Analytic Studies on the Helmholtz Spatial Solitons in Power-Law Optical Materials

With symbolic computation and Hirota method, analytic two-soliton solutions for the coupled nonlinear Schrodinger (CNLS) equations, which describe the propagation of spatial solitons in an AlGaAs slab waveguide, are derived. Two types of coefficient constraints of the CNLS equations to distinguish the elastic and inelastic interactions between spatial solitons are obtained for the first time in this paper. Asymptotic analysis is made to investigate the spatial soliton interactions. The inelastic interactions are studied under the obtained coefficient constraints of the CNLS equations. The influences of parameters for the obtained soliton solutions are discussed. All-optical switching and soliton amplification are studied based on the dynamic properties of inelastic interactions between spatial solitons. Numerical simulations are in good agreement with the analytic results. The presented results have applications in the design of birefringence-managed switching architecture.

Analytic studies on a generalized inhomogeneous higher-order nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain

In this paper, coupled higher-order nonlinear Schrödinger equations are studied for the ultrashort pulse propagation in such optical media as the multi-mode fibers and birefringent fibers. Through the Hirota method and symbolic computation, analytic mixed-type one- and two-soliton solutions are derived, and three sets of conditions for the non-singular solutions are given as well. Via the one-soliton solutions obtained, critical condition for the black and gray solitons is obtained analytically. Asymptotic analysis is carried out on the two-soliton solutions to derive the condition for the inelastic interaction. Evolution of the bound vector solitons, elastic and inelastic interactions between the two vector solitons are also investigated graphically. Moreover, after the inelastic interaction, the two bright solitons are observed to disappear, while the two dark ones form the new bound soliton.

In this article, the propagation and collision of vector solitons are investigated from the 3-coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of the Lax pair, infinitely-many conservation laws are obtained. Under an integrability constraint among the variable coefficients for the group velocity dispersion (GVD), nonlinearity and fibre gain/loss, and two mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation. Influence of the variable coefficients for the GVD and nonlinearity on the vector soliton amplitudes and velocities is analysed. Through the asymptotic and graphic analysis, bound states and elastic and inelastic collisions between the vector two solitons are investigated: Not only the elastic but also inelastic collision between the 2-bright-1-dark vector two solitons can occur, whereas the collision between the 1-bright-2-dark vector two solitons is always elastic; for the bound states, the GVD and nonlinearity affect their types; with the GVD and nonlinearity being the constants, collision period decreases as the GVD increases but is independent of the nonlinearity.

In the biased guest-host photorefractive polymer, the Manakov equations can be used to describe the optical soliton propagation and interaction. Hereby for such equations, via the Hirota method and symbolic computation, analytic soliton solutions in the bright-dark and dark-dark forms are obtained. Based on the choice of photorefractive polymer parameter and incident-optical-beam parameter, the bright-dark and dark-dark solitons as well as their interaction can occur in the polymer when the total intensity is much lower than the background illumination, and our analysis indicates that the incident light with different polarization directions influence little on the soliton propagation. γ, representing the soliton intensity far away from the soliton center, determines the appearance of bright or dark soliton under the background illumination. Through the graphic and asymptotic analysis on the two-soliton solutions along with the different γ, we find that there exist the elastic and inelastic interactions between the bright-dark solitons, while the interactions between the dark-dark solitons are always elastic.

Mixed-type soliton solutions for the N-coupled higher-order nonlinear schrödinger equation in optical fibers

We investigate coupled nonlinear Schrödinger equations (NLSEs) with variable coefficients and gain. The coupled NLSE is a model equation for optical soliton propagation and their interaction in a multimode fiber medium or in a fiber array. By using Hirota's bilinear method, we obtain the bright-bright, dark-bright combinations of a one-soliton solution (1SS) and two-soliton solutions (2SS) for an n-coupled NLSE with variable coefficients and gain. Crucial properties of two-soliton (dark-bright pair) interactions, such as elastic and inelastic interactions and the dynamics of soliton bound states, are studied using asymptotic analysis and graphical analysis. We show that a bright 2-soliton, in addition to elastic interactions, also exhibits multiple inelastic interactions. A dark 2-soliton, on the other hand, exhibits only elastic interactions. We also observe a breatherlike structure of a bright 2-soliton, a feature that become prominent with gain and disappears as the amplitude acquires a minimum value, and after that the solitons remain parallel. The dark 2-soliton, however, remains parallel irrespective of the gain. The results found by us might be useful for applications in soliton control, a fiber amplifier, all optical switching, and optical computing.

This study investigates dust acoustic waves (DAWs) in a space plasma composed of negatively charged dust grains, Cairns-Tsallis distributed electrons, and Cairns-distributed ions. The generalized polarization force (PF) arising from the interaction between negative dust particles and nonthermal ions is examined. Using the reductive perturbation method, the dynamics of the DAWs are described by a KortewegCde Vries equation, from which both single- and two-soliton solutions are derived. The two-soliton interactions and resulting phase shifts are analyzed via Hirota's bilinear method. Key findings indicate that the PF parameter decreases significantly with increasing nonthermality and decreases moderately with decreasing nonextensivity. Furthermore, the phase shifts of colliding solitons decrease with higher nonextensivity and increase with greater nonthermality. These results provide valuable insights into wave propagation and soliton behavior in mesospheric plasmas.

