arXiv subjects

Partha Guha

Publications and source records attributed to Partha Guha.

At least 37 records · Page 2Linked to original sources

arXiv2019

Analysis and comparative study of non-holonomic and quasi-integrable deformations of the Nonlinear Schrödinger Equation

The non-holonomic deformation of the nonlinear Schrödinger equation, uniquely obtained from both the Lax pair and Kupershmidt's bi-Hamiltonian [Phys. Lett. A 372, 2634 (2008)] approaches, is compared with the quasi-integrable deformation of the same system [Ferreira et. al. JHEP 2012, 103 (2012)]. It is found that these two deformations can locally coincide only when the phase of the corresponding solution is discontinuous in space, following a definite phase-modulus coupling of the non-holonomic inhomogeneity function. These two deformations are further found to be not gauge-equivalent in general, following the Lax formalism of the nonlinear Schrödinger equation. However, asymptotically they converge for localized solutions as expected. Similar conditional correspondence of nonholonomic deformation with a non-integrable deformation, namely, due to local scaling of the amplitude of the nonlinear Schrödinger equation is further obtained.

nlin.SI↗

arXiv2019

On Coupled Delayed Van der Pol-Duffing oscillators

We investigate the dynamics of a delay differential coupled Duffing-Van der Pol oscillator equation. Using the Lindstedt's method, we derive the in-phase mode solutions and then obtain the slow flow equations governing the stability of the in-phase mode by employing the two variable perturbation method. We solve the slow flow equations using series expansion and obtain conditions for Hopf bifurcation and studied stability of the in-phase mode. Finally, we studied stability and bifurcations of the origin. Our interest in this system is due to the fact that it is related to the coupled laser oscillators.

nlin.CD↗

arXiv2019

Reduction and Hamiltonian aspects of a model for virus-tumour interaction in oncolytic virotherapy

We analyse the Hamiltonian structure of a system of first-order ordinary differential equations used for modeling the interaction of an oncolytic virus with a tumour cell population. The analysis is based on the existence of a Jacobi Last Multiplier for the system and a time dependent first integral. For suitable conditions on the model parameters this allows for the reduction of the problem to a planar system of equations for which the time dependent Hamiltonian flows are described. The geometry of the Hamiltonian flows are finally investigated using the symplectic and cosymplectic methods.

q-bio.PE↗

arXiv2019

On purely nonlinear oscillators generalizing an isotonic potential

In this paper we consider a nonlinear generalization of the isotonic oscillator in the same spirit as one considers the generalization of the harmonic oscillator with a truly nonlinear restoring force. The corresponding potential being asymmetric we invoke the symmetrization principle and construct a symmetric potential in which the period function has the same value as in the original asymmetric potential. The period function is amplitude dependent and expressible in terms of the hypergeometric function and reduces to $2π$ when $α=1$, i.e., corresponding to the special case of an isotonic oscillator.

physics.class-ph↗

arXiv2019

On the quest for generalized Hamiltonian descriptions of $3D$-flows generated by curl of a vector potential

We study Hamiltonian analysis of three-dimensional advection flow $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ of incompressible nature $\nabla \cdot {\bf v} ={\bf 0}$ assuming that dynamics is generated by the curl of a vector potential $\mathbf{v} = \nabla \times \mathbf{A}$. More concretely, we elaborate Nambu-Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity $\mathbf{A} \cdot \nabla \times \mathbf{A}$. We present an example (satisfying $\mathbf{A} \cdot \nabla \times \mathbf{A} \neq 0$) which can be written as in the form of Nambu-Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying $\mathbf{A} \cdot \nabla \times \mathbf{A} = 0$) which we cannot able to write it in the form of a Nambu-Hamiltonian or bi-Hamiltonian system. On the hand, this second example can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations $\dot{x}_i = - ε_{ijk}{\partial η_j}/{\partial x_k}$.

math-ph↗

arXiv2019

Ihara Zeta Entropy

In this article, we introduce an entropy based on the formal power series expansion of the Ihara Zeta function. We find a number of inequalities based on the values of the Ihara zeta function. These new entropies are applicable in symbolic dynamics and the dynamics of billiards.

math-ph↗

arXiv2019

Sharma-Mittal Quantum Discord

We demonstrate a generalization of quantum discord using a generalized definition of von-Neumann entropy, which is Sharma-Mittal entropy; and the new definition of discord is called Sharma-Mittal quantum discord. Its analytic expressions are worked out for two qubit quantum states as well as Werner, isotropic, and pointer states as special cases. The R{é}nyi, Tsallis, and von-Neumann entropy based quantum discords can be expressed as limiting cases for of Sharma-Mittal quantum discord. We also numerically compare all these discords and entanglement negativity.

quant-ph↗

arXiv2019

Nonlocal transformations of the Generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systems

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov-Lewis invariant for a time dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Liénard equation, x + f(x)x_ + g(x) = 0, is mapped to that of the linear harmonic oscillator equation we obtain a relation between the functions f(x) and g(x) which is exactly similar to the condition derived in the context of isochronicity of the Liénard equation.

nlin.SI↗

arXiv2019

Branched Hamiltonians and time translation symmetry breaking in equations of the Lienard type

