We investigate the interactional behavior of three nonautonomous solitons in an optical fiber with inhomogeneous nature which can be described by nonlinear Schrödinger equation with nonautonomous ...term. A pair of matrices constructed with the aid of AKNS procedure for nonautonomous NLS equation with variable coefficients. Based on the Lax pair, three soliton solutions are generated by employing Darboux method. We show the switchable characteristics of nonautonomous three solitons by typical graphical illustration. The remarkable results presented in this study can be used to design the optical logic gate devices for optical computing.
A nonlinear Schrödinger equation with the combined effects of variable nonlinearity and generalized external potentials is investigated. Three soliton solutions are generated by means of Darboux ...method through constructed Lax pair. We attained two constraints related to gain or loss function for considered equation via compatibility condition. Using three soliton solutions, influences of the inhomogeneous nonlinearity and harmonic potential on soliton structures are analysed by properly tailoring the loss or gain parameter. Specifically, via inelastic collision among three solitons, soliton switching characteristics is observed. Additionally, we explore the Modulation instability (MI) through linear stability analysis (LSA) and impact of nonlinearity profile is examined. The trigonometric, exponential and constant values have been chosen for loss or gain parameter to study the effect on the MI gain spectrum.
In this work, the existence of nonlinear ring waves in the form of multi-solitons is investigated analytically by considering a non-isospectral Nonlinear Schrödinger (NLS) equation with external ...potential. In order to generate soliton solutions, Darboux transformation technique is employed based on the Lax pair. Due to inhomogeneous effect of non-isospectral parameter, we attained the various form of ring solitons. Moreover, transformation from bright ring to dark ring and dromion structure are also observed for the first time. This indicates that the structure of nonlinear wave is strongly dependent on the combined effects of non-isospectral functions and external potential. To illustrate the physical significant of the considered equation and obtained solutions, the 2D and 3D graphs are successfully plotted by selecting appropriate values of control parameters. Finally, we perform linear stability analysis on non-isospectral Nonlinear Schrödinger (NLS) equation to demonstrate the gain spectrum for different strength of nonlinearity.
This paper is devoted to investigate the integrability and linearizability problems around a singular point at the origin of a cubic three-dimensional Lotka-Volterra differential system with ...-resonance. A complete set of necessary conditions for both integrability and linearizability are given. For sufficiency part, we show that the system has two analytic first integrals at the origin. In particular, we use Darboux method of integrability, linearizable node and transformation technique to show that the system admits two independent first integrals.
The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the ...finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.
By using the Darboux method of integrability and solving linear partial differential equations, the whole classification of the integrals of motion of the reduced three-wave interaction system is ...obtained. Rigorous proof is given.
We provide a contact invariant characterization for equations of the formuxy+a(x, y, u)ux+b(x, y, u)uy+c(x, y, u)=0,uxy+a(x, y)ux+b(x, y)uy+c(x, y, u)=0,uxy+c(x, y, u)=0,uxy=0. We classify all ...equations of the form uxy+f(x, y, u, ux, uy)=0 for which the two Ovsiannikov's invariants are constants. These results include characterization of the Klein–Gordon equation uxy=u, the Liouville equation uxy=eu, and the class of Euler–Poisson–Darboux equations. It is shown that the wave equation uxy=0, Liouville equation, and the linear equation uxy=2u/(x+y)2 are the only variational equations Darboux integrable at level one. We also show that a hyperbolic Monge–Ampére equation Darboux integrable at level one is equivalent to an equation of type uxy+f(x, y, u, ux, uy)=0. We prove that the hyperbolic Fermi–Ulam–Pasta (FPU) equation uyy=κ(ux)2uxx is contact equivalent to a linear equation of type uxy=c(x+y)u and we classify all FPU equations Darboux integrable at level one. We also apply our results to equations of type uxy=F(u, ux) that describe pseudo-spherical surfaces.
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