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We work with a formulation of Noether-symmetry analysis which uses the properties of infinitesimal point transformations in the space-time variables to establish the association between symmetries and conservation laws of a dynamical system. Here symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of the damped harmonic oscillator representing...
In this paper we discuss symmetries of classes of wave equations that arise as a consequence of some Vaidya metrics. We show how the wave equation is altered by the underlying geometry. In particular, a range of consequences on the form of the wave equation, the symmetries and number of conservation laws, inter alia, are altered by the manifold on which the model wave rests. We find Lie and Noether...
In this paper a class of multi-dimensional Gordon-type equations are analysed using a multiplier and homotopy approach to construct conservation laws. The main focus is the analysis of the classical versions of the Gordon-type equations and obtaining higher-order variational symmetries and corresponding conserved quantities. The results are extended to the multi-dimensional Gordon-type equations with...
This paper provides a very brief introduction to three of Chandrasekhar’s famous books on Stellar Structure, Hydrodynamics and Black Holes. In particular we summarize Chandra’s treatment of the “Thermal Instability” which plays such a crucial role in the understanding of convection zones in stellar atmospheres. We also outline three important ideas in fluid dynamics which are inexplicably omitted...
In this paper, exact solutions of Benjamin–Bona–Mahony–Peregrine equation are obtained with power-law and dual power-law nonlinearities. The Lie group analysis as well as the simplest equation method are used to carry out the integration of these equations. The solutions obtained are cnoidal waves, periodic solutions and soliton solutions. Subsequently, the conservation laws are derived for the underlying...
In this letter, the Lie point symmetries of a class of Gordon-type wave equations that arise in the Milne space-time are presented and analysed. Using the Lie point symmetries, it is showed how to reduce Gordon-type wave equations using the method of invariants, and to obtain exact solutions corresponding to some boundary values. The Noether point symmetries and conservation laws are obtained for...
In this paper, the classical Lie group method is employed to obtain exact travelling wave solutions of the generalized Camassa–Holm Kadomtsev–Petviashvili (g-CH–KP) equation. We give the conservation laws of the g-CH–KP equation. Using the symmetries, we find six classical similarity reductions of g-CH–KP equation. Many kinds of exact solutions of the g-CH–KP equation are derived by solving the reduced...
In this study, we explore the modified form of ($$1+n$$ 1+n )-dimensional Zakharov–Kuznetsov equation, which is used to investigate the waves in dusty and magnetised plasma. It is proved that the equation follows the property of nonlinear self-adjointness. Lie point symmetries are calculated and conservation laws in the framework of the new general conservation theorem of Ibragimov are obtained. The...
In this paper, we study Klein–Gordon–Zakharov equations which describe the propagation of strong turbulence of the Langmuir wave in a high-frequency plasma. Using the symbolic manipulation tool Maple, the classifications of symmetry algebra are carried out, and the construction of several local non-trivial conservation laws based on a direct method of Anco and Bluman is illustrated. Starting with...
The degenerate coupled multi-Korteweg–de Vries equations for coupled multiplicity $$l=3$$ l=3 are studied. The equations, also known as three-field Kaup–Boussinesq equations, are considered for invariant analysis and conservation laws. The classical Lie’s symmetry method is used to analyse the symmetries of equations. Based on the Killing’s form, which is invariant of adjoint action, the full classification...
In this paper, the AB system with time-dependent coefficients for the ultrashort pulses in an inhomogeneous optical fibre or the marginally unstable baroclinic wave packets in an atmospheric or oceanic system is investigated via the Lie symmetry analysis. We obtain the Lie symmetries, reduced equations and group-invariant solutions. The nonlinear self-adjointness of the AB system is proved, and the...
This paper investigates the new coupled $$(2+1)$$ (2+1) -dimensional Zakharov–Kuznetsov (ZK) system with time-dependent coefficients for multiple types of exact solutions by using the Lie symmetry transformation method. Similarity transformation reduces the system of equations into ordinary differential equations and further, these are solved for solutions having bright, dark and singular solitons...
In this paper, we primarily investigate Lie symmetry analysis and exact solutions for the coupled integrable dispersionless equations. First of all, based on the Lie symmetry analysis, an optimal system of one-dimensional subalgebras is constructed. Furthermore, similarity reductions and group invariant solutions are given. Next, exact solutions of the reduced equations have been derived by the method...
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