Traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity

In this paper, we consider the existence of traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity. By relaxing the monotonicity of the upper solutions and allowing it greater than positive equilibrium point, we establish the existence of traveling wave solutions by means of Schauder's fixed point theorem. And then, we apply our results to delayed competition‐cooperation systems on higher dimensional lattices.     

Traveling wave solutions of nonlinear partial differential equations

We propose a simple algebraic method for generating classes of traveling wave solutions for a variety of partial differential equations of current interest in nonlinear science. This procedure applies equally well to equations which may or may not be integrable. We illustrate the method with two distinct classes of models, one with solutions including compactons in a class of models inspired by the Rosenau–Hyman, Rosenau–Pikovsky and Rosenau–Hyman–Staley equations, and the other with solutions including peakons in a system which generalizes the Camassa–Holm, Degasperis–Procesi and Dullin–Gotwald–Holm equations. In both cases, we obtain new classes of solutions not studied before.     

Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity. Based on two different mixed-quasi monotonicity reaction terms, we propose new conditions on the reaction terms and new definitions of upper and lower solutions. By using Schauder’s fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive diffusive Lotka-Volterra systems.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

Traveling wave solutions in a delayed lattice competition-cooperation system

In this paper, we first reduce the existence of traveling wave solutions in a delayed lattice competition-cooperation system to the existence of a pair of upper and lower solutions by means of Schauder’s fixed point theorem and the cross iteration scheme, and then we construct a pair of upper and lower solutions to obtain the existence and nonexistence of traveling wave solutions. We also consider the asymptotic behaviour of any nonnegative traveling wave solutions at negative infinity.     

Existence of traveling wave fronts of delayed lattice differential equations

Existence of traveling wave fronts for delayed lattice differential equations is established by Schauder fixed point theorem. The main result is applied to a delayed and discretely diffusive model for the population of Daphnia magna.     

Traveling wave fronts in a delayed lattice competitive system

This paper is concerned with the existence, asymptotic behavior, strict monotonicity, and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed lattice competitive system. We first prove the existence of traveling wave fronts by constructing upper and lower solutions and Schauder’s fixed point theorem, and then, for sufficiently small intraspecific competitive delays, prove that these traveling wave fronts decay exponentially at both infinities. Furthermore, for system without intraspecific competitive delays, the strict monotonicity and uniqueness of traveling wave fronts are established by means of the sliding method. In addition, we give the exact decay rate of the stronger competitor under some technique conditions by appealing to uniqueness.     

Global stability of traveling wave fronts for non-local delayed lattice differential equations

This paper is concerned with the global stability of traveling wave fronts of a non-local delayed lattice differential equation. By the comparison principle together with the semi-discrete Fourier transform, we prove that, all noncritical traveling wave fronts are globally stable in the form of t^−1/αe^−μt for some constants μ>0 and 0<α≤2, and the critical traveling wave fronts are globally stable in the algebraic form of t^−1/α.     

Traveling wave solutions in a delayed diffusive competition system

In this paper we first investigate the existence of traveling wave fronts in a delayed diffusive competition system by constructing a pair of upper and lower solutions. Then we consider the asymptotic behavior of traveling wave solutions at the minus/plus infinity by means of the bilateral Laplace transform. Finally, the monotonicity and uniqueness (up to the translation) of traveling wave solutions are also obtained by the strong comparison principle and the sliding method.     

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.     

Exact travelling wave solutions for a class of nonlinear partial differential equations

In this paper, the tanh-method is improved by means of a proper rational transformation based upon a coupled projective Riccati equations. The ansatz can be applied to find more and new exact solutions of the partial differential equations with the aid of symbolic computation system, Maple. We choose an example, which includes φ⁴ equation, Klein–Gordon equation, Duffing equation, Landau–Ginburg–Higgs equation and Sine–Gordon equation, to illustrate the method.     

Functional differential equations with certain monotonicity properties

The present paper is concerned with the study of retarded differential equations in n-space, which satisfy some monotonicity properties. Sufficient conditions are given guaranteeing that solutions are contained in the probability (n – 1)-simplex. Existence and uniqueness of constant solutions are proved. It is also shown that every solution converges to a constant as t → ∞.     

Traveling wave solutions of the n-dimensional coupled Yukawa equations

We discuss traveling wave solutions to the Yukawa equations, a system of nonlinear partial differential equations which has applications to meson–nucleon interactions. The Yukawa equations are converted to a six-dimensional dynamical system, which is then studied for various values of the wave speed and mass parameter. The stability of the solutions is discussed, and the methods of competitive modes is used to describe parameter regimes for which chaotic behaviors may appear. Numerical solutions are employed to better demonstrate the dependence of traveling wave solutions on the physical parameters in the Yukawa model. We find a variety of interesting behaviors in the system, a few of which we demonstrate graphically, which depend upon the relative strength of the mass parameter to the wave speed as well as the initial data.     

Traveling wave solutions of the generalized nonlinear evolution equations

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.     

Periodic solutions of hyperbolic partial differential equations

Summary Starting with the problem of finding a mass-distribution on a sphere, admitting in each point P a density, function of the point P and of the mass contained in a certain domain depending on P, the author proves the existence and the uniqueness of the solution of (5) under assumption (1), (2), (3). As a generalization, he also studies hyperbolic partial differential equations having solutions periodic in one of the variables with period depending on the others. To Prof. BeniaminoSegre on the occasion of his 70-th birthday. Entrata in Redazione il 24 marzo 1973.     

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite