Symmetry reductions for thin film type equations

The lubrication equation ut = (u nuxxx)x plays an important role in the study of the interface movements. In this work we analyze the generalizations of the above equation given by ut = (u nuxxx)x − kumux. By using Lie classical method the corresponding reductions are performed and some solutions are characterized

Generalized Camassa-Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa-Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively

Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation

This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie's symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning.The support of the Plan Propio de Investigacion de la Universidad de Cadiz is gratefully acknowledged. The authors also thank the referees for their suggestions to improve the quality of the paper

Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, allowing us to transform the partial differential equation into an ordinary differential equation by using the symmetry reductions. Furthermore, we determine the conservation laws of this equation by applying the multiplier method

Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation

For a generalized KdV-Burgers-Kuramoto equation we have studied conservation laws by using the multiplier method, and investigated its first-level and second level potential systems. Furthermore, the Lie point symmetries of the equation and the Lie point symmetries associated with the conserved vectors are determined. We obtain travellingwave reductions depending on the form of an arbitrary function. We present some explicit solutions: soliton solutions, kinks and antikinks

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin-Bona-Mahony-Burgers Equation in 2+1-Dimensions

The Benjamin-Bona-Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin-Bona-Mahony-Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G & PRIME;(u)& NOTEQUAL;0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov's method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.The authors acknowledge the financial support from Junta de Andalucia group FQM-201. The authors warmly thank the referees for their valuable comments and recommending changes that significantly improved this paper. In memory of Maria de los Santos Bruzon Gallego: thank you for dedicating your time and effort to care us and help us. You will always be our role model. May Maruchi rest in peace

Lie symmetries and equivalence transformations for the Barenblatt-Gilman model

In this paper we have considered the Barenblatt-Gilman equation which models the nonequilibrium countercurrent capillary impregnation. The equation of this model is a third-order equation and the unknown function concerns to the effective water saturation. We have applied the classical method to get the Lie group classification with respect to unknown function and we have constructed the equivalence transformations. We have also obtained the invariant solutions for some forms of the equation, including travelling wave solutions based on the Jacobi elliptic sine function

Simetrías potenciales de un modelo matemático que describe las vibraciones de una viga

En este trabajo presentamos un estudio, desde el punto de vista de la teoría de las simetrías potenciales clásicas y no clásicas para ecuaciones en derivadas parciales, del modelo que describe las vibraciones de una viga

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs