Recursive generation of isochronous Hamiltonian systems

We propose a simple procedure to identify the collective coordinate Q which is used to generate the isochronous Hamiltonian. The new isochronous Hamiltonian generates more and more isochronous oscillators, recursively

A Linear Approximation for the Excitation Energies of single and double analog states in the f_{7/2} shell

We find that the excitation energies of single analog states for odd-even nuclei in the f$_{7/2}$ shell with J=j=7/2$^{-}$ and the J=0$^{+}$ double analog states in the even-even nuclei are well described by the formulas $E^{*}(j,T+1) = b (T+X)$ and $E^{*}(0^{+},T+2) = 2b (T+X+0.5)$,respectively, where $T=\mid N-Z\mid /2$ is usually the ground state isospin. It is remarkable to note that the parameter X accounts for the departures from the symmetry energy based predictions.Comment: 8 pages and no figure

Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

We consider the nonlinear Helmholtz (NLH) equation describing the beam propagation in a planar waveguide with Kerr-like nonlinearity under non-paraxial approximation. By applying the Lie symmetry analysis, we determine the Lie point symmetries and the corresponding symmetry reductions in the form of ordinary differential equations (ODEs) with the help of the optimal systems of one-dimensional subalgebras. Our investigation reveals an important fact that in spite of the original NLH equation being non-integrable, its symmetry reductions are of Painlev\'e integrable. We study the resulting sets of nonlinear ODEs analytically either by constructing the integrals of motion using the modified Prelle-Singer method or by obtaining explicit travelling wave-like solutions including solitary and symbiotic solitary wave solutions. Also, we carry out a detailed numerical analysis of the reduced equations and obtain multi-peak nonlinear wave trains. As a special case of the NLH equation, we also make a comparison between the symmetries of the present NLH system and that of the standard nonlinear Schr\"odinger equation for which symmetries are long available in the literature.Comment: Accepted for publication in "Applied Mathematics and Computation". 18 pages, 15 figure