arXiv subjects

Partha Guha

Publications and source records attributed to Partha Guha.

At least 19 recordsLinked to original sources

arXiv2025

Quasi-harmonic spectra from branched Hamiltonians

We revisit the canonical quantization to assess the spectrum of the modified Emden equation $\ddot{x} + kx\dot{x} + \omega^2 x + \frac{k^2}{9}x^3 = 0$, which is an isochronous case of the Li\'enard-Kukles equation. While its classical isochronicity and canonical quantization, leading to polynomial solutions with an exactly-equispaced spectrum have been discussed earlier, including in the recent paper [Int. J. Theor. Phys. 64, 212 (2025)], the present study focuses on the quantization of its branched Hamiltonians. For small $k$, we show numerically that the resulting energy spectrum is no longer perfectly harmonic but only approximately equispaced, exhibiting quasi-harmonic behavior characterized by deviations from uniform spacing. Our numerical results are precisely validated by analytical calculations based on perturbation theory.

quant-ph↗

arXiv2025

Quasi-integrability from PT-symmetry

Parity and time-reversal (PT ) symmetry is shown as the natural cause of quasi-integrability of deformed integrable models, crucial to represent real physical systems as they posses various irregularities. The condition for asymptotic conservation of quasi-conserved charges appear as a direct consequence of the PT -symmetric phase of the system, ensuring definite PT -properties of the corresponding Lax pair as well as that of the anomalous contribution, consistent with the Wilson-loop criterion for integrability-like behavior. As a result, the quasi-deformed charge densities always acquire definite PT -properties suitable for the asymptotic conservation, as the Abelianization approach to construct them also preserves the definite PT -behavior of the Lax pair. This PT -symmetry based origin of quasi-conservation is general and has been demonstrated for quasi-deformations of multiple systems such as KdV, NLSE and non-local NLSE.

nlin.SI↗

arXiv2025

On novel Hamiltonian descriptions of some three-dimensional non-conservative systems

We present novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave-interaction problem and some well-known chaotic systems, namely, the Chen, L\"u, and Qi systems. We show that all of these systems can be described in a Hamiltonian framework in which the Poisson matrix $\mathcal{J}$ is supplemented by a resistance matrix $\mathcal{R}$. While such resistive-Hamiltonian systems are manifestly non-conservative, we construct higher-degree Poisson matrices via the Jordan product as $\mathcal{N} = \mathcal{J} \mathcal{R} + \mathcal{R} \mathcal{J}$, thereby leading to new bi-Hamiltonian systems. Finally, we discuss conformal Hamiltonian dynamics on Poisson manifolds and demonstrate that by appropriately choosing the underlying parameters, the reduced three-wave-interaction model as well as the Chen and L\"u systems can be described in this manner where the concomitant non-conservative part of the dynamics is described with the aid of the Euler vector field.

math-ph↗

arXiv2024

Generalized Li\'enard systems and isochronous connections

In this paper, we explore some classical and quantum aspects of the nonlinear Li\'enard equation $\ddot{x} + k x \dot{x} + \omega^2 x + (k^2/9) x^3 = 0$, where $x=x(t)$ is a real variable and $k, \omega \in \mathbb{R}$. We demonstrate that such an equation could be derived from an equation of the Levinson-Smith kind which is of the form $\ddot{z} + J(z) \dot{z}^2 + F(z) \dot{z} + G(z) = 0$, where $z=z(t)$ is a real variable and $\{J(z), F(z), G(z)\}$ are suitable functions to be specified. It can further be mapped to the harmonic oscillator by making use of a nonlocal transformation, establishing its isochronicity. Computations employing the Jacobi last multiplier reveal that the system exhibits a bi-Hamiltonian character, i.e., there are two distinct types of Hamiltonians describing the system. For each of these, we perform a canonical quantization in the momentum representation and explore the possibility of bound states. While one of the Hamiltonians is seen to exhibit an equispaced spectrum with an infinite tower of states, the other one exhibits branching but can be solved exactly in closed form for certain choices of the parameters.

