Computational soliton solutions to $$(2+1)$$ ( 2 + 1 ) -di
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Computational soliton solutions to (2 + 1)-dimensional Pavlov equation using Lie symmetry approach SACHIN KUMAR1
,∗ ,
MUKESH KUMAR2 and DHARMENDRA KUMAR3
1 Department
of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 007, India 2 Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211 004, India 3 Department of Mathematics, SGTB Khalsa College, University of Delhi, Delhi 110 007, India ∗ Corresponding author. E-mail: [email protected] MS received 12 June 2019; revised 19 September 2019; accepted 17 October 2019 Abstract. In this work, Lie symmetry analysis and one-dimensional optimal system for Pavlov equation are presented. All the possible vector fields, their commutative and adjoint relations are carried out under invariance property of Lie group theory. On the basis of optimal system, similarity reductions of Pavlov equation are obtained. A repeated process of similarity reductions transforms the Pavlov equation into ordinary differential equations, which generate invariant solutions. The obtained invariant solutions are supplemented by numerical simulation to analyse the physical behaviour. Thus, their parabolic, multisoliton, nonlinear, kink and antikink wave profiles are traced in results and discussions sections. Keywords. (2 + 1)-Dimensional Pavlov equation; optimal system; invariant solutions. PACS Nos 02.20.Sv; 02.20.Qs; 05.45.Yv; 02.30.Jr
1. Introduction Nonlinear partial differential equations (NPDEs) are widely used to study various phenomena arising in mathematical physics, ion-acoustic waves in plasmas, water surface gravity waves, condensed matter physics etc. [1,2]. The exact solutions of these NPDEs have a significant role to understand the dynamics and in the development of natural phenomena. Therefore, a variety of methods like multiple exp-function method [3], Hirota’s method [4], Jacobi elliptic function method [5], tanh–coth method [6], homogeneous balance method [7], Hirota’s bilinear method [8], the (G /G) -expansion method [9], solitary wave ansatz [10], F-expansion method [11], simplest equation method [2] etc. have been developed for obtaining exact solutions of the NPDEs. The main motive of this research is to attain some group invariant solution of Pavlov equation [1,12–21] := u yy = u t x + u y u x x − u x u x y ,
(1)
where u(x, y, t) is the amplitude of the relevant wave depending upon the space variables x, y and time t. Equation (1) was first derived as a symmetry reduction 0123456789().: V,-vol
of the second heavenly equation [13]. The Pavlov equation is a highly nonlinear partial differential equation. Therefore, finding its exact solutions is an arduous task. Besides the aforementioned references, a rich literature of integrable properties, symmetries and exact solutions of Pavlov equation is presented here. Pavlov [14] presented Benney-type moment chains and constructed new (2 + 1)-dimensional integrable hydrodynamic systems. Also, their hydrodynamical reductions and
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