Developing interaction potential for H (2H) → Cu(1 1 1) interaction system: A numerical study

In this study, we have used London–Eyring–Polanyi–Sato (LEPS) functional form as an interaction potential energy function to simulate H (2H) → Cu(1 1 1) interaction system. The parameters of the LEPS function are determined in order to analyze reaction dynamics via molecular dynamics computer simulations of the Cu(1 1 1) surface and H/(2H) system. Nonlinear least-squares method is used to find the LEPS parameters. For this purpose, we use the energy points which were calculated by a density-functional theory method with the generalized gradient approximation including exchange-correlation energy for various configurations of one and two hydrogen atoms on the Cu(1 1 1) surface. After the fitting procedures, two different parameters sets are obtained that the calculated root-mean-square values are close to each other. Using these sets, contour plots of the potential energy surfaces are analyzed for H → Cu(1 1 1) and 2H → Cu(1 1 1) interactions systems. In addition, sticking, penetration, and scattering sites on the surface are analyzed by using these sets.     

1 : 3-Resonance in a Hopf–Hopf bifurcation

We investigate the bifurcating solutions at a Hopf–Hopf interaction point with an internal 1 : 3 resonance. It turns out, that the transitions from single to mixed modes can be described by Duffing or Mathieu szenarios. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizations of PG(n − 1, q)\PG(k − 1, q) by Numerical and Polynomial Invariants

We show that the simple matroid PG(n − 1, q)\PG(k − 1, q), for n ≥ 4 and 1 ≤ k ≤ n − 2, is characterized by a variety of numerical and polynomial invariants. In particular, any matroid that has the same Tutte polynomial as PG(n − 1, q)\PG(k − 1, q) is isomorphic to PG(n − 1, q)\PG(k − 1, q).     

Three-dimensional modulation of electron-acoustic waves: 3 + 1 Davey–Stewartson system

In this paper we consider the three-dimensional modulation of an electron-acoustic wave by means of a multiple-scale perturbation method. We find that the governing equation for the amplitude that describes the asymptotic properties of the wave is a Davey–Stewartson system in three-space variables. We use the system to study the linear stability of the modulation wave and we compare our results with preceding studies by Kourakis and Shukla (2004) [8] on the oblique modulation problem based on a non-linear Schrödinger equation.     

1-Soliton solution of 1 + 2 dimensional nonlinear Schrödinger’s equation in power law media

This paper obtains the 1-soliton solution of the nonlinear Schrödinger’s equation in 1 + 2 dimensions for parabolic law nonlinearity. An exact soliton solution is obtained in closed form by the solitary wave ansatze.     

Explicit exact solutions for (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, we construct explicit exact solutions for the coupled Boiti–Leon–Pempinelli equation (BLP equation) by using a extended tanh method and symbolic computation system Mathematica. By means of the method, many new exact travelling wave solutions for the BLP system are successfully obtained. the extended tanh method can be applied to other higher-dimensional coupled nonlinear evolution equations in mathematical physics.     

Analytic integrability of the Bianchi class A cosmological models with 0 ⩽ k < 1

Many works study the integrability of the Bianchi class A cosmologies with k = 1, where k is the ratio between the pressure and the energy density of the matter. Here we characterize the analytic integrability of the Bianchi class A cosmological models when 0 ⩽ k < 1. We conclude that Bianchi types VI₀, VII₀, VIII and IX can exhibit chaos whereas Bianchi type I is not chaotic and Bianchi type II is at most partially chaotic.     

Pfaffian solutions and extended Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation

Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3 + 1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3 + 1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation.     

Links between some (2 + 1)-dimensional nonlinear evolution equations and Painlevé-II equations

Using the idea of transformation, some links between (2 + 1)-dimensional nonlinear evolution equations and the ordinary differential equations Painlevé-II equations has been illustrated. The Kadomtsev–Petviashvili (KP) equation, generalized (2 + 1)-dimensional break soliton equation and (2 + 1)-dimensional Boussinesq equation are researched. As a result, some new interesting results about these (2 + 1)-dimensional PDEs have been obtained, such as the exact solutions with arbitrary functions, rich rational solutions and the nontrivial Bäcklund transformations have been derived.     

A non-variational approach to the construction of new ‘higher-order’ conservation laws of the family of nonlinear equations α(ut + 3uux) + β(utxx + 2uxuxx + uuxxx) − γuxxx = 0

In this paper, we study and classify the conservation laws of the combined nonlinear KdV, Camassa–Holm, Hunter–Saxton and the inviscid Burgers equation which arises in, inter alia, shallow water equations. It is shown that these can be obtained by variational methods but the main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry generators and ‘multipliers’, particularly, the higher-order ones.     

On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2)

This paper is motivated from some recent papers treating the impulsive Cauchy problems for some differential equations with fractional order q ∈ (1, 2). A better definition of solution for impulsive fractional differential equation is given. We build up an effective way to find natural solution for such problems. Then sufficient conditions for existence of the solutions are established by applying fixed point methods. Four examples are given to illustrate the results.     

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

A new approach for solutions of the (2 + 1)-dimensional cubic nonlinear Schrödinger equation

A new method to solve the nonlinear evolution equations is presented, which combines the two kind methods – the tanh function method and symmetry group method. To demonstrate the method, we consider the (2 + 1)-dimensional cubic nonlinear Schrödinger (NLS) equation. As a result, some novel solitary solutions of the Schrödinger equation are obtained. And graphs of some solutions are displayed.     

New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation

The repeated homogeneous balance method is used to construct new exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain soliton-like and periodic-like solutions. This method is straightforward and concise, and it can be also applied to other nonlinear evolution equations.     

New (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources derived by the nonlinear variable separation method

By using the Hirota’s bilinear transformation method and direct variable separation assumption, a new (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources is derived for the first time. Correspondingly, a nonlinear variable separation solution included two lower-dimensional arbitrary functions is obtained.     

New soliton and periodic solutions of (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

With the aid of symbolic computation, the new generalized algebraic method is extended to the (1 + 2)-dimensional nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity for constructing a series of new exact solutions. Because of the dual-power law nonlinearity, the equation cannot be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we reduce the NLSE to an ordinary differential equation that is easy to solve and find a rich variety of new exact solutions for the equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions and rational function solutions. Numerical simulations are given for a solitary wave solution to illustrate the time evolution of the solitary creation. Finally, conditional stability of the solution in Lyapunov’s sense is discussed.     

Differential form method for finding symmetries of a (2 + 1)-dimensional Camassa–Holm system based on its Lax pair

In this paper, we use the differential form method to seek Lie point symmetries of a (2 + 1)-dimensional Camassa–Holm (CH) system based on its Lax pair. Then we reduce both the system and its Lax pair with the obtained symmetries, as a result some reduced (1 + 1)-dimensional equations with their new Lax pairs are presented. At last, the conservation laws for the CH system are derived from a direct method.