On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations

We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.

Lie symmetry analysis and invariant solutions for the(3+1)-dimensional Virasoro integrable model

Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.

Group Invariant Solutions of the Full Plastic Torsion of Rod with Arbitrary Shaped Cross Sections

Based on the theory of Lie group analysis,the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied.Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters.Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions.Moreover,physical explanations of each group invariant solution are discussed by all appropriate transformations.The methodology and solution techniques used belong to the analytical realm.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

Invariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoreticalmethod.The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators.Three invariant solutions are constructed.Their graphical figures are presented and physical meanings are elucidated in some cases.

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections

Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.

CONVERGENCE OF SOLUTIONS FOR RLC-NONLINEAR NETWORKS WITH TIME-VARYING ELEMENTS

This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions to prove the convergence of solutions under certain conditiolts. Under the assumption that every element value involving voltage source is asymptotically constallt, we establish four creteria for all solutiolls of such a system to converge to the set of equilibria of its limiting equations via LaSalle invariant principle.We also present two theorems on the existence of periodic solutions for periodically excited uonliltear circuits.This results generalize those of Brayton and Moser[1,2].

Lie Symmetries,1-Dimensional Optimal System and Optimal Reductions of(1+2)-Dimensional Nonlinear Schrodinger Equation

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.

On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs

We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous Monge-Ampère equation to first-order ODEs. Some classes of the invariant solutions are constructed.

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Spatial Gradient of the Height

The aim of this investigation is to determine the effect of fluid leak-off (suction) and fluid injection (blowing) at the horizontal base on the two-dimensional spreading under the gravity of a thin film of viscous incompressible fluid by studying the evolution of the streamlines in the thin film. It is assumed that the normal component of the fluid velocity at the base is proportional to the spatial gradient of the height of the film. Lie symmetry methods for partial differential equations are applied. The invariant solution for the surface profile is derived. It is found that the thin fluid film approximation is satisfied for weak to moderate leak-off and for the whole range of fluid injection. The streamlines are derived and plotted by solving a cubic equation numerically. For fluid injection, there is a dividing streamline originating at the stagnation point at the base which separates the flow into two regions, a lower region consisting mainly of rising fluid and an upper region consisting mainly of descending fluid. An approximate analytical solution for the dividing streamline is derived. It generates an approximate V-shaped surface along the length of the two-dimensional film with the vertex of each section the stagnation point. It is concluded that the fluid flow inside the thin film can be visualised by plotting the streamlines. Other models relating the fluid velocity at the base to the height of the thin film can be expected to contain a dividing streamline originating at a stagnation point and dividing the flow into a lower region of rising fluid and an upper region of descending fluid.

Study of exact analytical solutions and various wave profiles of a new extended(2+1)-dimensional Boussinesq equation using symmetry analysis

This paper systematically investigates the exact solutions to an extended(2+1)-dimensional Boussinesq equation,which arises in several physical applications,including the propagation of shallow-water waves,with the help of the Lie symmetry analysis method.We acquired the vector fields,commutation relations,optimal systems,two stages of reductions,and exact solutions to the given equation by taking advantage of the Lie group method.The method plays a crucial role to reduce the number of independent variables by one in each stage and finally forms an ODE which is solved by taking relevant suppositions and choosing the arbitrary constants that appear therein.Furthermore,Lie symmetry analysis(LSA)is implemented for perceiving the symmetries of the Boussinesq equation and then culminating the solitary wave solutions.The behavior of the obtained results for multiple cases of symmetries is obtained in the present framework and demonstrated through three-and two-dimensional dynamical wave profiles.These solutions show single soliton,multiple solitons,elastic behavior of combo soliton profiles,and stationary waves,as can be seen from the graphics.The outcomes of the present investigation manifest that the considered scheme is systematic and significant to solve nonlinear evolution equations.