manjit singh

The nonlinear variable coefficient Zakharov-Kuznetsov (Vc-ZK) equation is derived using reductive perturbation technique for ion-acoustic solitary waves in magnetized three-component dusty plasma having negatively charged dust particles, isothermal ions, and electrons. The equation is investigated for generalized symmetries using a recently proposed compatibility method. Some more general symmetries are obtained and group invariant solutions are also constructed for these symmetries. Besides this, the equation is also investigated for nontrivial local conservation laws

Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements‎. ‎Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$‎. ‎Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$‎. ‎In this paper‎, ‎we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and‎, ‎$mathcal{O}_q=langle trangle $ if $q=4t+1$‎, ‎where $q$ and $t$ are odd primes‎. ‎Further‎, ‎we determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$

Based on symbolic manipulation program Maple and using Riccati equation mapping method several explicit exact solutions including kink, soliton-like, periodic and rational solutions are obtained for (2+1)-dimensional variable coefficient Broer-Kaup system in quite a straightforward manner. The known solutions of Riccati equation are used to construct new solutions for variable coefficient Broer-Kaup system.

As an application of Hirota bilinear method, perturbation expansion truncated at different levels is used to obtain exact soliton solutions to (2+1)-dimensional nonlinear evolution equation in much simpler way in comparison to other existing methods. We have derived bilinear form of nonlinear evolution equation and using this bilinear form, bilinear Backlund transformations and construction of associated linear problem or Lax pair are presented in straightforward manner and finally for proposed nonlinear equation, explicit one, two and three soliton solutions are also obtained.