Shapere and Wilczek ( Phys. Rev. Lett. 109, 160402 and 200402 (2012)) have recently described certain singular Lagrangian systems which display spontaneous breaking of time translation symmetry. We begin by considering the standard Lienard equation for which a Lagrangian is constructed by using the method of Jacobi Last Multiplier. The velocity dependance of the Lagrangian is such that the momentum may exhibit multivaluedness thereby leading to the so called branched Hamiltonian. Next with a quadratic velocity dependance in the Lienard equation one can construct a Hamiltonian description involving a position dependent mass. We compute the Lagrangian and Hamiltonian of this system and show that the canonical Hamiltonian is single valued . However, we find that up to a constant shift, the square of this Hamiltonian describes systems giving rise to spontaneous time translation symmetry breaking provided the potential function is negative

nlin.SI↗

arXiv2019

Study of Non-Holonomic Deformations of Non-local integrable systems belonging to the Nonlinear Schrodinger family

The non-holonomic deformations of non-local integrable systems belonging to the Nonlinear Schrodinger family are studied using the Bi-Hamiltonian formalism as well as the Lax pair method. The non-local equations are first obtained by symmetry reductions of the variables in the corresponding local systems. The bi-Hamiltonian structures of these equations are explicitly derived. The bi-Hamiltonian structures are used to obtain the non-holonomic deformation following the Kupershmidt ansatz. Further, the same deformation is studied using the Lax pair approach and several properties of the deformation discussed. The process is carried out for coupled non-local Nonlinear Schrodinger and Derivative Nonlinear Schrodinger (Kaup Newell) equations. In case of the former, an exact equivalence between the deformations obtained through the bi-Hamiltonian and Lax pair formalisms is indicated

nlin.SI↗

arXiv2019

Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwell's equations in the curved spacetime. In particular, we see in very concrete terms how the magnetolectric susceptibility enters the Jacobi-Maupertuis-Randers-Finsler function.

gr-qc↗

arXiv2019

From Sharma-Mittal to von-Neumann Entropy of a Graph

In this article, we introduce the Sharma-Mittal entropy of a graph, which is a generalization of the existing idea of the von-Neumann entropy. The well-known R{é}nyi, Thallis, and von-Neumann entropies can be expressed as limiting cases of Sharma-Mittal entropy. We have explicitly calculated them for cycle, path, and complete graphs. Also, we have proposed a number of bounds for these entropies. In addition, we have also discussed the entropy of product graphs, such as Cartesian, Kronecker, Lexicographic, Strong, and Corona products. The change in entropy can also be utilized in the analysis of growing network models (Corona graphs), useful in generating complex networks.

cs.DM↗

arXiv2018

Superintegrable systems with position dependent mass: master symmetry and action-angle methods

We consider the issue of deriving superintegrable systems with position dependent mass (PDM) in two dimensions from certain known superintegrable systems using the recently introduced method of master symmetries and complex factorization by M. Ranada \cite{Rana1,Rana2,Rana3,Rana4}. We introduce a noncanonical transformation to map the Hamiltonian of the PDM systems to that of ordinary unit mass systems. We observe a duality between these systems. We also study Tsiganov's method \cite{Tsiganov1,Tsiganov2,Tsiganov3,Tsiganov4} to derive polynomial integrals of motion using addition theorems for the action-angle variables using famous Chebyshev's theorem on binomial differentials. We compare Tsiganov's method of generating an additional integral of motion with that of Ranada's master symmetry method.

nlin.SI↗

arXiv2018

Dynamical systems of null geodesics and solutions of Tomimatsu-Sato 2

We have studied optical metrics via null geodesics and optical-mechanical formulation of classical mechanics, and described the geometry and optics of mechanical systems with drag dependent quadratically on velocity. Then we studied null geodesics as a central force system, deduced the related Binet's equation applied the analysis to other solutions of Einstein's equations in spherically symmetric spaces, paying special attention to the Tomimatsu-Sato metric. Finally, we examined the dualities between different systems arising from conformal transformations that preserve the Jacobi metric.

physics.gen-ph↗

arXiv2018

Inhomogeneous Heisenberg Spin Chain and Quantum Vortex Filament as Non-Holonomically Deformed NLS Systems

Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits, corresponding to the inhomogeneous XXX Heisenberg spin chain [Balakrishnan, J. Phys. C 15, L1305 (1982)] and that of a thin vortex filament moving in a superfluid with drag [Shivamoggi, Eur. Phys. J. B 86, 275 (2013) 86; Van Gorder, Phys. Rev. E 91, 053201 (2015)], are shown to be particular non-holonomic deformations (NHDs) of the standard NLS system involving generalized parameterizations. Crucially, such NHDs of the NLS system are restricted to specific spectral orders that exactly complements NHDs of the original physical systems. The specific non-holonomic constraints associated with these integrodifferential generalizations additionally posses distinct semi-classical signature.

math-ph↗

arXiv2018

On a reduction of the generalized Darboux-Halphen system

The equations for the general Darboux-Halphen system obtained as a reduction of the self-dual Yang-Mills can be transformed to a third-order system which resembles the classical Darboux-Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux-Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.

nlin.SI↗

arXiv2017

Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy

The hierarchy of equations belonging to two different but related integrable systems, the Nonlinear Schrödinger and its derivative variant, DNLS are subjected to two distinct deformation procedures, viz. quasi-integrable deformation (QID) that generally do not reserve the integrability, only asymptotically integrable, and non-holonomic deformation (N HD) that does. QID is carried out generically for the NLS hierarchy while for the DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of the family. No QI anomaly is observed at the level of EOMs which suggests that at that level the QID may be identified as some integrable deformation. NHD is applied to the NLS hierarchy generally as well as with the specific focus on the NLS equation itself and the coupled KdV type NLS equation. For the DNLS hierarchy, the Kaup-Newell(KN) and Chen-Lee-Liu (CLL) equations are deformed non-holonomically and subsequently, different aspects of the results are discussed.

math-ph↗

arXiv2017

On Time-dependent Hamiltonian Realizations of Planar and Nonplanar Systems

In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our constructions with various examples.

math.DG↗