Special types of the nonlinear dispersive Zakharov–Kuznetsov equation with compactons,solitons, and periodic solutions

In this article, we study nonlinear dispersive special types of the Zakharov–Kuznetsov equation with positive and negative exponents. The approach depends mainly on the sine–cosine algorithm. Compactons, solitary patterns, solitons, and periodic solutions are formally derived.     

Approximate periodic solutions for the Helmholtz-Duffing equation

Approximate periodic solutions for the Helmholtz-Duffing oscillator are obtained in this paper. He’s Energy Balance Method (HEBM) and He’s Frequency Amplitude Formulation (HFAF) are adopted as the solution methods. Oscillation natural frequencies are analytically analyzed. Error analysis is carried out and accuracy of the solution methods is evaluated.     

Spatially heterogeneous periodic solutions of the Hutchinson equation with distributed saturation

Asymptotics of spatially heterogeneous periodic solutions of the spatially distributed Hutchinson equation with periodic boundary conditions are obtained in cases of symmetric and asymmetric saturations. The numerical analysis of a simplified model shows the multistability, i.e., the coexistence of periodic solutions in the form of travelling waves, among which there are no solutions with the obtained asymptotics.     

On the periodic solutions of the Matrix Riccati equation

We investigate the structure of the periodic orbits of timeinvariant matrix Riccati equations. Matrix Riccati equations are of critical importance in control, estimation, differential games, scattering theory, and in several other applications. It is therefore important to understand the principal features of the phase portraits of Riccati equations, such as the existence and structure of periodic solutions.     

Stability of the simplest periodic solutions in the Stuart-Landau equation with large delay

In this paper, the stability of the simplest periodic solutions of a complex equation with large delay with cubic nonlinearity depending on the parameters is investigated. Sufficient conditions for the stability and instability of the periodic solutions are found. The geometry of the regions of stability and instability in the plane of parameters that define the main part of the solution is described.     

On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

We investigate a periodic version of the Benjamin-Ono (BO) equation associated with a discrete Laplacian. We find some special solutions to this equation, and calculate the values of the first two integrals of motion _I1$I_1$ and _I2$I_2$ corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald q$q$-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to q=0$q=0$ in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions _In$I_n$ (n=1,2,…$n=1\text{,}2\text{,}\dots$), and successfully identify it with the eigenvalues of Macdonald operators in the limit q→0$q\to 0$, i.e. the limit where Macdonald polynomials tend to the Hall–Littlewood polynomials.     

非对称NNV方程新的广义孤波解和周期解

指数函数法是求解非线性发展方程的一种简单有效的方法,利用该方法并借助Mathematica软件的符号计算功能求解了非对称NNV方程,得到了新的孤波解和周期解。     

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

Numerical mountain pass periodic solutions of a nonlinear spring equation

In this paper, we show how the introduction of a nonlinear term in the classic spring model can produce dramatic results. We compute a large amplitude solution which is drastically different from the known linear, small amplitude solution. A dual variational formulation is given, recasting the problem as one in which saddle points correspond to solutions of the differential equation. Our computations are based on the numerical mountain pass algorithm developed by Choi and McKenna which was inspired by the theorems of Ambrosetti, Rabinowitz and Ekeland.     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.     

New soliton-like solutions and compacton-like periodic wave solutions of parametric type to a nonlinear dispersive equation

In this paper, by using the integral bifurcation method, we study a nonlinear dispersive equation. Some new soliton-like solutions and some compacton-like periodic wave solutions are obtained. Their dynamic characters are investigated and the profiles are given by the mathematical software Maple. From the graphs of some soliton-like solutions, we find that their profiles are transformable.     

Interaction of solitons in a Boussinesq equation with dissipation

We investigate numerically an equation of Boussinesq type with square and cubic nonlinearity. In the model equation, dissipation is added and we investigate the physical properties of the modified problem. The technique applied here is the Christov spectral method in L ²(?∞, ∞). In previous works of the author, it was found that this technique was effective, accurate and computationally efficient for problems of this kind. Localized solutions are obtained numerically for the case of the moving frame which are used as initial conditions for the time-dependent problem. We investigate the propagation, head-on and overcome interaction of solitons. The issue of the phase shift is introduced and is been evaluated numerically.     

The difference periodic Ricati equation for the periodic predictionproblem

Gives a comprehensive treatment of several important aspects of the discrete-time periodic Riccati equation (DPRE) arising from the prediction problem for linear discrete-time periodic systems. The authors analyze the symmetric periodic positive semidefinite (SPPS) solution of the DPRE under appropriate assumptions of stabilizability and detectability of the periodic system. Among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the SPPS solution and the stability of the resulting closed-loop system. Some of these results can be seen as extensions of the corresponding results for the time-invariant case; however, a number of them contain contributions to the time-invariant case as well. The paper also gives a numerical algorithm based on an iterative linearization procedure for computing the SPPS solution. The algorithm is a periodic version of Kleinman's algorithm for the time-invariant case     

Exact periodic,cusp, solitary and loop wave solutions of the EX-ROE

The extended reduced Ostrovsky equation (EX-ROE) is investigated by using the dynamical system theory. The bifurcation phase portraits are drawn in different regions of parameter plane. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakon, solitary loop waves and periodic loop waves are obtained. The dynamic characters of these solutions are investigated.     

Numerical solution of the discrete-time periodic Riccati equation

In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error. A method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with     

Exact periodic cross-kink wave solutions and breather type of two-solitary wave solutions for the (3 + 1)-dimensional potential-YTSF equation

In this paper, the (3+1)-dimensional potential-YTSF equation is investigated. Exact solutions with three-wave form including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave solutions are obtained using Hirota’s bilinear form and generalized three-wave approach with the aid of symbolic computation. Moreover, the properties for some new solutions are shown with some figures.