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Scalar-field cosmology can be regarded as one of the significant fields of research in recent years. This paper is dedicated to a thorough investigation of the symmetries and conservation laws of the geodesic equations associated to a specific exact cosmological solution of a scalar-field potential which was originally motivated by six-dimensional Einstein-Maxwell theory. The mentioned string inspired Friedmann-Robertson-Lama i ^ tre-Walker (FRLW) solution is one of the noteworthy solutions of Einstein field equations. For this purpose, first of all the Christoffel symbols and the corresponding system of geodesic equations are computed and then the associated Lie symmetries are totally analyzed. Moreover, the algebraic structure of the Lie algebra of local symmetries is briefly investigated and a complete classification of the symmetry subalgebras is presented. Besides by applying the resulted symmetry operators the invariant solutions of the system of geodesic equations are discussed. In addition, the Noether symmetries and the Killing vector fields of the geodesic Lagrangian are determined and the corresponding optimal system of one-dimensional subalgebras is constructed. Mainly, an entire set of local conservation laws is computed for our analyzed scalar-field cosmological solution. For this purpose, two distinct procedures are applied: the celebrated Noether’s theorem and the direct method which is fundamentally based on a systematic application of Euler differential operators which annihilate any divergence expression identically.

Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we find conservation laws for this equation in three physically interesting cases., Comment: Applied Mathematics and Mechanics(2011) - Springer

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained. Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore, the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.

In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system in mathematical physics, the Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation, is studied. By applying the basic Lie symmetry method for the KdV–ZK equation, the classical Lie point symmetry operators are obtained. Also, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The non-classical symmetries of the KdV–ZK equation are also investigated.

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lame-Helmholtz equation still remains to be tackled. Arscott and Moglich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution. Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species ℰci(sna), ℰsi (i=1,2,3,4) including the well known Lame functions Eci(sna), Esi(sna) as special cases. This is effected by deriving two integro-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann's idea of P function, we introduce D function to express their transformation properties.

Purpose: In this paper, a detailed analysis of an important nonlinear model system, the two dimensional generalized Kuramoto-Sivashinsky (2D gKS) equation, is presented by group analysis. Methods: The basic Lie symmetry method is applied in order to determine the general symmetry group of our analyzed nonlinear model. Results: The symmetry group of the equation and some results related to the algebraic structure of the Lie algebra of symmetries are obtained. Also, a complete classification of the subalgebras of the symmetry algebra is resulted. Conclusions: It is proved that the Lie algebra of symmetries admits no three dimensional subalgebra. Mainly, all the group invariant solutions and the similarity reduced equations associated to the infinitesimal symmetries are obtained.