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.     

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.     

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the (G′/G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.     

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.     

Bifurcations of travelling wave solutions in the (N + 1)-dimensional sine–cosine-Gordon equations

In this paper, the (N + 1)-dimensional sine–cosine-Gordon equations are studied. The existence of solitary wave, kink and anti-kink wave, and periodic wave solutions are proved, by using the method of bifurcation theory of dynamical systems. All possible bounded exact explicit parametric representations of the above travelling solutions are obtained.     

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.     

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.     

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.     

Generalized dromions of the (2+1)-dimensional nonlinear Schrdinger equations

IntroductionMost of the mathematical work in the realm of nonlinear phenomena refers to integrablenonlinear equation and their exact soluted. The existence of more generalized locajized solutions for the (2+l)-dimensional KdV ~boil] and the (2+1)-dimensional breaking solitonequationl'] apart from the basic dro~ ~.sl3'41 has given an impetus to search for amore general class of localized sol~ in Oafs (2+1)-dimensional nonlinear evolution equations. Recelltlyt stating from the symmeq constraint…     

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.     

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

We demonstrate that four solutions from 13 of the (3 + 1)-dimensional Kadomtsev–Petviashvili equation obtained by Khalfallah [1] are wrong and do not satisfy the equation. The other nine exact solutions are the same and all “new” solutions by Khalfallah can be found from the well known solution.     

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.     

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.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Cesàro matrix and (1, 1; r)-convex sequences

Positivity - This paper deals with (1, 1; r)-convexity of sequences. First, we prove several results on the sets of (1, 1; r)-convex sequences for various values...     

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.     

New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics

In this article, new extension of the generalized and improved (G′/G)-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV) equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.     

Applications of some transformations for several variable-coefficient nonlinear evolution equations from plasma physics,arterial mechanics,nonlinear optics and Bose–Einstein condensates

Variable-coefficient nonlinear evolution equations have occurred in such fields as plasma physics, arterial mechanics, nonlinear optics and Bose–Einstein condensates. This paper is devoted to giving some transformations to convert the original nonlinear evolution equations, e.g., the variable-coefficient nonlinear Schrödinger, generalized Gardner and variable-coefficient Sawada–Kotera equations to simpler ones or even constant-coefficient ones. Based on some constraints, we simplify the original equations and derive the associated chirp solitons, Lax pairs, and Bäcklund transformations from the original equations by means of the aforementioned transformations.     

Multiple soliton solutions for some (3+1 )-dimensional nonlinear models generated by the Jaulent–Miodek hierarchy

In this work we study four (3+1)-dimensional nonlinear evolution equations, generated by the Jaulent–Miodek hierarchy. We derive multiple soliton solutions for each equation by using the Hereman–Nuseir form, a simplified form of the Hirota’s method. The obtained soliton solutions are characterized by distinct phase shifts.     

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.