arXiv subjects

B. Talukdar

Publications and source records attributed to B. Talukdar.

14 recordsLinked to original sources

arXiv2017

On the correlation measure of two-electron systems

We make use of a Hylleraas-type wave function to derive an exact analytical model to quantify correlation in two-electron atomic/ionic systems and subsequently employ it to examine the role of inter-electronic repulsion in affecting (i) the bare (uncorrelated) single-particle position- and momentum-space charge distributions and (ii) corresponding Shannon's information entropies. The results presented for the first five members in the helium iso-electronic sequence, on the one hand, correctly demonstrate the effect of correlation on bare charge distributions and, on the other hand, lead us to some important results for the correlated and uncorrelated values of the entropies. These include the limiting behavior of the correlated entropy sum (sum of position- and momentum-space entropies) and geometrical realization for the variation of information entropies as a function of Z. We suggest that, rather than the entropy sum, individual entropies should be regarded as better candidates for the measure of correlation.

quant-ph↗

arXiv2015

Solitons, compactons and undular bores in Benjamin-Bona-Mahony-like systems

We examine the effect of dissipation on traveling waves in nonlinear dispersive systems modeled by Benjamin- Bona- Mahony (BBM)-like equations. In the absence of dissipation the BBM-like equations are found to support soliton and compacton/anticompacton solutions depending on whether the dispersive term is linear or nonlinear. We study the influence of increasing nonlinearity of the medium on the soliton- and compacton dynamics. The dissipative effect is found to convert the solitons either to undular bores or to shock-like waves depending on the degree of nonlinearity of the equations. The anticompacton solutions are also transformed to undular bores by the effect of dissipation. But the compactons tend to vanish due to viscous effects. The local oscillatory structures behind the bores and/or shock-like waves in the case of solitons and anticompactons are found to depend sensitively both on the coefficient of viscosity and solution of the unperturbed problem.

nlin.SI↗

arXiv2009

One-way coupled Van der Pol system

The equation of the Van der Pol oscillator, being characterized by a dissipative term, is non-Lagrangian. Appending an additional degree of freedom we bring the equation in the frame of action principle and thus introduce a one-way coupled system. As with the Van der Pol oscillator, the coupled system also involves only one parameter that controls the dynamics. The response system is described by a linear differential equation coupled nonlinearly to the drive system. In the linear approximation the equations of our coupled system coincide with those of the Bateman dual system (a pair of damped and anti-damped harmonic oscillators). The critical point of damped and anti-damped oscillators are stable and unstable for all physical values of the frictional coefficient $μ$. Contrarily, the critical points of the drive- (Van der Pol) and response systems depend crucially on the values of $μ$. These points are unstable for $μ> 0$ while the critical point of the drive system is stable and that of the response system is unstable for $μ< 0$. The one-way coupled system exhibits bifurcations which are different from those of the uncoupled Van der Pol oscillator. Our system is chaotic and we observe phase synchronization in the regime of dynamic chaos only for small values of $μ$.

nlin.SI↗

arXiv2009

Modified KdV hierarchy : Lax pair representation and bi-Hamiltonian structure

We consider equations in the modified KdV (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV equations. We also show that the complex modified KdV (cmKdV) equation follows from the action principle to have a Lagrangian representation. This representation not only provides a basis to write the cmKdV equation in the canonical form endowed with an appropriate Poisson structure but also help us construct a semianalytical solution of it. The solution obtained by us may serve as a useful guide for purely numerical routines which are currently being used to solve the cmKdV eqution.

nlin.SI↗

arXiv2008

On the supersymmetric nonlinear evolution equations

Supersymmetrization of a nonlinear evolution equation in which the bosonic equation is independent of the fermionic variable and the system is linear in fermionic field goes by the name B-supersymmetrization. This special type of supersymmetrization plays a role in superstring theory. We provide B-supersymmetric extension of a number of quasilinear and fully nonlinear evolution equations and find that the supersymmetric system follows from the usual action principle while the bosonic and fermionic equations are individually non Lagrangian in the field variable. We point out that B-supersymmetrization can also be realized using a generalized Noetherian symmetry such that the resulting set of Lagrangian symmetries coincides with symmetries of the bosonic field equations. This observation provides a basis to associate the bosonic and fermionic fields with the terms of bright and dark solitons. The interpretation sought by us has its origin in the classic work of Bateman who introduced a reverse-time system with negative friction to bring the linear dissipative systems within the framework of variational principle.

