Construction of Nth-order rogue wave solutions for Hirota equation by means of bilinear method
Gui Mu, Zhenyun Qin
本文利用双线性方法构造了Hirota方程的N阶Rouge波解。这个公式可以用行列式来表示.此外，一些有趣的动力学模式的流氓波展出。
In this work, we focus on the construction of Nth-rouge wave solutions for the Hirota equation by utilizing the bilinear method. The formula can be represented in terms of determinants. In addition, some interesting dynamic patterns of rogue waves are exhibited.
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We construct rogue wave solutions on the double periodic background for the Hirota equation through one fold Darboux transformation formula. We consider two types of double periodic solutions as seed solutions. We identify the squared eigenfunctions and eigenvalues that appear in the one fold Darboux transformation formula through an algebraic method with two eigenvalues. We then construct the desired solution in two steps. In the first step, we create double periodic waves as the background. In the second step, we build rogue wave solution on the top of this double periodic solution. We prese…
Modulation instability and controllable rogue waves with multiple compression points for periodically modulated coupled Hirota equations
基于调制不稳定性分析和广义达布变换，我们导出了变系数耦合Hirota方程的流氓波解族。给出了显式的一阶流氓波解，通过在耦合方程系数中选取足够大的周期调制振幅，得到了具有多个压缩点的暗-亮复合流氓波。此外，暗-亮和复合Peregrine梳是从多个压缩点生成的。对于二阶情形，分别提出了暗亮三姐妹、流氓波四联体和流氓波六联体结构，其中流氓波包含一个或多个压缩点。此外，还讨论了暗流氓波解具有多个压缩点的一些波动特征，如光强与连续波背景的差异、脉冲能量演化等。
Generalized Darboux transformation and higher-order rogue wave solutions of the coupled Hirota equations
研究了具有三阶色散、自陡和受激拉曼散射等高阶非线性效应的耦合Hirota方程的高阶反常波解.利用广义达布变换，得到了具有3N+1个自由参数的N阶流氓波解的统一表示。特别地，明确地给出了包含四阶多项式的一阶流氓波解和由八阶多项式组成的二阶流氓波解。通过数值计算，我们发现四个或六个基本流氓波可以共存于二阶流氓波。通过调整自由参数的值，可以得到圆形、四边形、三角形、直线和基本模式等不同的时空分布结构。此外，我们看到，九个或十二个基本流氓波可以同时出现在三阶流氓波。通过自由参数的适当选择，可以得到更为复杂的时空分布形态。讨论了异常波的振幅、最高峰的坐标位置等波浪特征。
High-order rogue waves for the Hirota equation
Hirota方程比非线性薛定谔方程更适合于深海波的近似。利用参数化达布变换构造了Hirota方程的高阶有理解.根据解的结构将这类解分为几类。
Nonautonomous discrete rogue waves and interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients
本文研究了变系数广义Ablowitz-Ladik-Hirota格点中的非自治离散流氓波解及其相互作用，这些非自治离散流氓波解具有复杂的时间传播特性，并超越了通常的离散流氓波.当隧道之间的耦合系数的幅度减少，这些非自治离散流氓波的解决方案成为本地化的时间后，他们传播了一些大的临界值。此外，我们发现，非自治离散流氓波之间的相互作用是弹性的。特别是，这些结果可以减少到通常的离散流氓波的解决方案时，忽略的增益或损失项。
Rogue waves of the Hirota and the Maxwell-Bloch equations
本文导出了飞秒脉冲在掺铒光纤中传输时的Hirota和Maxwell-Bloch（H-MB）系统的Darboux变换，并将其推广到该系统的$n$ -折叠达布变换的矩阵形式。这种$n$ -折叠达布变换蕴含了由$(E, p,η)$的已知解生成的$(E^{[n]},p^{[n]}, η^{[n]})$的$n$ -次新解的行列式表示. $(E^{[n]},p^{[n]} ,η^{[n]})$的行列式表示给出了H-MB系统的孤子解、位置子解和呼吸子解（亮呼吸子和暗呼吸子）。从呼吸子解出发，我们还构造了H-MB系统的亮和暗流氓波解，这是目前数学和物理学中最热门的话题之一。令人惊讶的是，$p\, and\, η$的流氓波解具有两个峰值，这是因为它们的分子和分母的顺序。同时，在固定时间和空间参数，改变另外两个未知参数$α$和$β$的情况下，首次生成了一个异常波形。
