Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters \(b,\ c\) and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.     

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System: Algebraic Structures,Bäcklund Transformation and Hidden Structural Symmetries

The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, {J. Math. Phys.} 16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Bäcklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2+1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.     

Diverse chirped optical solitons and new complex traveling waves in nonlinear optical fibers

This paper studies chirped optical solitons in nonlinear optical fibers. However, we obtain diverse soliton solutions and new chirped bright and dark solitons, trigonometric function solutions and rational solutions by adopting two formal integration methods. The obtained results take into account the different conditions set on the parameters of the nonlinear ordinary differential equation of the new extended direct algebraic equation method. These results are more general compared to Hadi et al(2018 Optik 172 545–53) and Yakada et al(2019 Optik197 163108).     

Rogue wave dynamics in barotropic relaxing media

In this work, we deal with a nonlinear wave equation, namely the Vakhnenko equation, which models the propagation of nonlinear wave in the barotropic relaxing media. Based on the homoclinic breather limit method, we seek rogue wave solution to the above equation. The results show that rogue wave or giant wave can exist in such a medium.     

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System:Algebraic Structures,Bcklund Transformation and Hidden Structural Symmetries

The prolongation structure methodologies of Wahlquist-Estabrook [H.D.Wahlquist and F.B.Estabrook,J.Math.Phys.16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system.Based on the obtained prolongation structure,a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed.A Lie-Algebra representation of some hidden structural symmetries of the previous system,its Bcklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived.In the wake of the previous results,we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + 1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation,which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.     

Traveling Wave-Guide Channels of a New Coupled Integrable Dispersionless System

In the wake of the recent investigation of new coupled integrable dispersionless equations by means of the Darboux transformation [Zhaqilao,et al.,Chin.Phys.B 18(2009) 1780],we carry out the initial value analysis of the previous system using the fourth-order Runge-Kutta's computational scheme.As a result,while depicting its phase portraits accordingly,we show that the above dispersionless system actually supports two kinds of solutions amongst which the localized traveling wave-guide channels.In addition,paying particular interests to such localized structures,we construct the bilinear transformation of the current system from which scattering amongst the above waves can be deeply studied.     

Jet bundle formalism towards the extension of the stretched rope equation

In this paper, based upon a jet bundle formalism, we investigate the connection and the curvature associated to the stretched rope equation. Following the prolongation structure analysis due to Wahlquist and Estabrook combined to Cartan-Ehresmann connection with structure group SL(3,R), we derive an interesting coupled system dubbed as the two-component stretched rope equation which physically and geometrically arises from the curve motion flow in Euclidean space.