Dimensional Boiti Leon Pempinelli Equation Research Articles

Lie symmetries, invariant solutions and phenomena dynamics of Boiti–Leon–Pempinelli system

The present work deals with highly nonlinear dispersive water equation named Boiti–Leon–Pempinelli (BLP) system, which is a well known two dimensional generalization of sinh-Gordon equation. The BLP system describes interaction of horizontal wave component propagating in plane in infinite narrow channel with constant depth. The Lie symmetry method is employed to derive infinitesimal generators under invariance conditions of one-parameter transformations. The invariance conditions ensure the reducibility of independent variables and preserve the characteristics of system. The commutative relations of infinitesimal vectors show infinite dimensional algebra of symmetries. Thereafter, similarity variables are derived using infinitesimal generators, which lead to first symmetry reductions. Thus, a repeated process of symmetry reductions provide equivalent system of ODEs. These ODEs are integrated and thus some invariant solutions of BLP system are constructed under parametric constraints. The derived solutions are general than previous researches [J Q Yu et al 2010 Exact solutions and conservation laws of (2 + 1)-dimensional Boiti–Leon–Pempinelli equation, Appl. Math. Comput. 216 2293–2300; M Kumar, R Kumar, 2014 On new similarity solutions of the boiti–leon–pempinelli system, Commun. Theor. Phys. 61 121–126; M Kumar et al 2015 Some more similarity solutions of the (2 + 1)-dimensional BLP system, Comput. Math. Appl. 70 212–221; J Fei et al 2015 Symmetry reduction and explicit solutions of the (2 + 1)-dimensional boiti–leon–pempinelli system, Appl. Math. Comput. 268 432–438; M Kumar et al 2021 Some more invariant solutions of (2 + 1)-water waves, Int. J. Appl. Comput. Math. 7 18] as they contain all four arbitrary functions involved in infinitesimals and several free parameters as well. The deduction of previous results obtained by taking particular choices of arbitrary functions and constants, ensure novelty and importance of these solutions as well as the authenticity and effectiveness of Lie symmetry method. To demonstrate the distinct physical structures of phenomena, these solutions are expanded via numerical simulation. Thus, doubly soliton, line soliton, multi soliton, soliton fusion and fission nature have been analyzed. The graphical representations are seemed useful to understand the dynamics of BLP system and propagation of dispersive water waves in infinite narrow channel.

Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics

In this study, a new method called improved Bernoulli sub-equation function method has been proposed. This method is based on the Bernoulli sub-ODE method. After we mention the general properties of proposed method, we apply this algorithm to the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation system. This gives us some new prototype solutions such as exponential and rational function solutions. Then, we have plotted two- and three-dimensional surfaces of analytical solutions. Finally, we have submitted a comprehensive conclusion.

Symmetry analysis and CTE solvability for the (2+1)-dimensional Boiti–Leon–Pempinelli equation

The residual symmetry is derived from the known truncated Painlevé expansion of the (2+1)-dimensional Boiti–Leon–Pempinelli (BLP) equation, and this kind of residual symmetry is localized by introducing two new variables. By applying the Lie point symmetry method to the enlarged system, a new type of finite symmetry transformation is derived. Moreover, it is proved that the (2+1)-dimensional BLP equation is consistent tanh expansion solvable, the exact interaction solutions between solitons and any other types of potential Burgers waves are also obtained, which include soliton-error function waves, soliton-periodic waves, and so on.

Localized structures for (2+1)-dimensional Boiti–Leon–Pempinelli equation

It is shown that Painleve integrability of (2+1)-dimensional Boiti–Leon–Pempinelli equation is easy to be verified using the standard Weiss–Tabor–Carnevale (WTC) approach after introducing the Kruskal’s simplification. Furthermore, by employing a singular manifold method based on Painleve truncation, variable separation solutions are obtained explicitly in terms of two arbitrary functions. The two arbitrary functions provide us a way to study some interesting localized structures. The choice of rational functions leads to the rogue wave structure of Boiti–Leon–Pempinelli equation. In addition, for the other choices, it is observed that two solitons may evolve into breather after interaction. Also, the interaction between two kink compactons is investigated.

New Exact Traveling Wave Solutions of Some Nonlinear Higher-Dimensional Physical Models

In this paper, some new traveling wave solutions of the (4 + 1)-dimensional Fokas equation, (3 + 1)-dimensional Jumbo–Miwa equation and (2 + 1)-dimensional Boiti–Leon–Pempinelli equation are obtained through the ( G ′ G ) -expansion technique. The key idea of this technique is to take full advantage of a Riccati equation involving two parameters and use its solutions in obtaining the traveling wave solutions. The results reveal that this technique is very effective and powerful for solving higher-dimensional nonlinear problems arising in mathematical physics.

Localized coherent structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation

A new mapping equation (coupled Riccati equations) method is used to obtain three kinds of variable separation solutions with two arbitrary functions of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. Based on the variable separation solution and by selecting appropriate functions, two type of multidromion excitations, that is, dromion lattice and multidromion solitoffs, are investigated. Moreover, we can discuss head-on collision and “chase and collision” phenomena between two multidromions.

Application of the Exp-function method to the (2+1)-dimensional Boiti–Leon–Pempinelli equation using symbolic computation

This paper deals with the so-called Exp-function method for studying a particular nonlinear partial differential equation (PDE): the (2+1)-dimensional Boiti–Leon–Pempinelli equation. The method is constructive and can be carried out in a computer with the aid of a computer algebra system. The obtained generalized solitary wave solutions contain more arbitrary parameters compared with the earlier works, and thus, they are wider. This means that our method is effective and powerful for constructing exact and explicit analytic solutions to nonlinear PDEs.

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.

Solitons, Bäcklund transformation, and Lax pair for the (2+1)-dimensional Boiti–Leon–Pempinelli equation for the water waves

Under investigation in this paper is the (2+1)-dimensional Boiti–Leon–Pempinelli (BLP) equation for the water waves. By virtue of the binary Bell polynomials and symbolic computation, the bilinear form for the BLP equation is obtained. Furthermore, soliton solutions are presented, and soliton interaction properties including the elastic, inelastic, and elastic-inelastic collisions are discussed by the graphical analysis. Besides, the Bäcklund transformation in the form of the binary Bell polynomials is derived. Via the Bäcklund transformation, the shock-wave solutions and Lax pair are both constructed.

Exact solutions and conservation laws of (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, we obtain the symmetry group theorem by using the modified CK’s direct method, and some new exact solutions of (2 + 1)-dimensional BLP equation. Also we derive the corresponding Lie algebra and the conservation laws of BLP equation.

A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper the (2 + 1)-dimensional Boiti–Leon–Pempinelli (BLP) equation will be studied. The tanh–coth method will be used to obtain exact travelling wave solutions for this equation. The Exp-function method will also be applied to the BLP equation to derive a new variety of travelling wave solutions with distinct physical structures.

Periodic structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, firstly, a new mapping method is used to obtain the variable separation solutions, with two arbitrary functions, of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. From the variable separation solution and by selecting appropriate functions, some novel Jacobian elliptic wave structure and periodic wave evolutional behaviors are investigated.

Novel interactions between semi-foldons of the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, firstly, the general projective Riccati equation method (PREM) is applied to derive variable separation solutions of the (2 + 1)-dimensional Boiti–Leon–Pempinelli (BLP) equation. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the variable separation solution and by selecting appropriate multi-valued functions, novel interactions between a special bell-like semi-foldon and a special peakon-like semi-foldon are investigated.

A new generalized algebra method and its application in the (2+1) dimensional Boiti–Leon–Pempinelli equation

Abstract In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics.