Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order $n$ for any positive integer $n,$ and we call these solutions, solutions of the order $n.$ Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order 10.

The Benjamin-Bona-Mahony (BBM) equation represents the unidirectional propagation of nonlinear dispersive long waves, which has a clear physical background, and is a more suitable mathematical and physical equation than the KdV equation. Therefore, the research on the BBM equation is very important. In this article, we put forward an effective algorithm, the modified hyperbolic function expanding method, to build the solutions of the BBM equation. We, by utilizing the modified hyperbolic function expanding method, obtain the traveling wave solutions of the BBM equation. When the parameters are taken as special values, the solitary waves are also derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The modified hyperbolic function expanding method is direct, concise, elementary and effective, and can be used for many other nonlinear partial differential equations.

In this paper, we study the local convergence of a three-step Newton-type method for solving nonlinear equations in Banach spaces under weaker hypothesis. More precisely, we derive the existence and uniqueness theorems, when the first-order derivative of nonlinear operator satisfies the $L$-average conditions instead of the usual Lipschitz condition, which have been discussed in the earlier study.

In this paper, based on the one-signed Green’s function and the compact results on the infinite interval, we obtain the existence and multiplicity of positive solutions for the boundary value problems $$\begin{cases} \Delta^2 x(n-1)-p(n)\Delta x(n-1)-q(n)x(n-1)+f(n,x(n))=0, &n\in\mathbb{N},\\  \alpha x(0)-\beta \Delta x(0)=0, & \lim\limits_{n\rightarrow\infty}x(n)=0  \end{cases}$$by the fixed point theorem in cones. The main results extend some results in the previous literature.

In this paper, we consider a stochastic spruce budworm differential model with time delay. Based on the nonnegative initial conditions, the existence and uniqueness of the global positive solution are easily found. Then, we obtain the ultimate boundedness of solution in mean under the same conditions. Furthermore, we verify that the sample Lyapunov exponent of solution is less than a positive constant. Finally, numerical examples are presented to show the consistency of the theoretical results.

In this paper, we investigate the existence of triple positive solutions of boundary value problems for high-order fractional differential equation at resonance with singularities by using the fixed point index theory and the Leggett-Williams theorem. The spectral theory and some new height functions are also employed to establish the existence of triple positive solutions. The nonlinearity involved is arbitrary fractional derivative, and permits singularity.

The paper studies the existence and uniqueness for impulsive fractional $q_k$-difference equations of initial value problems involving Riemann-Liouville fractional $q_k$-integral and $q_k$-derivative by defining a new $q$-shifting operator. In this paper, we obtain existence and uniqueness results for impulsive fractional $q_k$-difference equations of initial value problems by using the Schaefer’s fixed point theorem and Banach contraction mapping principle. In addition, the main result is illustrated with the aid of several examples.

In this paper, we consider Poincaré bifurcation from an elliptic Hamiltonian of degree four with two-saddle cycle. Based on the Chebyshev criterion, not only one case in the Liénard equations of type (3, 2) is discussed again in a different way from the previous ones, but also its two extended cases are investigated, where the perturbations are given respectively by adding $εy(d_0 + d_2v^{2n} )\frac{∂}{∂y}$ with $n ∈ \mathbb{N}^ +$ and $εy(d_0 + d_4v^4 + d_2v^{2n+4})\frac{∂}{∂y}$ with $n = −1$ or $n ∈ \mathbb{N}^+,$ for small $ε > 0.$ For the above cases, we obtain all the sharp upper bound of the number of zeros for Abelian integrals, from which the existence of limit cycles at most via the first-order Melnikov functions is determined. Finally, one example of double limit cycles for the latter case is given.

We investigate the flocking conditions of a group coupling system with time delays, in which the communication between particles includes inter-group and intra-group interactions, and the time delay comes from the theory of moving object observation. As an effective model, we introduce a system of nonlinear functional differential equations to describe its dynamic evolution mechanism. By constructing two differential inequalities on velocity and velocity fluctuation from a continuity argument, and using the Lyapunov functional approach, we present some sufficient conditions for the existence of asymptotic flocking solutions to the coupling system, in which an upper bound of the delay allowed by the system is quantitatively given to ensure the emergence of flocking behavior. All results are novel and can be illustrated by using some specific numerical simulations.

We study the set of periods of the homogeneous polynomial maps $f:\mathbb{R}^n → \mathbb{R}^n$ and $f : \mathbb{C}^n → \mathbb{C}^n$ of degree $m > 1.$ For these complex maps, we also describe the number of invariant straight lines through the origin by $f^k$ for $k = 1, 2, . . .$ and the dynamics of $f^k$ over them.

A stochastic two-prey-one-predator model with Beddington-DeAngelis functional response and Lévy jumps is proposed and investigated in this paper. First of all, we prove the existence and uniqueness of the global positive solution, and stochastic ultimate boundedness of the solution. Next, under a simple assumption, by using Itô formula and other important inequalities, some sufficient conditions are established to ensure the extinction and persistence in the mean of the system. The results show that neither strong white noise nor Lévy noise is conducive to the persistence of the population. Finally, the theoretical results are verified by numerical simulations.

In this paper, we investigate a rational difference equation of higher order for the existence of nonoscillatory solutions. To prove our main results, we use an inclusion theorem stated and proved in [4]. In this way, we give an answer of an open problem formulated in [3].

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.