Noether symmetries and the quantization of a Liénard-type nonlinear oscillator

The classical quantization of a Liénard-type nonlinear oscillator is achieved by a quantization scheme (M. C. Nucci. Theor. Math. Phys., 168:994–1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation in the momentum space as given in (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light on the apparently remarkable connection with the linear harmonic oscillator.     

Finite Morse index solutions of exponential problems

We prove that the problem −Δu=eu$-\mathrm{\Delta }u={\mathrm{e}}^u$ has no negative finite Morse index solution on R³${\mathbb{R}}^3$ and give some applications to bounded domain problems.     

Density Theorems for Complete Minimal Surfaces in {\mathbb{R}}^{3}

In this paper we have proved several approximation theorems for the family of minimal surfaces in that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C ^k convergence on compact sets, for any . As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in with uncountably many ends. This research is partially supported by MEC-FEDER Grant no. MTM2004 - 00160.     

Energetics during proton transfer process in hydrated zündel ion complex

Zündel ion (H₅O) is one of the two important structures formed during the proton transfer process in aqueous system. This work reports microsolvation of Zündel ion using density functional theory based B3LYP method with aug‐cc‐pVTZ basis set. Interaction of Zündel ion with four water molecules in its first solvation shell is studied using many‐body analysis approach. A change in many‐body energies and their contribution to the binding energy of a complex during the proton transfer process from donor to acceptor water molecule in Zündel ion‐4H₂O complex is obtained. For the hydrated Zündel ion complex, the contribution from total two‐body, three‐body, four‐body, five‐body, and relaxation energy to the binding energy is 84.7, 14, 6.87, 1.6, and 4%, respectively, at B3LYP/aug‐cc‐pVTZ level. Relaxation energy and total five‐body energy have repulsive contribution to the binding energy of a hydrated Zündel ion complex. It is found that the relaxation energy and binding energy of a Zündel‐4H₂O complex is the maximum and minimum, respectively, when a shared proton is at equal distance from oxygen atom of donor and acceptor water molecules. A significant change in two‐body, three‐body, and four‐body energies for which Zündel ion is one of the many‐body terms is observed during the proton transfer process. A change in total two‐body, total three‐body, total four‐body, and relaxation energy is about 2.6, 1.8, 0.4, and 1.1%, respectively, during the proton transfer process. A change in two‐body, three‐body, and four‐body interaction energies between water molecules is very small during the proton transfer process. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces

A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( or (where d ₂ is the two-dimensional dilation algebra), while for those admitting (where represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes ).     

Symmetry reductions and solutions for pollutant diffusion in a cylindrical system

This study is devoted to investigating transient coupled fluid flow and mass transfer partial differential equations (PDEs) describing pollutant transport in cylindrical coordinates. Symmetry analysis of the system of coupled PDEs is performed and some large Lie algebras are obtained for some special cases of the arbitrary and special choices of constants, and the source term. Optimal systems are constructed for all the admitted symmetries. We perform reductions for different choices of the source term. In some cases invariant solution is sought, however some cases resulted in coupled systems of highly nonlinear ordinary differential equations (ODEs). Imposing realistic boundary conditions and considering a constant source term, we then use the Adomain decomposition techniques to solve the boundary value problem.     

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A₀ acting on a Hilbert space H singular perturbations of the form A₀+V, are studied under some additional requirements of symmetry imposed on the initial operator A₀ and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A₀+V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L₂(R³) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.     

Symmetries and similarity solutions for the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane

We perform a complete analysis of all the Lie point symmetries admitted by the equation describing the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane. We then investigate the existence of group-invariant solutions for particular values of the power-law parameter β.