Helical solitons in vector modified Korteweg–de Vries equations

We study existence of helical solitons in the vector modified Korteweg–de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.     

Periodic Solutions to KdV-Burgers-Kuramoto Equation

In this paper, a new special ansatz solution, where elliptic equation satisfied by elliptic functions is taken as an intermediate transformation, is applied to solve the KdV-Burgers-Kuramoto equation, and many more new periodic solutions are obtained, including solutions expressed in terms of Jacobi elliptic functions, solution expressed in terms of Weierstrass elliptic function.     

Integrable discrete static Hamiltonian with a parametric double-well potential

A one-dimensional discrete conservative Hamiltonian with a generalized form of the Schmidt potential, is constructed with the help of a non-integrable discrete Hamiltonian whose parametrized double-well potential can be reduced to the ?⁴ potential. The new conservative Hamiltonian is completely integrable in the discrete static regime, and the associate exact nonlinear solution is shown to coincide with the continuum nonlinear periodic solution of the non-integrable Hamiltonian. Numerical simulations and nonlinear stability analysis suggest that the discrete mapping derived from the completely integrable Hamiltonian undergoes a bifurcation which does not leads to the chaotic phase with randomly pinned states, but instead to a phase where real solutions become rare forming a cluster of periodic points around an elliptic fixed point.     

构造孤子方程的Weierstrass椭圆函数解的一个新方法

利用具有Weierstrass椭圆函数解的方程，首先获得了投影Riccati方程的两组新解.由于投影Riccati方程可用于多种具孤子解的非线性演化方程的求解，因而得到了一个可以构造这些方程的Weierstrass椭圆函数解的新方法. 关键词： Weierstrass椭圆函数解 投影Riccati方程 非线性演化方程     

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff(CBS) equation. This model describes the(2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation(ESE) method is applied to the model, and a variety of novel solitarywave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration(VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme's performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.     

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.     

Symbolic Computation of Extended Jacobian Elliptic Function Algorithm for Nonlinear Differential-Different Equations

The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schr ödinger equation. When the modulous m→1 or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.     

变系数广义KdV方程新的类孤波解和精确解

用普通KdV方程作变换，构造变系数广义KdV方程的解，获得了变系数广义KdV方程新的Jacobi椭圆函数精确解、类孤波解、三角函数解和Weierstrass椭圆函数解. 关键词： KdV方程 变系数广义KdV方程 类孤波解 精确解     

The Toy Top,an Integrable System of Rigid Body Dynamics

Abstract

A toy top is defined as a rotationally symmetric body moving in a constant gravitational field while one point on the symmetry axis is constrained to stay in a horizontal plane. It is an integrable system similar to the Lagrange top. Euler-Poisson equations are derived. Following Felix Klein, the special unitary group SU(2) is used as configuration space and the solution is given in terms of hyperelliptic integrals. The curve traced by the point moving in the horizontal plane is analyzed, and a qualitative classification is achieved. The cases in which the hyperelliptic integrals degenerate to elliptic ones are found and the corresponding solutions are given in terms of Weierstrass elliptic functions.     

Evaluating the robustness of rogue waves under perturbations

Rogue waves, and their periodic counterparts, have been shown to exist in a number of integrable models. However, relatively little is known about the existence of these objects in models where an exact formula is unattainable. In this work, we develop a novel numerical perspective towards identifying such states as localized solutions in space-time. Importantly, we illustrate this methodology for different perturbations of nonlinear Schrödinger models. In particular, in addition to benchmarking known solutions (and confirming their numerical propagation under controllable error) this enables the continuation of such solutions over parametric variations to non-integrable models. As a result, we can answer in the positive the question about the parametric robustness of Peregrine-like waveforms and even of generalizations thereof on a cnoidal wave background.     

Time Dependent Ginzburg-Landau方程的Weierstrass椭圆函数解

将文[22]中提出的求解非线性演化方程的Weierstrass椭圆函数解的一个新方法应用于Time Dependent Ginzburg-Landau方程,获得了该方程的一些新的双周期解,并在退化情形下得到了一些新的精确孤波解.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

A Generalized Method and Exact Solutions in Bose-Einstein Condensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schrfdinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function＇s soliton solutions, but also Jacobi elliptic function solutions and Weierstra.ss elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient.     

Approximate Bound States Solution of the Hellmann Potential

The Hellmann potential, which is a superposition of an attractive Coulomb potential -a/r and a Yukawa potential b e^-δr/r, is often used to compute bound-state normalizations and energy levels of neutral atoms. By using the generalized parametric Nikiforov-Uvarov (NU) method, we have obtained the approximate analytical solutions of the radial Schrödinger equation (SE) for the Hellmann potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented, which show good agreement with a numerical amplitude phase method and also those previously obtained by other methods. As a particular case, we find the energy levels of the pure Coulomb potential.     

Weierstrass Elliptic Function Solutions to Nonlinear Evolution Equations

This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation. Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.     

On the use of two classical series expansion methods to determine the vibration of harmonically excited pure cubic oscillators

An analytical approach to determine the steady-state response of a damped and undamped harmonically excited oscillator with no linear term and with cubic non-linearity is presented. The governing equation is transformed into a form suitable for the application of a classical series expansion technique. The Linstedt–Poincaré method and the method of multiple scales are then used to determine the amplitude-frequency response and approximate solution for the response at the excitation frequency. The results obtained are compared with numerical solutions and analytical solutions found in the literature for the case when there is strong non-linearity.     

(3+1)维 KdV 方程新的精确解和孤子解

用普通Korteweg-de Vries(KdV)方程作变换,构造(3 1)维KdV方程的解,获得了新的孤子解、Jaoobi椭圆函数解、三角函数解和Weierstrass椭圆函数解.     

A Generalized Method and Exact Solutions in Bose-Einstein Condensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schr(o)dinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function's soliton solutions, but also Jacobi elliptic function solutions and Weierstrass elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

New Weierstrass Semi-rational Expansion Method to Doubly Periodic Solutions of Soliton Equations

Based on the Weierstrass elliptic function equation, a new Weierstrass semi-rational expansion method and its algorithm are presented. The main idea of the method changes the problem solving soliton equations into another one solving the corresponding set of nonlinear algebraic equations. With the aid of Maple, we choose the modified KdV equation, (2+1)-dimensional KP equation, and (3+1)-dimensional Jimbo-Miwa equation to illustrate our algorithm. As a consequence, many types of new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover the corresponding new Jacobi elliptic function solutions and solitary wave solutions are also presented as simple limits of doubly periodic solutions.