Articles written in Pramana – Journal of Physics
Volume 94 All articles Published: 4 August 2020 Article ID 0116 Research Article
In the present article, Lie group of point transformations method is successfully applied to study the invariance properties of the (2 + 1)-dimensional Pavlov equation. Applying the Lie symmetry method, we strictly obtain the infinitesimals, vector fields, commutation relation and several interesting symmetry reductions of the equation. The explicit exact solutions are derived under some limiting conditions imposed on the infinitesimals $\xi$, $\phi$, $\tau$ and $\eta$. Then, the Pavlov equation is transformed into a number of nonlinear ODEs through several symmetry reductions. These new exact solutions are more general and entirely different from the work of Kumar $et al$ (Pramana - J. Phys. 94: 28 (2020)). The obtained invariant solutions are examined analytically as well as physically through numerical simulation by giving free alternative values of arbitrary functions and constants. Consequently, graphical representations of all these solutions are studied and demonstrated in 3D-graphics and the corresponding contour plots. Interestingly, the solution profiles show the annihilation of three-dimensional parabolic profile, doubly soliton and elastic multisolitons and nonlinear wave nature form.
Volume 95 All articles Published: 27 March 2021 Article ID 0051 Research Article
In the present work, abundant group-invariant solutions of (2 + 1)-dimensional Bogoyavlenskii–Schieff equation have been investigated using Lie symmetry analysis. The Lie infinitesimal generators, all the geometric vector fields, their commutative and adjoint relations are provided by utilising the Lie symmetry method. The Lie symmetry method depends on the invariance criteria of Lie groups, which results in the reduction of independent variables by one. A repeated process of Lie symmetry reductions, using the double, triple and septuple combinations between the considered vectors, converts the Bogoyavlenskii–Schieff (BS) equation into nonlinear ordinary differential equations (ODEs) which furnish numerous explicit exact solutions with the help of computerised symbolic computation. The obtained group-invariant solutions are entirely new and distinct from the earlier established findings. As far as possible, a comparison of our reported results with the previous findings is given. The dynamical behaviour of solutions is discussed both analytically as well as graphically via their evolutionary wave profiles by considering suitable choices of arbitrary constants and functions. To ensure rich physical structures, the exact closed-form solutions are supplemented via numerical simulation, which produce some bright solitons, doubly solitons, parabolic waves, U-shaped solitons and asymptotic nature.
Volume 95, 2021
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