Двухкомпонентное бризерное решение уравнения Хироты

Using the generalized perturbation reduction method the Hirota equation is transformed to the coupled nonlinear Schr¨odinger equations. A solution of the Hirota equation in the form of the two-component vector breather oscillating with the sum and difference of the frequencies and the wave numbers which coincide with the 0π pulse of the self-induced transparency is obtained.

Controllable molecule waves in the femtosecond regime

We investigate the nonautonomous molecule waves of the inhomogeneous Hirota equation describing the propagation of femtosecond pulses in inhomogeneous fibers. By employing the characteristic line analysis, the breather molecules of the inhomogeneous Hirota equation under different cases of dispersion and nonlinear modulation are obtained. We find that the inhomogeneous coefficients d2(z) and d3(z) have influences on the distance between atoms. In addition, we introduce the state transitions to the nonautonomous breather molecules and reveal that there is no full-transition mode for the inhomogeneous Hirota equation.

Multi-Elliptic Rogue Wave Clusters of the Nonlinear Schrodinger Equation on Different Backgrounds

In this work we analyze the multi-elliptic rogue wave clusters as new solutions of the nonlinear Schr\"odinger equation (NLSE). Such structures are obtained on uniform backgrounds by using the Darboux transformation scheme of order $n$ with the first $m$ evolution shifts that are equal, nonzero, and eigenvalue-dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of $n-2m$ order appears at the origin of the $(x,t)$ plane and can be considered as the central rogue wave of the cluster. We show that the high-intensity narrow peak, with characteristic intensity distribution in its vicinity, is enclosed by $m$ ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a periodic background, we utilize the modified Darboux transformation scheme to build these solutions on a Jacobi elliptic dnoidal background. We analyze the minor semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the $(x,t)$ plane is violated when the shift values are increased above a threshold. We apply the same analysis on Hirota equation, to examine the influence of a free real parameter and Hirota operator on the cluster appearance. The same analysis can be extended to the infinite hierarchy of extended NLSEs.

Solving Schrödinger–Hirota Equation in a Stochastic Environment and Utilizing Generalized Derivatives of the Conformable Type

This work is devoted to providing new kinds of deterministic and stochastic solutions of one of the famous nonlinear equations that depends on time, called the Schrödinger–Hirota equation. A new and straightforward methodology is offered to extract exact wave solutions of the stochastic nonlinear evolution equations (NEEs) with generalized differential conformable operators (GDCOs). This methodology combines the features of GDCOs, some instruments of white noise analysis, and the generalized Kudryashov scheme. To demonstrate the usefulness and validity of our methodology, we applied it to extract diversified exact wave solutions of the Schrödinger–Hirota equation, particularly in a Wick-type stochastic space and with GDCOs. These wave solutions can be turned into soliton and periodic wave solutions that play a main role in numerous fields of nonlinear physical sciences. Moreover, three-dimensional, contour, and two-dimensional graphical visualizations of some of the extracted solutions are exhibited with some elected functions and parameters. According to the results, our new approach demonstrates the impact of random and conformable factors on the solutions of the Schrödinger–Hirota equation. These findings can be applied to build new models in plasma physics, condensed matter physics, industrial studies, and optical fibers. Furthermore, to reinforce the importance of the acquired solutions, comparative aspects connected to some former works are presented for these types of solutions.