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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

非线性振动、非线性波与Jacobi椭圆函数(续)

5　正弦戈登方程、扭结、反扭结波、呼吸孤立波正弦戈登方程 (sing -Gordon方程 ,简记为SG方程 )φxx- φtt=sin φ (38)是自然科学和应用科学中出现的另一个重要的非线性发展方程 (演化方程 ) .在晶体位错的传播、磁旋波在铁磁材料中的传播、约瑟夫森 (Josephson)结中继传输线等乃至生物物理的诸多问题中都应用了这一方程 .本文从单摆弹簧链模型这一特例引入SG方程 .考虑如图 3所示的N个相连弹簧 -单摆运动方程 (称为单摆弹簧链 ) .N个单摆悬挂在横置的诸弹簧上 ,每个摆的长度和质量均相同 ,每个螺旋弹簧的长度和劲度系数也相同 ,则第i…     

非线性振动、非线性波与Jacobi椭圆函数

介绍用较易懂且简捷的Jacobi椭圆函数解法求非线性振动与非线性波的解析解,并以单摆,达芬(Duffing)振子,KdV方程,正弦戈登(Gordon)方程(SG方程),非线性薛定谔方程(NLS方程)的椭圆函数解,钟形孤立波解,扭结与反扭结波解,呼吸子解,扭结波与反扭结波迎头碰撞及包络型孤立子波解等重要实例,给出了说明．     

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation.In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions contain polynomials of the fourth order rather than the second order. Moreover, dark-brightrogue wave solutions of the coupled Hirota equations are given, and interactions between Peregrine solitons and dark-bright solitons are analyzed. The results further reveal the dynamical properties of rogue waves for the coupled Hirota equations.