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Study the behaviour and the evolution of the cosmological field equations in an homogeneous and anisotropic spacetime with two scalar fields coupled in the kinetic term. Specifically, the kinetic energy for the scalar field Lagrangian is that of the Chiral model and defines a two-dimensional maximally symmetric space with negative curvature. For the background space we assume the locally rotational spacetime which describes the Bianchi I, the Bianchi III and the Kantowski-Sachs anisotropic spaces. We work on the $H$% -normalization and we investigate the stationary points and their stability. For the exponential potential we find a new exact solution which describes an anisotropic inflationary solution. The anisotropic inflation is always unstable, while future attractors are the scaling inflationary solution or the hyperbolic inflation. For scalar field potential different from the exponential, the de Sitter universe exists.
We solve the group classification problem for the $2+1$ generalized quantum Zakharov-Kuznetsov equation. Particularly we consider the generalized equation $u_{t}+fleft( uright) u_{z}+u_{zzz}+u_{xxz}=0$, and the time-dependent Zakharov-Kuznetsov equat ion $u_{t}+delta left( tright) uu_{z}+lambda left( tright) u_{zzz}+varepsilon left( tright) u_{xxz}=0$% . Function $fleft( uright) $ and $delta left( tright) ,~lambda left( tright) $,~$varepsilon left( tright) $ are determine in order the equations to admit additional Lie symmetries. Finally, we apply the Lie invariants to find similarity solutions for the generalized quantum Zakharov-Kuznetsov equation.
We apply the Painleve Test for the Benney and the Benney-Gjevik equations which describe waves in falling liquids. We prove that these two nonlinear 1+1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic so lutions in terms of Laurent expansions are presented.
We apply the theory of Lie symmetries in order to study a fourth-order $1+2$ evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the spec ific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined.
The 1 + 2 dimensional Bogoyavlensky-Konopelchenko Equation is investigated for its solution and conservation laws using the Lie point symmetry analysis. In the recent past, certain work has been done describing the Lie point symmetries for the equati on and this work seems to be incomplete (Ray S (2017) Compt. Math. Appl. 74, 1157). We obtained certain new symmetries and corresponding conservation laws. The travelling-wave solution and some other similarity solutions are studied.
We consider the generalized Painleve--Ince equation, begin{equation*} ddot{x}+alpha xdot{x}+beta x^{3}=0 end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are relate d as $beta =alpha ^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlev{e} test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlev{e}--Ince equation fails at the Painlev{e} test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlev{e}--Ince equation is integrable is terms of Lie symmetries and of the Painlev{e} test.
A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a g eneralized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painleve Series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a Left Painleve Series and one given by a Right Painleve Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.
The integrability of the $Lambda-$Einstein-nonlinear $SU(2)$ $sigma$-model with nonvanishing cosmological charge is studied. We apply the method of singularity analysis of differential equations and we show that the equations for the gravitational fi eld are integrable. The first few terms of the solution are presented.
The Szekeres system is studied with two methods for the determination of conservation laws. Specifically we apply the theory of group invariant transformations and the method of singularity analysis. We show that the Szekeres system admits a Lagrangi an and the conservation laws that we find can be derived by the application of Noethers theorem. The stability for the special solutions of the Szekeres system is studied and it is related with the with the Left or Right Painleve Series which describes the expansions.
89 - M.C. Nucci , P.G.L. Leach 2008
In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of advancement of the use of ${cal PT} $ symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Diracs theory by providing an Hamiltonian which is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order cite {Nucci} that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.
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