Novel nonlinear wave equation: Regulated rogue waves and accelerated soliton solutions

A new exactly solvable ($1+1$)-dimensional complex nonlinear wave equation exhibiting rich analytic properties has been introduced. A rogue wave (RW), localized in space–time like Peregrine RW solution, though richer due to the presence of free parameters is discovered. This freedom allows to regulate amplitude and width of the RW as needed. The proposed equation allows also an intriguing topology changing accelerated dark soliton solution in spite of constant coefficients in the equation.     

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Enabling arbitrary wavelength frequency combs on chip

A necessary condition for generation of bright soliton Kerr frequency combs in microresonators is to achieve anomalous group velocity dispersion (GVD) for the resonator modes. This condition is hard to implement in the visible as well as ultraviolet since the majority of optical materials are characterized with large normal GVD in these wavelength regions. We overcome this challenge by borrowing ideas from strongly dispersive coupled systems in solid state physics and optics. We show that photonic compound ring resonators can possess large anomalous GVD at any desirable wavelength, even if each individual resonator is characterized with normal GVD. Based on this concept, we design a mode‐locked frequency comb with thin‐film silicon nitride compound ring resonators in the vicinity of the rubidium D₁ line (794.6 nm) and propose to use this optical comb as a flywheel for chip‐scale optical clocks.

     

A system of coupled partial differential equations exhibiting both elevation and depression rogue wave modes

Analytical solutions are obtained for a coupled system of partial differential equations involving hyperbolic differential operators. Oscillatory states are calculated by the Hirota bilinear transformation. Algebraically localized modes are derived by taking a Taylor expansion. Physically these equations will model the dynamics of water waves, where the dependent variable (typically the displacement of the free surface) can exhibit a sudden deviation from an otherwise tranquil background. Such modes are termed ‘rogue waves’ and are associated with ‘extreme and rare events in physics’. Furthermore, elevations, depressions and ‘four-petal’ rogue waves can all be obtained by modifying the input parameters.     

On Gradient Solitons of the Ricci–Harmonic Flow

In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.     

亮孤子在宇称时间对称波导中的传输和控制

基于光波在宇称-时间(PT)对称波导中传输的理论模型, 数值研究了亮孤子在呈高斯分布的PT对称克尔非线性平板波导中的传输和控制. PT对称波导, 要求波导的折射率分布呈偶对称, 而增益/损耗分布呈奇对称. 结果表明: 当波导的折射率分布强度为正时, PT对称波导的中心折射率最大, 即使没有自聚焦克尔非线性效应, PT对称波导也可以束缚光波, 形成波浪形光束且长距离传输; 当折射率分布强度为负时, PT对称波导的中心折射率最小, 光波的传输方向发生偏移. 而增益/损耗分布可控制光波的偏移方向: 增益/损耗分布强度为正, 光波向左偏移; 强度为负, 光波向右偏移; 强度为零时, 光波被分为两束. 且当折射率分布强度为负时, 可以很好地抑制相邻亮孤子间的相互作用. 该研究结果可为未来PT对称波导在全光控制方面的应用提供一定的理论依据.     

(1+2)维各向同性介质中的旋转椭圆空间光孤子

从(1+2)维非局域非线性薛定谔方程出发, 通过坐标变换得到了旋转坐标系下的非局域非线性薛定谔方程. 假设响应函数为高斯型, 用虚时间法数值求解了旋转坐标系下的非局域非线性薛定谔方程的静态孤子解, 迭代出了不同非局域程度条件下的静态椭圆孤子数值解. 最后采用分步傅里叶算法, 以迭代的孤子解作为初始输入波形, 模拟了在不同的非局域程度条件下, (1+2)维椭圆空间光孤子的旋转传输特性. 强非局域时, 椭圆光孤子的长轴方向和短轴方向波形都是高斯型, 其他的非局域程度下, 不是高斯型. 由此表明：(1+2)维椭圆光孤子对非局域程度依赖性很强. 旋转角速度和功率均与非局域程度以及孤子的椭圆度有关.     

Soliton and other solutions to the (1+2)-dimensional chiral nonlinear Schrödinger equation

The (1+2)-dimensional chiral nonlinear Schrödinger equation (2D-CNLSE) as a nonlinear evolution equation is considered and studied in a detailed manner. To this end, a complex transform is firstly adopted to arrive at the real and imaginary parts of the model, and then, the modified Jacobi elliptic expansion method is formally utilized to derive soliton and other solutions of the 2D-CNLSE. The exact solutions presented in this paper can be classified as topological and nontopological solitons as well as Jacobi elliptic function solutions.     

Darboux transformation and exact multisolitons for a matrix coupled dispersionless system

We present a matrix coupled dispersionless(CD) system. A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair. We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants. By using properties of the quasideterminants, we obtain some exact solutions, including bright and dark-type solitons, rogue wave and breather solutions of the matrix CD system. Furthermore, it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system, sine-Gordon equation and Maxwell-Bloch system.     

Zero-ϕeff non-Bragg gap solitons in 1D Kerr polaritonic/metamaterial heterostructures

A theoretical study of one-dimensional heterostructures composed of alternate layers of a Kerr polaritonic material and a linear dispersive metamaterial is performed. For frequency values at the edges of the non-Bragg zero-ϕ_eff gap of the heterostructure in the linear regime, a switching from very low to high transmission states is obtained and localized gap solitons of various orders are found, depending on the particular value of the incident power. Soliton solutions are shown to be robust with respect to absorption effects and a study is presented for gap soliton phases at the top and bottom of the zero-ϕ_eff gap in the case of defocusing and focusing nonlinearities.