The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq-Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq-Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink （soliton） solutions and multiple-singular-kink （soliton） solutions are derived for the two equations.

Soliton Solutions of Coupled KdV System from Hirota's Bilinear Direct Method

与 Hirota 的双线性的直接方法，我们学习特殊联合 KdV 系统获得它的新 soliton 答案。然后，我们进一步与阴谋详细讨论 soliton 进化，相应结构，和有趣的交互现象。作为结果，我们在相互作用以后发现那， solitons 做有弹性的碰撞并且没有他们包括除了阶段移动的精力，速度和形状的物理数量的交换。

Integrable discretization of soliton equations via bilinear method and Backlund transformation

We present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota’s bilinear method.This approach is mainly based on the compatibility between an integrable system and its B¨acklund transformation.We apply this procedure to several equations,including the extended Korteweg-deVries(Kd V)equation,the extended Kadomtsev-Petviashvili(KP)equation,the extended Boussinesq equation,the extended Sawada-Kotera(SK)equation and the extended Ito equation,and obtain their associated semidiscrete analogues.In the continuum limit,these differential-difference systems converge to their corresponding smooth equations.For these new integrable systems,their B¨acklund transformations and Lax pairs are derived.

An Augmented Lagrangian based Semismooth Newton Method for a Class of Bilinear Programming Problems

This paper proposes a semismooth Newton method for a class of bilinear programming problems(BLPs)based on the augmented Lagrangian,in which the BLPs are reformulated as a system of nonlinear equations with original variables and Lagrange multipliers.Without strict complementarity,the convergence of the method is studied by means of theories of semismooth analysis under the linear independence constraint qualification and strong second order sufficient condition.At last,numerical results are reported to show the performance of the proposed method.

High-Order Soliton Solutions and Hybrid Behavior for the (2 + 1)-Dimensional Konopelchenko-Dubrovsky Equations

In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.

(3+1)维广义非线性发展方程的双线性Backlund变换与精确解

基于Hirota双线性方法和试探函数法,研究一个(3+1)维广义非线性发展方程的双线性Backlund变换和精确解问题。用Hirota双线性法,构造(3+1)维广义非线性发展方程的双线性形式和双线性Backlund变换。基于双线性形式和双线性Backlund变换,利用试探函数法与符号计算系统Mathematica,获得(3+1)维广义非线性发展方程的多种精确解,包括呼吸波解、复合型解、Lump周期解和孤子解,并分析解的相互作用情况。

(1+1)维的具有时间与空间色散的五阶方程

研究了一个新的(1+1)维的具有时空色散可积五阶方程,推广了方程的形式,并探讨了色散关系及其各自的系数情况.利用简化Hirota方法推导了方程的可积条件,导出了多孤子解.此外,利用指数型初始解构造方程的变换,得出双线性方程,继而利用Hirota双线性导数法导出单孤子解、双孤子解及N孤子解.结合双线性方程给出了不同双线性交换算子的双线性Bäcklund变换,继而导出不同条件下的双曲行波解、周期行波解等行波解.

A Modified Homogeneous Balance Method and Its Applications

A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance method.Generalized Boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour method.This approach is also applicable to a large variety of nonlinear evolution equations.

Soliton Solutions and Bilinear Bcklund Transformation for Generalized Nonlinear Schrdinger Equation with Radial Symmetry

Investigated in this paper is the generalized nonlinear Schrdinger equation with radial symmetry.Withthe help of symbolic computation, the one-, two-, and N-soliton solutions are obtained through the bilinear method.Bcklund transformation in the bilinear form is presented, through which a new solution is constructed.Graphically, wehave found that the solitons are symmetric about x=0, while the soliton pulse width and amplitude will change alongwith the distance and time during the propagation.

基于结构特征的地震剖面二维可视化方法

针对山地高陡复杂地质构造层位倾角较大,传统的可视化渲染方法破坏了地层的连续性,影响可视化效果。提出一种基于结构特征的地震剖面二维可视化方法。首先,采用邻域加权平均滤波法对地震勘探数据做预处理,提出正切值过滤法进行特征优化选择;其次,采用一种改进的相关算法来提取最具有明显结构特征的序列;最后,以结构特征序列为约束条件,进行二维渲染,输出地震剖面的二维可视化图像。实验结果表明,该方法结构特征明显,具有更好的连续性,提高了可视化的效果。

Trajectory equation of a lump before and after collision with other waves for generalized Hirota–Satsuma–Ito equation

Based on the Hirota bilinear and long wave limit methods,the hybrid solutions of m-lump with n-soliton and nbreather wave for generalized Hirota–Satsuma–Ito(GHSI)equation are constructed.Then,by approximating solutions of the GHSI equation along some parallel orbits at infinity,the trajectory equation of a lump wave before and after collisions with n-soliton and n-breather wave are studied,and the expressions of phase shift for lump wave before and after collisions are given.Furthermore,it is revealed that collisions between the lump wave and other waves are elastic,the corresponding collision diagrams are used to further explain.

