the bifurcation loop disappears if the linear coupling is very strong. Collisions between moving solitons are studied too. Symmetric solitons always collide elastically, while collisions between asymmetric solitons turns them into breathers, that subsequently undergo dynamical symmetrization.Comment: to be published in journal Mathematics and Computers in Simulation, the special issue on "Nonlinear Waves: Computation and Theory

Symmetric and asymmetric solitons in dual-core couplers with competing quadratic and cubic nonlinearities

We consider the model of a dual-core spatial-domain coupler with chi^(2) and chi^(3) nonlinearities acting in two parallel cores. We construct families of symmetric and asymmetric solitons in the system with self-defocusing chi^(3) terms, and test their stability. The transition from symmetric to asymmetric soliton branches, and back to the symmetric ones proceeds via a bifurcation loop. A pair of stable asymmetric branches emerge from the symmetric family via a supercritical bifurcation; eventually, the asymmetric branches merge back into the symmetric one through a reverse bifurcation. The existence of the loop is explained by means of an extended version of the cascading approximation for the chi^(2) interaction, which takes into regard the XPM part of the chi(3) interaction. When the inter-core coupling is weak, the bifurcation loop features a concave shape, with the asymmetric branches losing their stability at the turning points. In addition to the two-color solitons, which are built of the fundamental-frequency (FF) and second-harmonic (SH) components, in the case of the self-focusing chi^(3) nonlinearity we also consider single-color solitons, which contain only the SH component, but may be subject to the instability against FF perturbations. Asymmetric single-color solitons are always unstable, whereas the symmetric ones are stable, provided that they do not coexist with two-color counterparts. Collisions between tilted solitons are studied too

Symmetry breaking of spatial Kerr solitons in fractional dimension

We study symmetry breaking of solitons in the framework of a nonlinear fractional Schr\"{o}dinger equation (NLFSE), characterized by its L\'{e}vy index, with cubic nonlinearity and a symmetric double-well potential. Asymmetric, symmetric, and antisymmetric soliton solutions are found, with stable asymmetric soliton solutions emerging from unstable symmetric and antisymmetric ones by way of symmetry-breaking bifurcations. Two different bifurcation scenarios are possible. First, symmetric soliton solutions undergo a symmetry-breaking bifurcation of the pitchfork type, which gives rise to a branch of asymmetric solitons, under the action of the self-focusing nonlinearity. Second, a family of asymmetric solutions branches off from antisymmetric states in the case of self-defocusing nonlinearity through a bifurcation of an inverted-pitchfork type. Systematic numerical analysis demonstrates that increase of the L\'{e}vy index leads to shrinkage or expansion of the symmetry-breaking region, depending on parameters of the double-well potential. Stability of the soliton solutions is explored following the variation of the L\'{e}vy index, and the results are confirmed by direct numerical simulations.Comment: 22 pages, 12 figure

Stabilization of one-dimensional Townes solitons by spin-orbit coupling in a dual-core system

It was recently demonstrated that two-dimensional Townes solitons (TSs) in two-component systems with cubic self-focusing, which are normally made unstable by the critical collapse, can be stabilized by linear spin-orbit coupling (SOC), in Bose-Einstein condensates and optics alike. We demonstrate that one-dimensional TSs, realized as optical spatial solitons in a planar dual-core waveguide with dominant quintic self-focusing, may be stabilized by SOC-like terms emulated by obliquity of the coupling between cores of the waveguide. Thus, SOC offers a universal mechanism for the stabilization of the TSs. A combination of systematic numerical considerations and analytical approximations identifies a vast stability area for skew-symmetric solitons in the system's main (semi-infinite) and annex (finite) bandgaps. Tilted ("moving") solitons are unstable, spontaneously evolving into robust breathers. For broad solitons, diffraction, represented by second derivatives in the system, may be neglected, leading to a simplified model with a finite bandgap. It is populated by skew-antisymmetric gap solitons, which are nearly stable close to the gap's bottom.Comment: to be published in CNSNS (Communications in Nonlinear Science and Numerical Simulation

Solitons and nonlinear dynamics in dual-core optical fibers

The article provides a survey of (chiefly, theoretical) results obtained for self-trapped modes (solitons) in various models of one-dimensional optical waveguides based on a pair of parallel guiding cores, which combine the linear inter-core coupling with the intrinsic cubic (Kerr) nonlinearity, anomalous group-velocity dispersion, and, possibly, intrinsic loss and gain in each core. The survey is focused on three main topics: spontaneous breaking of the inter-core symmetry and the formation of asymmetric temporal solitons in dual-core fibers; stabilization of dissipative temporal solitons (essentially, in the model of a fiber laser) by a lossy core parallel-coupled to the main one, which carries the linear gain; and stability conditions for PT (parity-time)-symmetric solitons in the dual-core nonlinear dispersive coupler with mutually balanced linear gain and loss applied to the two cores.Comment: A chapter to appear in Handbook of Optical Fibers (G.-D. Peng, Editor: Springer, 2018