quant-ph↗

arXiv2022

Non-holonomic and Quasi-integrable deformations of the AB Equations

For the first time, both non-holonomic and quasi-integrable deformations are obtained for the AB system of coupled equations. The AB system models geophysical and atmospheric fluid motion along with ultra-short pulse propagation in nonlinear optics and serves as a generalization of the well-known sine-Gordon equation. The non-holonomic deformation retains integrability subjected to higher-order differential constraints whereas the quasi-AB system, which is partially deviated from integrability, is characterized by an infinite subset of quantities (charges) that are conserved only asymptotically given the solution possesses definite space-time parity properties. Particular localized solutions to both these deformations of the AB system are obtained, some of which are qualitatively unique to the corresponding deformation, displaying similarities with physically observed excitations.

math-ph↗

arXiv2021

Relativistic formulation of curl force, relativistic Kapitza equation and trapping

In this present communication the relativistic formulation of the curl forces with saddle potentials has been performed. In particular, we formulated the relativistic version of the Kapitza equation. The dynamics and trapping phenomena of this equation have been studied both theoretically and numerically. The numerical results show interesting characteristics of the charged particles associated with the particle trapping and escaping in the relativistic domain. In addition, the relativistic generalization of the Kapitza equation associated with the monkey saddle has also been discussed.

math-ph↗

arXiv2021

Application of regularization maps to quantum mechanical systems in 2 and 3 dimensions

We extend the Levi-Civita (L-C) and Kustaanheimo-Stiefel (K-S) regularization methods that maps the classical system where a particle moves under the combined influence of $\frac{1}{r}$ and $r^2$ potentials to a harmonic oscillator with inverted sextic potential and interactions to corresponding quantum mechanical counterparts, both in 2 and 3 dimensions. Using the perturbative solutions of the Schrödinger equation of the later systems, we derive the eigen spectrum of the Hydrogen atom in presence of an additional harmonic potential. We have also obtained the mapping of a particle moving in the shifted harmonic potential to H-atom using Bohlin-Sundman transformation, for quantum regime. Exploiting this equivalence, the solution to the Schrödinger equation of the former is obtained from the solutions of the later.

math-ph↗

arXiv2021

Regularization of central forces with damping in two and three-dimensions

Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping co-efficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in 2 and 3-dimensions, we show that the regularized equation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin-Sudman transformation.

math-ph↗

arXiv2021

Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping

In this present communication the integrable modulation problem has been applied to study parametric extension of the Kapitza rotating shaft problem, which is a protypical example of curl force as formulated by Berry and Shukla in [J. Phys. A 45 305201 (2012)] associated with simple saddle potential. The integrable modulation problems yield parametric time dependent integrable systems. The Hamiltonian and first integrals of the linear and nonlinear parametric Kapitza equation (PKE) associated with simple and monkey saddle potentials have been given. The construction has been illustrated by choosing $ω(t) = a + b \cos t$ and that maps to Mathieu type equations, which yield Mathieu extension of PKE. We study the dynamics of these equations. The most interesting finding is the mixed mode of particle trapping and escaping via the heteroclinic orbits depicted with the parametric Mathieu-Kapitza equation which are absent in the case of non parametric cases.

math-ph↗

arXiv2021

$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations

We examine $3D$ flows $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ admitting vector identity $M\mathbf{v} = \nabla \times \mathbf{A}$ for a multiplier $M$ and a potential field $\mathbf{A}$. It is established that, for those systems, one can complete the vector field $\mathbf{v}$ into a basis fitting an $\mathfrak{sl}(2)$-algebra. Accordingly, in terms of covariant quantities, the structure equations determine a set of equations in Maurer-Cartan form. This realization permits one to obtain the potential field as well as to investigate the (bi-)Hamiltonian character of the system. The latter occurs if the system has a time-independent first integral. In order to exhibit the theoretical results on some concrete cases, three examples are provided, namely the Gulliot system, a system with a non-integrable potential, and the Darboux-Halphen system in symmetric polynomials.

