具有强吸收的非Newtonian扩散方程的行波解及交界面

本文我们首先找到了有限行波解,然后研究了具有强吸收项的非Newtonian扩散方程交界面的短时间行为. 我们将表明交界面的短时间行为依赖于初始浓度, 准确的说,找到了初始浓度的一个临界值,用它可区分交界面是热前面还是冷前面.     

A note on travelling-wave solutions to Lax’s seventh-order KdV equation

Ganji and Abdollahzadeh [D.D. Ganji, M. Abdollahzadeh, Appl. Math. Comput. 206 (2008) 438–444] derived three supposedly new travelling-wave solutions to Lax’s seventh-order KdV equation. Each solution was obtained by a different method. It is shown that any two of the solutions may be obtained trivially from the remaining solution. Furthermore it is noted that one of the solutions has been known for many years.     

A Mickens-type monotone discretization for bounded travelling-wave solutions of a Burgers–Fisher partial differential equationa*

We present a nonlinear method to approximate solutions of a Burgers–Huxley equation with generalized advection factor and logistic reaction. The equation under investigation possesses travelling-wave solutions that are temporally and spatially monotone functions; the travelling-wave fronts considered are bounded and connect asymptotically the stationary solutions of the model. For the linear regime, the method is consistent of first order in time and second order in space. In the nonlinear scenario, we investigate conditions under which bounded initial profiles evolve into bounded new approximations. The main results report on parametric conditions that guarantee the boundedness, the positivity and the monotonicity preservation of the method. As a consequence, our recursive method is capable of preserving the temporal and the spatial monotonicity of the solutions. We provide simulations that show that, indeed, our technique preserves the positivity, the boundedness and the temporal and spatial monotonicity of solutions.     

Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest.     

Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model

We study the global existence of weak solutions to a reduced gravity two-and-a-half layer model appearing in oceanic fluid dynamics in two-dimensional torus. Based on Faedo–Galerkin method and weak convergence method, we construct the global weak solutions which are renormalized in velocity variable, where the technique of renormalized solutions was introduced by Lacroix-Violet and Vasseur (2018). Besides, we prove that the renormalized solutions are weak solutions, which satisfy the basic energy inequality and Bresch–Desjardins entropy inequality, but not the Mellet–Vasseur type inequality. In the proof, we use the reduced gravity two-and-a-half layer model with drag forces and capillary term as approximate system. It should be pointed out that only when the capillary term vanishes, we prove the existence of renormalized solution to the approximation system, which is different from Lacroix-Violet and Vasseur (2018) with the quantum potential.     

A study on a two‐wave mode Kadomtsev–Petviashvili equation: conditions for multiple soliton solutions to exist

We establish a two‐wave mode equation for the integrable Kadomtsev–Petviashvili equation, which describes the propagation of two different wave modes in the same direction simultaneously. We determine the necessary conditions that make multiple soliton solutions exist for this new equation. The simplified Hirota's method will be used to conduct this work. We also use other techniques to obtain other set of periodic and singular solutions for the two‐mode Kadomtsev‐Petviashvili equation. Copyright © 2017 John Wiley & Sons, Ltd.     

Symmetries and similarity solutions for the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane

We perform a complete analysis of all the Lie point symmetries admitted by the equation describing the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane. We then investigate the existence of group-invariant solutions for particular values of the power-law parameter β.     

Some new and general solutions to the compound KdV-Burgers system with nonlinear terms of any order

In this letter, a new Riccati equation expansion method is presented for constructing exact travelling-wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the Riccati equation to construct exact travelling-wave solutions of nonlinear partial differential equations. As a result, some more generalized solutions, which contain triangular periodic solutions, exp function solutions and the soliton-like solutions, are obtained.     

Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups.     

Travelling waves in a reaction-diffusion system modelling fungal mycelia

We discuss travelling-wave solutions of a system of coupledreaction—diffusion equations used by the authors to describethe macroscopic behaviour of fungal mycelia. Such systems havebeen used in a multitude of applications; and, in particular,Merkin and Needham (1990, Proc. R. Soc. Lond. A 430, 315–45)have studied a certain formulation as a model for generic isothermalchemical reactions. We show that an alternative analysis providesmore complete results in ascertaining the conditions under whichtravelling-wave solutions exist and that it allows a wider rangeof parameter values to be considered; this is essential to theapplication considered in the present case. Numerical investigationsof travelling-wave solutions and the related initial-value problemare included to motivate and extend the analysis.     

