Solitary Waves for Cubic-Quintic Nonlinear Schroedinger Equation with Variable Coefficients

cubic-quintic 非线性的 Schrodinger 方程(CQNLS ) 在光纤维和原子水动力学起重要作用。由为 cubic-quintic 使用同类的平衡原则，钟类型，纽结类型，代数学的独居的波浪，和三角法的旅行波浪，有可变系数(vCQNLS ) 的非线性的 Schrodinger 方程在一套辅助高顺序的平常的微分方程的帮助下被导出(亚方程为短) 。在这篇论文使用的方法可能帮助一个为 otherhigh 顺序导出准确答案非线性的进化方程，和表演同类的平衡原则的新申请。

Exact Meromorphic Stationary Solutions of the Cubic-Quintic Swift-Hohenberg Equation

In this paper, we study an ODE of the form b0u（4）＋b1u′′＋b2u＋b3u3＋b4u5=0, ′= d/dz＇ , which includes, as a special case, the stationary case of the cubic-quintic Swift-Hohenberg equation. Based on Nevanlinna theory and Painleve analysis, we first show that all its meromorphic solutions are elliptic or degenerate elliptic. Then we obtain them all explicitly by the method introduced in [7].

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrdinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansion technique are used to construct exactself-similar solutions to the cubic-quintic nonlinear Schrodinger equation.Such solutions exist under certain conditions,and impose constraints on the functions describing dispersion,nonlinearity,and the external potential.Some simpleself-similar waves are presented.

SOME DYNAMICAL BEHAVIOR OF THE STUART-LANDAU EQUATION WITH A PERIODIC EXCITATION

The lock-in periodic solutions of the Stuart-Landau equation with a periodic excitation are studied. Using singularity theory, the bifurcation behavior of these solutions with respect to the excitation amplitude and frequency are investigated in detail, respectively.The results show that the universal unfolding with respect to the excitation amplitude possesses codimension 3. The transition sets in unfolding parameter plane and the bifurcation diagrams are plotted under some conditions. Additionally, it has also been proved that the bifurcation problem with respect to frequence possesses infinite codimension. Therefore the dynamical bifurcation behavior is very complex in this case. Some new dynamical phenomena are presented, which are the supplement of the results obtained by Sun Liang et al.

Hybrid Dispersive Optical Solitons in Nonlinear Cubic-Quintic-Septic Schrödinger Equation

Certain hybrid prototypes of dispersive optical solitons that we are looking for can correspond to new or future behaviors, observable or not, developed or will be developed by optical media that present the cubic-quintic-septic law coupled, with strong dispersions. The equation considered for this purpose is that of non-linear Schrödinger. The solutions are obtained using the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning’ functions. Some of the obtained solutions show that their existence is due only to the Kerr law nonlinearity presence. Graphical representations plotted have confirmed the hybrid and multi-form character of the obtained dispersive optical solitons. We believe that a good understanding of the hybrid dispersive optical solitons highlighted in the context of this work allows to grasp the physical description of systems whose dynamics are governed by nonlinear Schrödinger equation as studied in this work, allowing thereby a relevant improvement of complex problems encountered in particular in nonliear optaics and in optical fibers.

Nonautonomous solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrdinger equation with time-and space-modulated coefficients

A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.

A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions

It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer et al. recently prove that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois cannot explain these results. On the other hand, Gauss et al. proved the fundamental theorem of algebra. The theorem declared that there were n solutions for the n degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solutions of cubic equation as a premise to calculate the parameters of his equation. Therefore, Abel’s proof is a logical circular argument and invalid. Besides, Abel confused the variables with the coefficients (constants) of algebraic equations. An expansion with 14 terms was written as 7 terms, 7 terms were missing. We prefer to consider Galois’s theory as a hypothesis rather than a proof. Based on that permutation group S5 had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In order to prove the effectiveness of radical extension group of automorphism mapping for the cubic and quartic equations, in the Galois’s theory, some algebraic relations among the roots of equations were used to replace the root itself. This violated the original definition of automorphism mapping group, led to the confusion of concepts and arbitrariness. For the general cubic and quartic algebraic equations, the actual solving processes do not satisfy the tower structure of Galois’s solvable group. The resolvents of cubic and quartic equations are proved to have no symmetries of Galois’s soluble group actually. It is invalid to use the solvable group theory to judge whether the high degree equation has a radical solution. The conclusion of this paper is that there is only the Sn symmetry for the n degree algebraic equations. The symmetry of Galois’s solvable group does not exist. Mathematicians should get rid of the constraints of Abel and Galois’s theories, keep looking for the radical solutions of high degree equations.

Solutions of the Schrdinger Equation for PT-Symmetric Coupled Quintic Potentials in Two Dimensions

This paper deals with the solutions of time independent Schrodinger wave equation for a two-dimensionalVT-symmetric coupled quintic potential in its most general form.Employing wavefunction ansatz method,generalanalytic expressions for eigenvalues and eigenfunctions for first four states are obtained.Solutions of a particular caseare also presented.

Some Exact Solutions of Variable Coefficient Cubic-Quintic Nonlinear Schrdinger Equation with an External Potential

由构造适当转变和一条扩大椭圆形的亚方程途径，我们与一个外部潜力发现可变系数 cubic-quintic 的一些准确答案非线性的 Schrödinger 方程，它包括钟和纽结侧面独居的波浪答案，单个答案，三角形的周期的波浪答案等等。

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

When analysing the thermal conductivity of magnetic fluids, the traditional Sharma-Tasso-Olver (STO) equation is crucial. The Sharma-Tasso-Olive equation’s approximate solution is the primary goal of this work. The quintic B-spline collocation method is used for solving such nonlinear partial differential equations. The developed plan uses the collocation approach and finite difference method to solve the problem under consideration. The given problem is discretized in both time and space directions. Forward difference formula is used for temporal discretization. Collocation method is used for spatial discretization. Additionally, by using Von Neumann stability analysis, it is demonstrated that the devised scheme is stable and convergent with regard to time. Examining two analytical approaches to show the effectiveness and performance of our approximate solution.

Solitons for the cubic-quintic nonlinear Schrdinger equation with varying coefficients

In this paper,by means of similarity transfomations,we obtain explicit solutions to the cubic-quintic nonlinear Schr顜僤inger equation with varying coefficients,which involve four free functions of space.Four types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.

Applications of Extended Hyperbolic Function Method for Quintic Discrete Nonlinear SchrSdinger Equation

由使用扩大夸张函数方法，我们学习了 quinticdiscrete 非线性的 Schroedinger 方程并且获得了新准确局部性的解决方案包括分离明亮的 soliton 解决方案，黑暗 soliton 答案，bright 和黑暗 soliton 答案，轮流出现的阶段bright soliton 答案，轮流出现的阶段黑暗 soliton 答案，并且轮流出现阶段bright 和黑暗 soliton 答案，如果一种特殊关系对方程的系数约束力。

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic-quintic nonlinear Schrdinger （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 higher-order 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.

Artificial perturbation for solving the Korteweg-de Vries equation

A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the question of convergence of this approach is completely guaranteed here, because a limited number of term included in the series can describe a sufficient exact solution. Comparisons with the solutions of the quintic spline, and finite difference are presented.

Sinusoidal Time-Dependent Power Series Solution to Modified Duffing Equation

Duffing equation,, has a standard well-known ex-act solution [1]. Approximate solutions to this equation also are available [2]. Reference [3] [4] introduces a sinusoidal time-dependent Power Series solution. Applying this method successfully we investigate the approximate solution of the modified Duffing equations,, for n = 4 and 5. Symbolic manipulative utilities of a Computer Algebra System (CAS) specifically Mathematica [5] extensively is used investigating the results.

非Kerr光纤中亮孤子的稳定性与相互作用

非Kerr光纤中的亮孤子的演化可以用具有三次-五次竞争非线性项的非线性薛定谔方程来描述.为数值求解该方程的初值问题,本文将无界区域截断为有界区域,根据亮孤子在远场的渐近行为构造了合理的边界条件,从而将该初值问题转换为初边值问题.对这个初边值问题,本文分别提出了Crank-Nicolson有限差分(Crank-Nicolson Finite Difference, CNFD)格式和时间分裂有限差分(Time-Splitting Finite Difference, TSFD)格式.这两种格式在空间和时间维度上都具有二阶精度,其中CNFD格式是全隐格式,可以守恒离散能量和质量,TSFD是线性隐式格式,可以守恒离散质量.在以数值算例验证两种方法的计算效率后,本文用TSFD格式研究了非Kerr光纤中亮孤子的稳定性与相互作用.

Linear, Cubic and Quintic Coordinate-Dependent Forces and Kinematic Characteristics of a Spring-Mass System

By combining a pair of linear springs we devise a nonlinear vibrator. For a one dimensional scenario the nonlinear force is composed of a polynomial of odd powers of position-dependent variable greater than or equal three. For a chosen initial condition without compromising the generality of the problem we analyze the problem considering only the leading cubic term. We solve the equation of motion analytically leading to The Jacobi Elliptic Function. To avoid the complexity of the latter, we propose a practical, intuitive-based and easy to use alternative semi-analytic method producing the same result. We demonstrate that our method is intuitive and practical vs. the plug-in Jacobi function. According to the proposed procedure, higher order terms such as quintic and beyond easily may be included in the analysis. We also extend the application of our method considering a system of a three-linear spring. Mathematica [1] is being used throughout the investigation and proven to be an indispensable computational tool.

Spatiotemporal Similaritons in (3+l)-Dimensional Inhomogeneous Nonlinear Medium with Cubic-Quintic Nonlinearity

我们获得准确空间与时间的 similaritons 到一(3+1 ) 维的不同类的非线性的 Schr ? dinger 方程，它与分布式的分散和获得在 cubic-quintic 非线性媒介描述光脉搏的繁殖。当某个相容性条件满足时，在椭圆形的方程的如此的自我类似的波浪和答案之间的一对一的通讯被发现。基于准确解决方案，我们在二种典型 soliton 控制系统讨论自我类似的 cnoidal 波浪和啁啾的 similaritons 的进化行为。而且，在啁啾的 similaritons 之间的比较和啁啾免费的 solitons 被给。

Does a Restricted Quintic Polynomial in Minimum Time Step (Planck Time Interval) Being Solvable in a Galois Theory Sense Affect the Closing of a Wormhole Throat if (Kaluza Klein Theory) Is Assumed and Impact Admissible Gravitational Wave Polarization?

In a prior paper, the d = 1 to d = 7 sense of AdS/CFT solutions were described in general whereas we did not introduce commentary as to GW polarization of gravitational radiation from a worm hole. We will discuss GW polarization, for d = 1 and in addition say concrete facts as to the strength of the GW radiation, and admissible frequencies. First off, the term Δt is for the smallest unit of time step. Note that in the small Δt limit for d = 1 we avoid any imaginary time no matter what the sign of Ttemp is. And when d = 1 in order to have any solvability one would need X = Δt assumed to be infinitesimal. To first approximation, we set X = Δt as being of Planck time, 10-31 or so seconds, in duration.

Orthogonality and Quintic Functional Equations

Using the fixed point method, we prove the Hyers Ulam stability of an orthogonally quintic functional equation in Banach spaces and in non-Archimedean Banach spaces.