nlin.SI↗

arXiv2007

Lagrangian Approach to Dispersionless KdV Hierarchy

We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.

nlin.SI↗

arXiv2007

Bright solitons in asymmetrically trapped Bose-Einstein condensate

We study the dynamics of bright solitons in a Bose-Einstein condensate (BEC) confined in a highly asymmetric trap. While working within the f ramework of a variational approach we carry out the stability analysis o f BEC solitons against collapse. When the number of atoms in the soliton exceeds a critical number $N_c$, it undergoes the so called primary col lapse. We find an analytical expression for $N_c$ in terms of appropriat e experimental quantities that are used to produce and confine the conde nsate. We further demonstrate that, in the geometry of the problem consi dered, the width of the soliton varies inversely as the number of consti tuent atoms.

nlin.SI↗

arXiv2007

Embedded soliton solutions : A variational study

We use a variational method to construct soliton solutions for systems characterized by opposing dispersion and competing nonlinearities at fundamental and second harmonics. We show that both ordinary and embedded solitons tend to gain energy when the second harmonic field becomes weaker than the first harmonic field.

nlin.SI↗

arXiv2007

Inverse problem of variational calculus for nonlinear evolution equations

We couple a nonlinear evolution equation with an associated one and derive the action principle. This allows us to write the Lagrangian density of the system in terms of the original field variables rather than Casimir potentials. We find that the corresponding Hamiltonian density provides a natural basis to recast the pair of equations in the canonical form. Amongst the case studies presented the KdV and modified KdV pairs exhibit bi-Hamiltonian structure and allow one to realize the associated fields in physical terms

nlin.SI↗

arXiv2006

Matter-wave bright solitons of $^7$Li gas in an expulsive potential

A Lagrangian based method is used to derive an analytical model for studying the dynamics of matter-wave bright soliton created in a harmonic potential which is attractive in the transverse direction and expulsive in the longitudinal direction. By means of sech trial functions and a Ritz optimization procedure, evolution eqautions are constructed for width, amplitude and nonlinear frequency chirp of the propagating soliton of the atomic condensate. Our eqaution for the width is an exact agreement with that of Carr and Castin $[ Phys. Rev. A {\bf{66}}, 063602 (2002)]$, obtained by more detailed analysis. In agreement with the experiment of Paris group $[ Science {\bf{296}}, 1290 (2002)]$, the expulsive potential is found to cause the soliton to explode for $N|a_s|=0.98$, $N$ being the number of atoms in the condensate and $a_s$, the scattering length of the atom-atom interaction.

nlin.SI↗

arXiv2006

Remarks on the conserved densities of the Camassa-Holm equation

It is pointed out that the higher-order symmetries of the Camassa-Holm (CH) equation are nonlocal and nonlocality poses problems to obtain higher-order conserved densities for this integrable equation (J. Phys. A: Math. Gen. 2005, {\bf 38} 869-880). This difficulty is circumvented by defining a nolinear hierarchy for the CH equation and an explicit expression is constructed for the nth-order conserved density.

nlin.SI↗

arXiv2006

Embedded solitons in the third-order nonlinear Schrödinger equation

We derive a straightforward variational method to construct embedded soliton solutions of the third-order nonlinear Schödinger equation and analytically demonstrate that these solitons exist as a continuous family. We argue that a particular embedded soliton when perturbed may always relax to the adjacent one so as to make it fully stable.

nlin.SI↗

arXiv2006

On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy

A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type recursion operator is then employed to construct a hierarchy of Lagrangian equations. It is explicitly demonstrated that the constructed system of equations has a Lax representation and two compatible Hamiltonian structures. The homogeneous balance method is used to derive analytic soliton solutions of the third- and fifth-order equations.

nlin.SI↗

arXiv1998

Dimensional versus cut-off renormalization and the nucleon-nucleon interaction

The role of dimensional regularization is discussed and compared with that of cut-off regularization in some quantum mechanical problems with ultraviolet divergence in two and three dimensions with special emphasis on the nucleon-nucleon interaction. Both types of renormalizations are performed for attractive divergent one- and two-term separable potentials, a divergent tensor potential, and the sum of a delta function and its derivatives. We allow energy-dependent couplings, and determine the form that these couplings should take if equivalence between the two regularization schemes is to be enforced. We also perform renormalization of an attractive separable potential superposed on an analytic divergent potential.

nucl-th↗