Superposition formulas of multi-solution to a reduced(3+1)-dimensional nonlinear evolution equation

We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.

Soliton propagation for a coupled Schrödinger equation describing Rossby waves

We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.

Analytical three-periodic solutions of Korteweg-de Vries-type equations

Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for the first time.Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions,the soliton solution,the one-and the two-periodic solutions.Furthermore,it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.

Interaction solutions for the second extended(3+1)-dimensional Jimbo–Miwa equation

Based on the Hirota bilinear method,the second extended(3+1)-dimensional Jimbo–Miwa equation is established.By Maple symbolic calculation,lump and lump-kink soliton solutions are obtained.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions.Furthermore,periodiclump wave solution is derived via the ansatz including hyperbolic and trigonometric functions.Finally,3D plots,2D curves,density plots,and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.

Interaction solutions and localized waves to the(2+1)-dimensional Hirota-Satsuma-Ito equation with variable coefficient

This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.

Nondegenerate solitons of the(2+1)-dimensional coupled nonlinear Schrodinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

基于Hirota方法探求非零边界条件下MNLS/DNLS方程的孤子解

Hirota双线性导数变换处理非线性偏微分方程,是一种比反散射变换更为方便的直接方法。本文展示了Hirota双线性导数变换法应用于求解非线性可积方程的一般手续,以非零驻波边界条件下修正的非线性薛定谔(MNLS)方程为例,探求其孤子解;再通过简单的参数归零法直接得到导数非线性薛定谔(DNLS)方程在非零常数边界条件下的相应孤子解,亮/暗孤子解随时间和空间变量的演化也通过图像加以演示,所得孤子解与反散射方法得到的结果一致相符。

高速公路交通事故时空影响动态效应的传播分析

为探究并量化高速公路交通事故的时空传播机理,克服传统交通波理论在此类问题中考虑维度单一的缺陷,首先,基于微波雷达检测器采集的动态交通流数据,分析高速公路事故前后的速度变化态势;进而,引入速度变化率作为事故影响的度量指标,采用双线性插值法构建速度变化率轮廓图,并根据Savitzky-Golay滤波拟合法拟合事故影响时空区域的外围轮廓;最后,在速度变化率的阈值分别取20%、30%和40%时,计算并分析交通事故时空影响的各量化指标,包括事故影响开始时间、事故影响结束时间、事故影响持续时间、事故影响最近距离、事故影响最远距离、事故影响空间范围、事故影响传播速度以及事故影响消散速度。结果表明:速度变化率的阈值取值越小,则事故影响的开始时间越早,结束时间越晚,持续时间越长,且事故影响的距离越长;速度变化率阈值取值越大,则事故影响的开始时间越晚,结束时间越早,持续时间越短,且事故影响的距离越短;不同速度变化率阈值下的事故影响传播速度随时间的变化趋势有差异。文中方法操作性强、辨识度高,能够为高速公路事故发生后的实时交通管控和引导提供理论支撑。

Fuzzy Varying Coefficient Bilinear Regression of Yield Series

We construct a fuzzy varying coefficient bilinear regression model to deal with the interval financial data and then adopt the least-squares method based on symmetric fuzzy number space. Firstly, we propose a varying coefficient model on the basis of the fuzzy bilinear regression model. Secondly, we develop the least-squares method according to the complete distance between fuzzy numbers to estimate the coefficients and test the adaptability of the proposed model by means of generalized likelihood ratio test with SSE composite index. Finally, mean square errors and mean absolutely errors are employed to evaluate and compare the fitting of fuzzy auto regression, fuzzy bilinear regression and fuzzy varying coefficient bilinear regression models, and also the forecasting of three models. Empirical analysis turns out that the proposed model has good fitting and forecasting accuracy with regard to other regression models for the capital market.