On the investigation of nonlinear waves of the Hirota and the Maxwell–Bloch equation in nonlinear optics

In this paper, considering the Hirota and Maxwell-Bloch (H-MB) equations which is governed by femtosecond pulse propagation through two-level doped fibre system, we construct the Darboux transformation of this system through linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.     

Nonlinear Bloch waves

The nonlinear Bloch theorem for the temporal and spatial Schrödinger solitons in dispersive and nonlinear periodic structures is proved. It is shown that bright and dark solitary nonlinear Bloch waves exist only under certain conditions and that the parameter functions describing dispersion and nonlinearity periodic inhomogeneities cannot be chosen independently.     

Dark and composite rogue waves in the coupled Hirota equations

The intriguing dark and composite rogue wave dynamics in a coupled Hirota system are unveiled, based on the exact explicit rational solutions obtained under the assumption of equal background height. It is found that a dark rogue wave state would occur as a result of the strong coupling between two field components with large wavenumber difference, and there would appear plenty of composite structures that are attributed to the specific wavenumber difference and the free choice of three independent structural parameters. The coexistence of different fundamental rogue waves in such a coupled system is also demonstrated.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Cylindrical electromagnetic waves in a nonlinear nondispersive medium: Exact solutions of the Maxwell equations

The excitation and propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium are analyzed. It is assumed that the medium lacks a center of symmetry and that the dependence of the electric displacement on the electric field can be approximated by an exponential function. For this case, a method for integrating the system of the Maxwell equations is proposed. Exact solutions to a set of nonlinear electromagnetic field equations are obtained by this method. It is shown that nonlinear effects described by these solutions can become apparent under experimental conditions.     

Resonant nonlinear optics of backward waves in negative-index metamaterials

The extraordinary properties of resonant four-wave mixing of backward and ordinary electromagnetic waves in doped negative-index materials are investigated. The feasibility of independent engineering of the negative refractive index and the nonlinear optical response as well as quantum control of the nonlinear propagation process in such composites is shown due to the coherent energy transfer from a control field to a signal field. Laser-induced transparency, quantum switching, frequency-tunable narrow-band filtering, amplification, and realizing a miniature mirrorless optical parametric generator of the entangled backward and ordinary waves are among the possible applications of the investigated processes.     

A variational approach to nonlinear evolution equations in optics

A tutorial review is presented of the use of direct variational methods based on Rayleigh-Ritz optimization for finding approximate solutions to various nonlinear evolution equations. The practical application of the approach is demonstrated by some illustrative examples in connection with the nonlinear Schrödinger equation.     

Nonlinear electron density waves and nonlinear acoustic waves in carbon nanotubes

Problems in nonlinear properties of carbon nanotubes with a strong interaction of electrons are considered if the electron mobility, the Coulomb repulsion of electrons at one site of a carbon nanotube, and variations in the distance between neighboring sites owing to acoustic oscillations are taken into account. The possible occurrence of nonlinear acoustic lattices in carbon nanotubes that are small in diameter is investigated.     

High-order rogue waves for the Hirota equation

The Hirota equation is better than the nonlinear Schrödinger equation when approximating deep ocean waves. In this paper, high-order rational solutions for the Hirota equation are constructed based on the parameterized Darboux transformation. Several types of this kind of solutions are classified by their structures.     

Stable standing waves of nonlinear Klein-Gordon equations

In this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear Klein-Gordon equations.     

Complexiton solutions for two nonlinear partial differential equations via modification of simplified Hirota method

In this study, we obtain complexiton solutions of Sawada–Kotera equation and ninth-order KdV equation. For this cause, we employ Wazwaz and Zhaqilao’s method which can be regarded as generalization of simplified Hirota method through extension real parameters into complex parameters. Special conditions to distinguish complexiton, soliton, and soliton–complexiton interaction solutions from each other are given.     

One invariance of the Hirota equations and its consequences

Moscow State University. Presented at the International Workshop Squeezing, Groups, and Quantum Mechanics, Baku, Azerbaijan, September 16–22, 1991.     

Nonlinear optics of backward waves and extraordinary features of plasmonic nonlinear-optical microdevices

Extraordinary properties and unique features of coherent nonlinear-optical coupling of ordinary and backward electromagnetic waves in negative-index metamaterials are described and numerically simulated. The possibility for design of unique microscopic photonic devices and all-optical data processing chips is shown.     

Symmetry of the Maxwell equations

An analysis is made of the properties of the ring allowed by the Maxwell system, whose important generators are first-order differential operators.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 33–38, November, 1969.     

大尺度浅水波方程中相互调制的非线性波

基于一般的浅水波方程, 根据大尺度正压大气的特点, 得到无量纲的控制大尺度大气的动力学非线性方程组. 利用多尺度法, 由无量纲的动力学方程组导出了扰动位势的非线性控制方程. 采用椭圆方程构造该扰动位势控制方程的解, 获得了扰动位势和速度的多周期波与冲击波(爆炸波) 并存的解析解. 扰动位势的解表明经向和纬向具有不同周期和波长的周期波, 且都受纬向孤波的调制; 速度的解表明大尺度大气流动存在气旋和反气旋周期性分布的现象. 关键词： 浅水波方程 大尺度正压大气 解析解 非线性波     

Phase measurement of waves that obey nonlinear equations

We consider the problem of phase retrieval for classical and quantum wave fields that obey a wide class of nonlinear wave equations. This problem is tackled by means of a suitable generalization of existing methods based on the linear transport-of-intensity equation. The extension of these ideas to systems of coupled nonlinear wave equations is also given.