Closed form invariant solutions of (2+1)-dimensional extended shallow water wave equation via Lie approach
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Closed form invariant solutions of (2+1)-dimensional extended shallow water wave equation via Lie approach Mukesh Kumara
, Kumari Manju
Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj 211004, India Received: 9 July 2020 / Accepted: 1 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the present letter, some invariant solutions of (2+1)-dimensional extended shallow water wave equation are studied. We reduce the equation to an ordinary differential equation(ODE) via the Lie symmetry method. The similarity transformation method is used to determine the group-invariant solutions for the equation. The method plays an important role to reduce the number of independent variables by one in each stage and finally forms an ODE. On the other hand, solutions provide rich physical configuration due to the presence of some distinct arbitrary constants and functions. Furthermore, numerical simulation has been done to trace out the derived results which illustrate the dynamic behavior of these obtained solutions.
1 Introduction 1.1 Scope In nonlinear sciences, the studies on the complexity of nonlinear evolution equations (NLEEs) and its challenges have arisen great attention of many scientists, researchers, engineers and mathematicians during the past few decades. Many of the physical phenomena are modeled by nonlinear partial differential equations (NLPDEs) [1–27] with time t as one of the independent variables. In fluid dynamics, Navier–Stokes and Euler equations can be seen as the best result of this. A wide range of relevant theories and methods have been developed to know about the central issues of field, for example Hirota bilinear method [5], binary Bell Polynomials [2,4,5], Hereman’s simplified method [1], Cole–Hopf transformation method [1], similarity method [14–27], GG method [3], Lax pair and Bäcklund transformation [4]. In this research work, we mainly aim at the following (2 + 1)-dimensional extended shallow water wave equation (ESWW): u yt + u x x x y − 3u x x u y − 3u x u x y + au x y = 0
(1.1)
where u(x, y, t) is the amplitude expressed in terms of longitudinal(x), transversal(y), time domain(t) and ‘a’ is any real parameter. Subscripts are for partial derivatives. The equation plays a very crucial role in applied mathematics, physics, oceanography and engineering
a e-mail: [email protected] (corresponding author)
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fields. The extension term here is au x y , which does not affect the integrability of the model; it is the only effect seen in dispersion relation [1]. Equation (1.1) is also known as the first extended shallow water wave equation, and the form of second extended shallow water wave equation is expressed as: u xt + u x x x y − 2u x x u y − 4u x u x y + au x y = 0.
(1.2)
For y = x and a = 0, Eqs. (1.1) and (1.2) both reduce to potential KdV equation. If only a = 0, Eq. (1.1) reduces to (2+1)-dime
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