math.DS↗

arXiv2021

On Quasi-integrable Deformation Scheme of The KdV System

We put forward a general approach to quasi-deform the KdV equation by deforming the corresponding Hamiltonian. Following the standard Abelianization process based on the inherent $sl(2)$ loop algebra, an infinite number of anomalous conservation laws are obtained, which yield conserved charges if the deformed solution has definite space-time parity. Judicious choice of the deformed Hamiltonian leads to an integrable system with scaled parameters as well as to a hierarchy of deformed systems, some of which possibly being quasi-integrable. As a particular case, one such deformed KdV system maps to the known quasi-NLS soliton in the already known weak-coupling limit, whereas a generic scaling of the KdV amplitude $u \to u^{1+\epsilon}$ also goes to possible quasi-integrability under an order-by-order expansion. Following a generic parity analysis of the deformed system, these deformed KdV solutions need to be parity-even for quasi-conservation which may be the case here following our analytical approach. From the established quasi-integrability of RLW and mRLW systems [Nucl. Phys. B 939 (2019) 49-94], which are particular cases of the present approach, exact solitons of the quasi-KdV system could be obtained numerically.

nlin.SI↗

arXiv2021

On Geometrical Couplings of Dissipation and Curl Forces

In this paper, we present several geometric ways to incorporate gyroscopic and dissipative forces to curl forces. We first present a proper metriplectic geometry. Then, using the Herglotz principle and generalized Euler-Lagrange equation, we propose a formulation of dissipative radial curl forces. Finally, we extend our result to azimuthal curl force using Galley's method, which leads to a natural formulation for Lagrangian and Hamiltonian dynamics of generic non-conservative systems.

math-ph↗

arXiv2021

On a geometric description of time dependent singular Lagrangians with applications to biological systems

We consider certain analytical features of a stochastic model that can explain among other things competition among species and simultaneous predation on the competing species from a geometric perspective which allows for a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which in turn allows for the construction of a Lagrangian. The Lagrangian is of singular nature so that construction of the Hamiltonian via a Legendre transformation is not possible. A Hamiltonian description of the model therefore requires the introduction of Dirac brackets. Explicit results are presented for the "Kill the winner" model and its reductions.

q-bio.PE↗

arXiv2021

Eisenhart lift and Randers-Finsler formulation for scalar field theory

We study scalar field theory as a generalization of point particle mechanics using the Polyakov action, and demonstrate how to extend Lorentzian and Riemannian Eisenhart lifts to the theory in a similar manner. Then we explore extension of the Randers-Finsler formulation and its principles to the Nambu-Goto action, and describe a Jacobi Lagrangian for it.

physics.class-ph↗

arXiv2020

A System of Billiard and Its Application to Information-Theoretic Entropy

In this article, we define an information-theoretic entropy based on the Ihara zeta function of a graph which is called the Ihara entropy. A dynamical system consists of a billiard ball and a set of reflectors correspond to a combinatorial graph. The reflectors are represented by the vertices of the graph. Movement of the billiard ball between two reflectors is represented by the edges. The prime cycles of this graph generate the bi-infinite sequences of the corresponding symbolic dynamical system. The number of different prime cycles of a given length can be expressed in terms of the adjacency matrix of the oriented line graph. It also constructs the formal power series expansion of Ihara zeta function. Therefore, the Ihara entropy has a deep connection with the dynamical system of billiards. As an information-theoretic entropy, it fulfils the generalized Shannon-Khinchin axioms. It is a weakly decomposable entropy whose composition law is given by the Lazard formal group law.

math-ph↗

arXiv2019

Construction of the classical time crystal Lagrangians from Sisyphus dynamics and duality description with the Liénard type equation

We explore the connection between the equations describing Sisyphus dynamics and the generic Liénard type or Liénard equation from the viewpoint of branched Hamiltonians. The former provides the appropriate setting for classical time crystal being derivable from a higher order Lagrangian. However it appears the equations of Sisyphus dynamics have a close relation with the Liénard-II equation when expressed in terms of the `velocity' variable. Another interesting feature of the equations of Sisyphus dynamics is the appearance of velocity dependent "mass function" in contrast to the more commonly encountered position dependent mass. The consequences of such mass functions seem to have connections to cosmological time crystals .

nlin.SI↗

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↗