具有两个二次代数曲线解的三次系统的代数极限环

本文证明了具有椭圆和抛物线解的三次系统可以存在代数极限环，纠正了文[4]的主要结果.     

A note on “New travelling wave solutions to the Ostrovsky equation”

In a recent paper by Ya?ar [E. Ya?ar, New travelling wave solutions to the Ostrovsky equation, Appl. Math. Comput. 216 (2010), 3191-3194], ‘new’ travelling-wave solutions to the transformed reduced Ostrovsky equation are presented. In this note it is shown that some of these solutions are disguised versions of known solutions.     

Multi-solitonic solutions for the variable-coefficient variant Boussinesq model of the nonlinear water waves

For the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water, we consider a variable-coefficient variant Boussinesq (vcvB) model with symbolic computation. We construct the connection between the vcvB model and a variable-coefficient Ablowitz-Kaup-Newell-Segur (vcAKNS) system under certain constraints. Using the N-fold Darboux transformation of the vcAKNS system, we present two sets of multi-solitonic solutions for the vcvB model, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Dynamics of those solutions are analyzed and graphically discussed, such as the parallel solitonic waves, shape-changing collision, head-on collision, fusion-fission behavior and elastic-fusion coupled interaction.     

Monotone iterative method of upper and lower solutions applied to a multilayer combustion model in porous media

This paper presents a new nonlinear reaction–diffusion–convection system coupled with a system of ordinary differential equations that models a combustion front in a multilayer porous medium. The model includes heat transfer between the layers and heat loss to the external environment. A few assumptions are made to simplify the model, such as incompressibility; then, the unknowns are determined to be the temperature and fuel concentration in each layer. When the fuel concentration in each layer is a known function, we prove the existence and uniqueness of a classical solution for the initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. We construct monotone iterations of upper and lower solutions and prove that these iterations converge to a unique solution for the problem, first locally and then, in time, globally.     

NONEXISTENCE OF PSEUDO-SELF-SIMILAR SOLUTIONS TO INCOMPRESSIBLE EULER EQUATIONS

In this paper we study a generalization of self-similar solutions. We show that just as for the solutions to the Navier-Stokes equations these supposedly singular solution reduce to the zero solution.In this paper we study a generalization of self-similar solutions. We show that just as for the solutions to the Navier-Stokes equations these supposedly singular solution reduce to the zero solution.     

Traveling wave solutions of competitive–cooperative Lotka–Volterra systems of three species

This paper is concerned with the existence of traveling front solutions for competitive–cooperative Lotka–Volterra systems of three species. By converting the system into a monotone system, we show that under certain assumptions on the parameters appearing in the system, traveling front solutions exist. Also, exact traveling front solutions, which are polynomials in the hyperbolic tangent function, are given explicitly in certain parameter regimes.     

Stratified laminar flows in a circular pipe: New analytical solutions in terms of elementary functions

Laminar stratified two-phase flows in circular pipes are studied. Under the assumption that the wall-liquid wettability properties of the phases are identical, new exact analytical solutions in terms of elementary functions are constructed. The solutions satisfy the Navier–Stokes equations exactly in fluids and the boundary conditions on the pipe walls and the interface for the two cases: the pipe is horizontal and the capillarity forces dominate the gravity ones, the pipe is inclined and the volumetric quantities of liquids in the pipe are the same. For the second case, the capillarity and gravity forces can be arbitrary, but if the gravity forces dominate the capillarity ones, the assumption about the equal wall-liquid tensions of the phases can be withdrawn.     

Painlevé analysis,lump-kink solutions and localized excitation solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated. The integrability test is performed yielding a positive result. Through the Painlevé–Bäcklund transformation, we derive four types of lump-kink solutions composed of two quadratic functions and N exponential functions. It is shown that fission and fusion interactions occur in the lump-kink solutions. Furthermore, a new variable separation solution with two arbitrary functions is obtained, the localized excitations including lumps, dromions and periodic waves are analyzed by some graphs.     

Multiplicity of supercritical fronts for reaction–diffusion equations in cylinders

We study multiplicity of the supercritical traveling front solutions for scalar reaction–diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.     

Solitary-wave solutions to a dual equation of the Kaup–Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.