Nonlinear self-adjointness and invariant solutions of a 2D Rossby wave equation

Abstract

Abstract:The paper investigates the nonlinear self-adjointness of the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane. It is a particular form of Rossby equation which does not possess variational structure and it is studied using a recently method developed by Ibragimov. The conservation laws associated with the infinite-dimensional symmetry Lie algebra models are constructed and analyzed. Based on this Lie algebra, some classes of similarity invariant solutions with nonconstant linear and nonlinear shears are obtained. It is also shown how one of the conservation laws generates a particular wave solution of this equation.

Solutions of the Bullough–Dodd Model of Scalar Field through Jacobi-Type Equations

“…The solution (34) depicts a new class of six-parameter solutions of (18). For the particular choice of parameters ω Case IV: Let us assume now that the parameters h i , i = 0, 4 are:…”

“…Important progress has been achieved and many powerful and effective methods have been proposed for deriving explicit solutions of nonlinear equations: the inverse scattering [8], the Hirota bilinear transformation [9][10][11], the Jacobi elliptic function method [12,13], the generalized Kudryashov method [14,15], the dynamical system approach and the bifurcation method of the phase plane [16], the (G /G)-expansion method [17,18] and its extension to the functional expansion method [19,20], the Lie symmetry method [21][22][23][24][25] and the generalized conditional symmetry approach [26], various extended tanh methods [27][28][29], as well as other tools of investigation for the nonlinear dynamical models [30][31][32][33][34].…”

Similarity Solutions and Conservation Laws for the Beam Equations: A Complete Study

“…The main motivation behind using the Ibragimov's approach is to obtain the conservation laws to deduce certain special solutions for the Beam equations following the methodology specified by Cimpoiasu [32], where the author have used the nonlinear self-adjointness method to compute solutions for the Rossby waves. The Noether's theorem can be easily applied to obtain the conserved terms but it is our intuition that the non-local conserved terms as obtained using the Ibragimov's method can contribute in obtaining new solutions in a different subspace of the complex plane.…”

Similarity solutions and conservation laws for the Beam Equations: a complete study

“…The main motivation behind using the Ibragimov's approach is to obtain the conservation laws to deduce certain special solutions for the Beam equations following the methodology specified by Cimpoiasu [7] where the author have used the nonlinear self-adjointness method to compute solutions for the Rossby waves. The Noether's theorem can be easily applied to obtain the conserved terms but it is our intuition that the non-local conserved terms as obtained using the Ibragimov's method can contribute in obtaining new solutions in a different subspace of the complex plane.…”