Symbolic computation on the bright soliton solutions for the generalized coupled nonlinear Schrödinger equations with cubic–quintic nonlinearity

Under investigation in this paper are the coupled nonlinear Schrödinger (CNLS) equations, which can be used to govern the optical-soliton propagation and interaction in such optical media as the multimode fibers, fiber arrays, and birefringent fibers. By taking the 3-CNLS equations as an example for the N-CNLS ones (N≥3), we derive the analytic mixed-type two- and three-soliton solutions in more general forms than those obtained in the previous studies with the Hirota method and symbolic computation. With the choice of parameters for those soliton solutions, soliton interactions and complexes are investigated through the asymptotic and graphic analysis. Soliton interactions and complexes with the bound dark solitons in a mode or two modes are observed, including that (i) the two bright solitons display the breatherlike structures while the two dark ones stay parallel, (ii) the two bright and dark solitons all stay parallel, and (iii) the states of the bound solitons change from the breatherlike structures to the parallel one even with the distance between those solitons smaller than that before the interaction with the regular one soliton. Asymptotic analysis is also used to investigate the elastic and inelastic interactions between the bound solitons and the regular one soliton. Furthermore, some discussions are extended to the N-CNLS equations (N>3). Our results might be helpful in such applications as the soliton switch, optical computing, and soliton amplification in the nonlinear optics.

Under investigation in this paper is a (3 + 1)-dimensional generalized nonlinear Schrödinger equation with the distributed coefficients for the spatiotemporal optical solitons or light bullets. Through the symbolic computation and Hirota method, one- and two-soliton solutions are derived. We also present the Bäcklund transformation, through which we derive the soliton solutions. When the gain/loss coefficient is the monotonically decreasing function for the propagation coordinate [Formula: see text], amplitude for the spatiotemporal optical soliton or light bullet decreases along [Formula: see text], while when the gain/loss coefficient is the monotonically increasing function for [Formula: see text], amplitude for the spatiotemporal optical soliton or light bullet increases along [Formula: see text]. Directions of the solitons are different because the signs of imaginary parts of the frequencies are adverse. Based on the two-soliton solutions, elastic and inelastic collisions between the two spatiotemporal optical solitons or light bullets are derived under different conditions presented in the paper.

Dark solitonic excitations and collisions from a fourth-order dispersive nonlinear Schrödinger model for the alpha helical protein

Studied in this paper are the vector bright solitons of the coupled higher-order nonlinear Schrödinger system, which describes the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber. With the help of auxiliary functions, we obtain the bilinear forms and construct the vector bright one- and two-soliton solutions via the Hirota method and symbolic computation. Two types of vector solitons are derived. Single-hump, double-hump, and flat-top solitons are displayed. Elastic and inelastic interactions between the Type-I solitons, between the Type-II solitons, and between the two combined types of the solitons are revealed, respectively. Especially, from the interaction between a Type-I soliton and a Type-II soliton, we see that the Type-II soliton exhibits the oscillation periodically before such an interaction and becomes the double-hump soliton after the interaction, which is different from the previously reported.

Spatial optical solitons are self-trapped optical beams that propagate without changing their spatial shape. This behaviour is caused by the competing effects of diffraction and selfocusing in a non-linear medium. Zakharov and Shaba(1) explained the connection between self-trapping and soliton theory. They also showed the complete analogy between temporal solitons, in wich the nonlinear phase modulation balances dispersion and spatial solitons in wich the non-linear index profile balances diffraction.

The possibility of self-trapping of optical beams due to an intensity-dependent refractive index was recognized in the early days of nonlinear optics. However, it was soon realized that in a three-dimensional medium, in which light diffracts in two transverse dimensions, self-trapping is not stable and leads to catastrophic collapse and filamentation. Stable self-trapping was then found to be feasible in two-dimensional media, in which the optical beam diffracts only in one transverse direction. Subsequently, the connection between self-trapping and soliton theory, and a complete analogy between spatial and temporal solitons were established. Whereas the formation of temporal solitons requires a balance between dispersion and nonlinear phase modulation, spatial solitons owe their existence to the balancing of diffraction with wavefront curvature induced by the nonlinear refractive index profile of the propagation medium. To observe a spatial soliton one must limit diffraction to one transverse direction, which can be achieved in a planar optical waveguide. The first experiments of this type were conducted using a multimode liquid waveguide (CS 2 confined between a pair of glass slides). Formation of spatial optical solitons in single-mode planar glass waveguides was reported shortly afterwards. Kerr nonlinearity The simplest nonlinearity capable of producing self-trapping (leading to soliton formation in a planar waveguide) is a Kerr nonlinearity, obtained when the refractive index of the medium has an intensity-dependent term of the formwhere I = | E | 2 is the electric field intensity of the optical beam.