The third-order nonlinear Schrodinger equation (alias the Hirota equation) is investigated via deep leaning neural networks, which describes the strongly dispersive ion-acoustic wave in plasma and the wave propagation of ultrashort light pulses in optical fibers, as well as broader-banded waves on deep water. In this paper, we use the physics-informed neural networks (PINNs) deep learning method to explore the data-driven solutions (e.g., soliton, breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and unperturbated (a 2% noise) training data are considere…
In this paper the coupled Maxwell-Bloch equations which describe the propagation of two optical pulses in an optical medium with coherent three-level atoms are studied by Darboux transformation. The general nth-order rogue wave solution involving two different choices of multiple roots for the spectral characteristic equation and the multiparametric nth-order semirational solution are both obtained in terms of Schur polynomials. The explicit rogue wave solutions and semirational solutions from first to second order are provided. In contrast to the known Peregrine soliton, dark and four-petaled…
Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing the unidirectional propagation of shallow-water waves and perform Painlevé analysis to understand its integrability nature. We construct the explicit form of higher-order rogue wave solutions by adopting Hirota's bilinearization and generalized polynomial functions. Further, we explore their dynamics in detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants in their m…
High-order rogue waves of a long wave-short wave model
长波-短波模型描述了长波和短波之间的相互作用。利用Hirota双线性方法和KP降阶法计算了由行列式表示的高阶有理解。结果表明，短波的基本流氓波可分为亮型、中型和暗型三种，而长波的流氓波则总是亮型。高阶异常波对应于基本异常波的叠加。调制不稳定性分析表明，基带调制不稳定的条件下，一个不稳定的连续波背景对应的扰动与无穷小的频率，符合存在的罗格波解决方案的条件。
Modulation instability, rogue wave and spectral analysis are investigated for the nonlinear Schrodinger equation with the higher-order terms. The modulation instability distribution characteristics from the sixth-order to the eighth-order nonlinear Schrodinger equations are studied. Higher-order dispersion terms are closely related to the distribution of modulation stability regime, and n-order dispersion term corresponds to $n-2$ modulation stability curves in the modulation instability band. Based on the generalized Darboux transformation method, the higher-order rational solutions are const…
Optical rogue waves in the generalized inhomogeneous higher-order nonlinear Schrodinger equation with modulating coefficients
高阶色散和非线性效应（别名{ \it微扰项}）如三阶色散、自陡、自频移等在超短光脉冲传输的研究中起着重要的作用。考虑了具有空间和时间调制参数的广义高阶非线性薛定谔方程的光学流氓波解及其相互作用。给出了一个适当的变换，将广义高阶NLS方程化为可积的常系数Hirota方程。这种变换使我们能够将广义高阶NLS方程的某些类精确解与可积Hirota方程的各种解联系起来。特别是，我们说明的方法，在两个最低阶的合理解决方案的广田方程作为播种功能，以产生流氓波解决方案本地化的时间，有复杂的演变在空间中有或没有微分增益或损失项。我们简单地分析了获得的光学流氓波的物理机制的基础上，这些限制。最后，数值分析了所得Rogue波解的稳定性。所获得的异常波解为非线性光学及其他非线性科学领域如玻色-爱因斯坦凝聚体和海洋等的相关实验和应用提供了可能
Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions
本文构造了非线性薛定谔方程的一个广义达布变换。给出了相应的$N$ -折叠达布变换的求和公式和行列式.作为应用，我们得到了聚焦非线性薛定谔方程和Hirota方程的$N$阶流氓波解的紧致表示。特别是，一般的三阶流氓波的动力学进行了讨论，并表现出有趣的结构。
Generalized Darboux transformation and higher-order rogue wave solutions to the Manakov system
在本文中，我们构造了一个广义递归达布变换的聚焦向量非线性薛定谔方程称为Manakov系统。我们将这种广义递归达布变换应用于该系统的Lax对，通过直接迭代规则生成具有相同谱参数的N阶矢量广义流氓波解。因此，我们讨论了一阶、二阶和三阶矢量广义流氓波解，并对这些特征作了说明。我们表明，高阶流氓波的解决方案取决于其自由参数的值。
Dynamics of rogue waves on a multi-soliton background in a vector nonlinear Schrodinger equation
利用Darboux-dressing变换和渐近展开法导出了向量非线性Schrodinger方程（Manakov系统）的一般高阶反常波。包含3 N个自由参数的N阶半有理解以分离变量形式表示。这些解决方案表现出流氓波的多孤子背景。结果表明，两分量系统中的异常波结构比单分量系统中的异常波结构更丰富。研究这些结果对于理解和预测光学、流体动力学、玻色-爱因斯坦凝聚体和金融等非线性复杂系统中的反常波现象具有重要意义。
In this paper, a modified symbolic computation approach is proposed. The multiple rogue wave solutions of a generalized (2+1)-dimensional Boussinesq equation are obtained by using the modified symbolic computation approach. Dynamics features of these obtained multiple rogue wave solutions are displayed in 3D and contour plots. Compared with the original symbolic computation approach, our method does not need to find Hirota bilinear form of nonlinear system.
General high-order rogue waves of the (1+1)-dimensional Yajima-Oikawa system
利用Hirota的双线性方法和KP族降阶技巧，得到了（1+1）维Yajima-Oikawa（YO）系统的一般高阶流氓波解。这些流氓波的解决方案提出的行列式中的元素是代数表达式。详细研究了自由参数不同时一阶和高阶流氓波的动力学。结果表明，基本（一阶）流氓波可以分为三种不同的模式：明亮，中间和黑暗的。高阶异常波对应于基本异常波的叠加。特别是，与非线性Schodinger方程相比，由于YO系统不具有伽利略不变性，因此对于一阶和高阶的异常波，都存在一个控制异常波形态的重要参数α。
Novel periodic solutions and rogue waves of the defocusing scalar and coupled Ablowitz-Ladik systems on a nonzero background
本文应用Hirota双线性方法，在背景振幅0<ρ<1的假设下，研究了散焦色散区域中标量和耦合Ablowitz-Ladik系统，发现了新型周期解和怪波解。
This work deals with the dynamics of higher-order rogue waves in a new integrable (2+1)-dimensional Boussinesq equation governing the evolution of high and steep gravity water waves. To achieve this objective, we construct rogue wave solutions by employing Bell polynomial and Hirota's bilinearization method, along with the generalized polynomial function. Through the obtained rogue wave solutions, we explore the impact of various system and solution parameters in their dynamics. Primarily, these parameters determine the characteristics of rogue waves, including the identification of their type…
Vector rogue wave patterns associated with generalized Hermite and Okamoto polynomials
本文在多分量非线性薛定谔和Hirota方程中，利用广义Hermite和Okamoto多项式的根，揭示了大参数下两类怪波模式，并通过数值验证。
In this paper, the PT -symmetric version of the Maccari system is introduced, which can be regarded as a two-dimensional generalization of the defocusing nonlocal nonlinear Schrodinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long-wave limit, and Kadomtsev-Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari system are derived for the first time. Simultaneously, a new nonlocal Davey-Stewartson-type equation is derived. Solutions for breathers and breathers on top of periodic line waves are obtai…
Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Backlund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function, a family of deformed soliton and deformed breather solutions are presented with the improved Hirotas bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue wa…
Explicit lump and line rogue wave solutions to a modified Hietarinta equation
本文研究修正的Hietarinta方程在2+1维下的lump波和怪波解，借助计算机代数系统求解并分析其动力学行为。
Periodic and rational solutions of the reduced Maxwell-Bloch equations
研究了描述短光脉冲在具有共振非简并跃迁的介质中传输的简化Maxwell-Bloch方程。利用达布变换得到了一般的N阶周期解，并从所得到的一般周期解的两种不同的极限情形，分别得到了具有紧致行列式表示的N阶退化周期解和含多个自由参数的N阶有理解.从一阶到二阶，这些解决方案的显式表达式。文中给出了RMB方程的四个分量在二阶有理解中的典型非线性波图案，即单峰、双峰双谷、双峰和单谷结构。这种有理解对应于约化Maxwell-Bloch方程中的异常波。
Integrable nonlocal Hirota equations
我们构造了几个新的对应于广田方程非局部版本的可积系统，广田方程是高阶非线性薛定谔方程的一个特殊例子。新模型的可积性是通过提供它们的拉克斯对或零曲率条件的显式形式来建立的。在这种结构中产生的两个相容性方程被发现是相互相关的奇偶变换$\mathcal{P}$，由时间反演$\mathcal{T}$或$\mathcal{PT}$ -变换可能结合共轭。我们构造这些模型的显式多孤子解采用Hirota的直接方法以及达布-克拉姆变换。这些模型的非局部性质允许修改这些解决方案的程序作为新的系统也具有不同的参数依赖性和不同的定性行为的新类型的解决方案。多孤子解的类型多种多样，如空间非局域的、时间非局域的时间晶体型的、在时间/空间上具有局域结构的规则的或Rogues波型的。
Two (2 + 1)-dimensional integrable nonlocal nonlinear Schrodinger equations: Breather, rational and semi-rational solutions
最近，Fokas（eq. (18)in Nonlinear 29}，319324（2016））。根据这种模式，在不同的约化条件下，这里引入了两个具有特定奇偶时间对称性的可积方程[方程2和3]。对于eq. 2.利用Hirota双线性方法，得到了两类周期解的解析表达式。在长波极限下，这两个周期解分别转化为流氓波解和半有理解。RWs具有线形，而半理性状态表示建立在周期性线波背景之上的RWs。类似地，半合理的解决方案，包括一条线RW和线呼吸器推导。对于eq. 3、三种解析解，\textit{viz} .，利用Hirota方法得到了呼吸子、块和代表块、周期线波和呼吸子的半有理解。它们的动力学进行了分析，并通过三维绘图演示。值得注意的是，EQ。通过一定的变换，方程2可化为（1+1）维\textquotedblleft逆空间非局部NLS方程。最后，总结了方程2与方程3解的主要区别。
Families of rational and semi-rational solutions of the partial reverse space-time nonlocal Mel'nikov equation
受Ablowitz，Mussliman和Fokas工作的启发，引入了一个部分逆时空非定域Mel'nikov方程。该方程提供了非局部薛定谔-布辛尼斯克方程的二维类似物。利用Hirota的双线性方法，得到了孤立子、呼吸子以及由呼吸子和周期线波组成的混合解。进一步，取这些孤子解的长波极限，导出了非局部Mel'nikov方程的有理解和半有理解。理性的解决方案是一堆。半有理解是由块、呼吸子和周期线波组成的混合解。在适当的参数约束下，从非局部Mel'nikov方程的解出发，得到了非局部Schrodinger-Boussinesq方程的基本异常波和半有理解.
Families of exact solutions of a new extended (2+1)-dimensional Boussinesq equation
最近，J.Y.提出了$(2+1)$维[ $(2+1)d$ ] Boussinesq方程的一个新的变形. Zhu，arxiv：1704.02779v2，2017;参见等式（三）、本文首先利用行波方法导出了推广的$(2+1)d$ Boussinesq方程的亮型和暗型单孤子解。其次，利用Hirota双线性方法和长波极限得到了$N$ -孤子、呼吸子和有理解。解析地得到了两类非奇异有理解：（i）W型线波形式的Rogue波解和（ii）块型解。还提出了两类一般的半有理解。所得到的半有理解如下：（iii）一阶块和亮单孤子解的混合和（iv）一阶块和一阶呼吸子的混合。
An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper.We will refer to this integrable extension as the (4+1)-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the (4 + 1)-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained h…
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