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.     

Continuous cocycles on locally compact groups: II

In this second paper we continue our development of elementary methods for computing continuous complex-valued solutions of the 2-cocycle functional equation on locally compact groups. The first paper dealt primarily with solvable groups. Here we consider some classical motion groups of mathematical physics, including the Euclidean, Galilean, Lorentz and Poincaré groups.     

Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation

The behavior of solutions to the biharmonic equation is well-understood in smooth domains. In the past two decades substantial progress has also been made for the polyhedral domains and domains with Lipschitz boundaries. However, very little is known about higher order elliptic equations in the general setting. In this paper we introduce new integral identities that allow to investigate the solutions to the biharmonic equation in an arbitrary domain. We establish: (1) boundedness of the gradient of a solution in any three-dimensional domain; (2) pointwise estimates on the derivatives of the biharmonic Green function; (3) Wiener-type necessary and sufficient conditions for continuity of the gradient of a solution. Mathematics Subject Classification (2000)  35J40, 35J30, 35B65     

Doubly periodic wave and folded solitary wave solutions for (2 + 1)-dimensional higher-order Broer–Kaup equation

A general solution including three arbitrary functions is obtained for the (2 + 1)-dimensional high-order Broer–Kaup equation by means of WTC truncation method. From the general solution, doubly periodic wave solutions in terms of the Jacobian elliptic functions with different modulus and folded solitary wave solutions determined by appropriate multiple valued functions are obtained. Some interesting novel features and interaction properties of these exact solutions and coherent localized structures are revealed.     

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.     

Trilinearization and localized coherent structures and periodic solutions for the (2 + 1) dimensional K-dV and NNV equations

In this paper, using a novel approach involving the truncated Laurent expansion in the Painlevé analysis of the (2 + 1) dimensional K-dV equation, we have trilinearized the evolution equation and obtained rather general classes of solutions in terms of arbitrary functions. The highlight of this method is that it allows us to construct generalized periodic structures corresponding to different manifolds in terms of Jacobian elliptic functions, and the exponentially decaying dromions turn out to be special cases of these solutions. We have also constructed multi-elliptic function solutions and multi-dromions and analysed their interactions. The analysis is also extended to the case of generalized Nizhnik–Novikov–Veselov (NNV) equation, which is also trilinearized and general class of solutions obtained.     

A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.     

Several classes of exact solutions to the 1 + 1 Born–Infeld equation

We obtain closed-form exact solutions to the 1 + 1 Born–Infeld equation arising in nonlinear electrodynamics. In particular, we obtain general traveling wave solutions of one wave variable, solutions of two wave variables, similarity solutions, multiplicatively separable solutions, and additively separable solutions. Then, putting the Born–Infeld model into correspondence with the minimal surface equation using a Wick rotation, we are able to construct complex helicoid solutions, transformed catenoid solutions, and complex analogues of Scherk’s first and second surfaces. Some of the obtained solutions are new, whereas others are generalizations of solutions in the literature. These exact solutions demonstrate the fact that solutions to the Born–Infeld model can exhibit a variety of behaviors. Exploiting the integrability of the Born–Infeld equation, the solutions are constructed elegantly, without the need for complicated analytical algorithms.     

On shifted periodic solutions of two nonlinear equations

Using the linear superposition approach, we find periodic solutions with shifted periods and velocities of the (2 + 1)-dimensional modified Zakharov–Kuznetsov equation and the (3 + 1)-dimensional Kadomtsev–Petviashvili equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure of generating solutions of nonlinear evolution equations is successful as a consequence of some cyclic identities satisfied by the Jacobi elliptic functions which reduce by 2 (or a larger even number) the degree of cyclic homogeneous polynomials in Jacobi elliptic functions.     

A systematic method for solving differential-difference equations

A systematic method for searching travelling-wave solutions to differential-difference equations (DDEs) is proposed in the paper. First of all, we introduce Bäcklund transformations for the standard Riccati equation which generate new exact solutions by using its simple and known solutions. Then we introduce a kind of formal polynomial solutions to DDEs and further determine the explicit forms by applying the balance principle. Finally, we work out exact solutions of the DDEs via substituting the form solutions and solving over-determined algebraic equations with the help of Maple. As illustrative examples, we obtain the travelling-wave solutions of the (2 + 1)-dimensional Toda lattice equation, the discrete modified KdV (mKdV) equation, respectively.     

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.     

A study on the (2+1)-dimensional and the (2+1)-dimensional higher-order Burgers equations

The (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional higher-order Burgers equation are investigated. The Cole–Hopf transformation method is used to carry out this study. Multiple-kink solutions are formally derived for each equation.     

Generalized extended tanh-function method and its application

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like, period-form solutions of nonlinear evolution equations (NEEs). Compared with most of the existing tanh-function method, extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By using this method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Make use of the method, we study the (3 + 1)-dimensional potential-YTSF equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions’ soliton-like solutions, singular soliton-like solutions, periodic form solutions.     

New exact solutions to the Zakharov–Kuznetsov equation and its generalized form

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons 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 note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

Exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili by Zhang [Huiqun Zhang, A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation, Appl. Math. Comput. 216 (2010) 2771-2777] are considered. To look for “new types of exact solutions travelling wave solutions” of equation Zhang has used the G′/G-expansion method. We demonstrate that there is the general solution for the reduction by Zhang from the (2+1)-dimensional Kadomtsev-Petviashvili equation and all solutions by Zhang are found as partial cases from the general solution.     

SECOND COHOMOLOGY GROUP OF GENERALIZED WITT TYPE LIE ALGEBRAS AND CERTAIN REPRESENTATIONS

ABSTRACT

We determine the second cohomology groups of Lie algebras of generalized Witt type which are some Lie algebras defined by Passman and Jordan, more general than those defined by Dokovic and Zhao, and slightly more general than those defined by Xu. Among all the 2-cocycles, there is a special one we think interesting. Using this 2-cocycle, we define the so-called Virasoro-like algebras. Then we give a class of their representations.     

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.     

Chaotic and fractal patterns for interacting nonlinear waves

Using an appropriate reduction method, a quite general new integrable system of equations 2 + 1 dimensions can be derived from the dispersive long-wave equation. Various soliton and dromion solutions are obtaining by selecting some types of solutions appropriately. The interaction between the localized solutions is completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. The arbitrariness of the functions included in the general solution implies that approximate lower dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic line soliton patterns and chaotic dromion patterns can appear in the solution. In a similar way, fractal dromion patterns and stochastic fractal excitations also exist for appropriate choices of the boundary conditions and/or initial conditions.     

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.     

Redundant exact solutions of nonlinear differential equations

We analyze the paper by Wazwaz and Mehanna [Wazwaz AM, Mehanna MS. A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation. Appl Math Comput 2010;217:1484–90]. The authors claim that they have found exact solutions of the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation using the tanh–coth method and the Exp-function method. We demonstrate that two of their solutions are incorrect. All the others can be simplified and they are the partial cases of the well-known solution. Wazwaz and Mehanna made a number of typical mistakes in finding exact solutions of nonlinear differential equations. Taking the results of this paper we introduce the definition of redundant exact solutions for the nonlinear ordinary differential equations.