We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of t
he Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schr${o}$dinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering we show that the class of systems can also be exactly solvable and are also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Li${e}$nard type nonlinear oscillators, which admit position dependent mass Hamiltonians.
We consider systems of ordinary differential equations (ODEs) of the form ${cal B}{mathbf K}=0$, where $cal B$ is a Hamiltonian operator of a completely integrable partial differential equation (PDE) hierarchy, and ${mathbf K}=(K,L)^T$. Such systems,
whilst of quite low order and linear in the components of $mathbf K$, may represent higher-order nonlinear systems if we make a choice of $mathbf K$ in terms of the coefficient functions of $cal B$. Indeed, our original motivation for the study of such systems was their appearance in the study of Painleve hierarchies, where the question of the reduction of order is of great importance. However, here we do not consider such particular cases; instead we study such systems for arbitrary $mathbf K$, where they may represent both integrable and nonintegrable systems of ordinary differential equations. We consider the application of the Prelle-Singer (PS) method --- a method used to find first integrals --- to such systems in order to reduce their order. We consider the cases of coupled second order ODEs and coupled third order ODEs, as well as the special case of a scalar third order ODE; for the case of coupled third order ODEs, the development of the PS method presented here is new. We apply the PS method to examples of such systems, based on dispersive water wave, Ito and Korteweg-de Vries Hamiltonian structures, and show that first integrals can be obtained. It is important to remember that the equations in question may represent sequences of systems of increasing order. We thus see that the PS method is a further technique which we expect to be useful in our future work.
We construct photon modulated coherent states of a generalized isotonic oscillator by expanding the newly introduced superposed operator through Weyl ordering method. We evaluate the parameter $A_3$ and the $s$-parameterized quasi probability distrib
ution function to confirm the non - classical nature of the states. We also calculate the identities related with the quadrature squeezing to explore the squeezing property of the states. Finally, we investigate the fidelity between the photon modulated coherent states and coherent states to quantify the non-Gaussianity of the states. The constructed states and their associated non - classical properties will add further knowledge on the potential.
In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li$acute{e}$nard type equation, $ddot {x} + f(x){dot {x}}^{2} + g(x)= 0$, where $f(x)$ and $g(x)$ are arbitrary functions of $x$. The
symmetry analysis gets divided into two cases, $(i)$ the maximal (eight parameter) symmetry group and $(ii)$ non-maximal (three, two and one parameter) symmetry groups. We identify the most general form of the quadratic Li$acute{e}$nard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the nonmaximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.
We explore squeezed coherent states of a 3-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered $X_1$-Laguer
re polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three mode squeezed coherent states resolve the identity operator. We evaluate Mandels $Q$-parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Further, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate Wigner function of these three mode squeezed coherent states and demonstrate squeezing property explicitly.
We construct exact solutions for a system of two nonlinear partial differential equations describing the spatio-temporal dynamics of a predator-prey system where the prey per capita growth rate is subject to the Allee effect. Using the $big(frac{G}{G
}big)$ expansion method, we derive exact solutions to this model for two different wave speeds. For each wave velocity we report three different forms of solutions. We also discuss the biological relevance of the solutions obtained.
We carry out an exact quantization of a PT symmetric (reversible) Li{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is
invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schr{o}dinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.
In this paper, we investigate the integrability aspects of a physically important nonlinear oscillator which lacks sufficient number of Lie point symmetries but can be integrated by quadrature. We explore the hidden symmetry, construct a second integ
ral and derive the general solution of this oscillator by employing the recently introduced $lambda$-symmetry approach and thereby establish the complete integrability of this nonlinear oscillator equation from a group theoretical perspective.
In this article, we construct loop soliton solutions and mixed soliton - loop soliton solution for the Degasperis-Procesi equation. To explore these solutions we adopt the procedure given by Matsuno. By appropriately modifying the $tau$-function give
n in the above paper we derive these solutions. We present the explicit form of one and two loop soliton solutions and mixed soliton - loop soliton solutions and investigate the interaction between (i) two loop soliton solutions in different parametric regimes and (ii) a loop soliton with a conventional soliton in detail.
