Publications (165)218.68 Total impact
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ABSTRACT: Rationally extended shape invariant potentials in arbitrary Ddimensions are obtained by using point canonical transformation (PCT) method. The boundstate solutions of these exactly solvable potentials can be written in terms of X_m Laguerre or X_m Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.12/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We consider (2+1) and (1+1) dimensional longwave shortwave resonance interaction systems. We construct an extensive set of exact periodic solutions of these systems in terms of Lam\'e polynomials of order one and two. The periodic solutions are classified into three categories as similar, mixed, superposed elliptic solutions. We also discuss the hyperbolic solutions as limiting cases.Physics Letters A 09/2014; · 1.63 Impact Factor 
Article: Successive phase transitions and kink solutions in ϕ^{8}, ϕ^{10}, and ϕ^{12} field theories.
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ABSTRACT: We obtain exact solutions for kinks in ϕ^{8}, ϕ^{10}, and ϕ^{12} field theories with degenerate minima, which can describe a secondorder phase transition followed by a firstorder one, a succession of two firstorder phase transitions and a secondorder phase transition followed by two firstorder phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higherorder field theories have kink solutions with algebraically decaying tails and also asymmetric cases with mixed exponentialalgebraic tail decay, unlike the lowerorder ϕ^{4} and ϕ^{6} theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the coexistence of up to three distinct kinks (for a ϕ^{12} potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higherorder field theories have specific cases in which only nonlinear phonons are allowed. For the ϕ^{10} field theory, which is a quasiexactly solvable model akin to ϕ^{6}, we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit.Physical review. E, Statistical, nonlinear, and soft matter physics. 08/2014; 90(21):023208. 
Article: Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PTSymmetric Nonlinear Equations
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ABSTRACT: For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schr\"odinger equation (NLSE), nonlocal AblowitzLadik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only $\dn(x,m)$ and $\cn(x,m)$ but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lam\'e polynomials of order 1, but it also admits solutions in terms of Lam\'e polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.05/2014;  [Show abstract] [Hide abstract]
ABSTRACT: For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions $\cn(x,m)$ and $\dn(x,m)$ with modulus $m$, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schr\"odinger equation, MKdV, a mixed KdVMKdV system, a mixed quadraticcubic nonlinear Schr\"odinger equation, the AblowitzLadik equation, the saturable nonlinear Schr\"odinger equation, $\lambda \phi^4$, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of $\dn^2(x,m)$, it also admits solutions in terms of $\dn^2(x,m) \pm \sqrt{m} \cn(x,m) \dn(x,m)$, even though $\cn(x,m) \dn(x,m)$ is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.03/2014; 55(3).  [Show abstract] [Hide abstract]
ABSTRACT: We obtain exact solutions for kinks in $\phi^{8}$, $\phi^{10}$ and $\phi^{12}$ field theories with degenerate minima, which can describe a secondorder phase transition followed by a firstorder one, a succession of two firstorder phase transitions and a secondorder phase transition followed by two firstorder phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higherorder field theories have kink solutions with algebraicallydecaying tails and also asymmetric cases with mixed exponentialalgebraic tail decay, unlike the lowerorder $\phi^4$ and $\phi^6$ theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the coexistence of up to three distinct kinks (for a $\phi^{12}$ potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higherorder field theories have specific cases in which only nonlinear phonons exist. For the $\phi^{10}$ field theory, which is a quasiexactly solvable (QES) model akin to $\phi^6$, we are also able to obtain analytically the classical free energy as well as the probability distribution function in the thermodynamic limit.02/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We demonstrate a kind of linear superposition for a large number of nonlinear equations which admit elliptic function solutions, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions cn(x,m) and dn(x,m), then it also admits solutions in terms of their sum as well as difference, i.e. dn(x,m)±mcn(x,m). Further, we also show that whenever a nonlinear equation admits a solution in terms of dn2(x,m), it also has solutions in terms of dn2(x,m)±mcn(x,m)dn(x,m) even though cn(x,m)dn(x,m) is not a solution of that nonlinear equation. Finally, we obtain similar superposed solutions in coupled theories.Physics Letters A 11/2013; 377(39):27612765. · 1.63 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the rationally extended exactly solvable Eckart potentials which exhibit extended shape invariance property. These potentials are isospectral to the conventional Eckart potential. The scattering amplitude for these rationally ex tended potentials is calculated analytically for the generalized mth (m = 1, 2, 3, ...) case by considering the asymptotic behavior of the scattering state wave functions which are written in terms of some new polynomials related to the Jacobi polyno mials. As expected, in the m = 0 limit, this scattering amplitude goes over to the scattering amplitude for the conventional Eckart potential.Physics Letters A. 09/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We study the statistical mechanics of the onedimensional discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. 84, 3740 (2000)] regarding the statistical mechanics of the onedimensional DNLS equation with a cubic nonlinearity. As in this earlier study, we identify the spontaneous creation of localized excitations with a discontinuity in the partition function. The fact that this phenomenon is retained in the saturable DNLS is nontrivial, since in contrast to the cubic DNLS whose nonlinear character is enhanced as the excitation amplitude increases, the saturable DNLS, in fact, becomes increasingly linear as the excitation amplitude increases. We explore the nonlinear dynamics of this phenomenon by direct numerical simulations.Physical Review E 04/2013; 87(41):044901. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the recently discovered, one parameter family of exactly solvable shape invariant potentials which are isospectral to the generalized P\"oschlTeller potential. By explicitly considering the asymptotic behaviour of the Xm Jacobi polynomials associated with this system (m = 1, 2, 3, ...), the scattering amplitude for the one parameter family of potentials is calculated explicitly.Physics Letters B 03/2013; 723(4). · 4.57 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We demonstrate a kind of linear superposition for a large number of nonlinear equations, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions $\cn(x,m)$ and $\dn(x,m)$, then it also admits solutions in terms of their sum as well as difference, i.e. $\dn(x,m) \pm \sqrt{m}\, \cn(x,m)$. Further, we also show that whenever a nonlinear equation admits a solution in terms of $\dn^2(x,m)$, it also has solutions in terms of $\dn^2(x,m) \pm \sqrt{m}\, \cn(x,m)\, \dn(x,m)$ even though $\cn(x,m)\, \dn(x,m)$ is not a solution of that nonlinear equation. Finally, we obtain similar superposed solutions in coupled theories.02/2013;  [Show abstract] [Hide abstract]
ABSTRACT: The scattering amplitude for the recently discovered exactly solvable shape invariant potential, which is isospectral to the generalized P\"oschlTaylor potential, is calculated explicitly by considering the asymptotic behavior of the $X_{1}$ Jacobi exceptional polynomials associated with this system.Annals of Physics 12/2012; 331. · 3.32 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider nonlinear Dirac equations (NLDE's) in the 1+1 dimension with scalarscalar selfinteraction g^{2}/κ+1(Ψ[over ¯]Ψ)^{κ+1} in the presence of various external electromagnetic fields. We find exact solutions for special external fields and we study the behavior of solitarywave solutions to the NLDE in the presence of a wide variety of fields in a variational approximation depending on collective coordinates which allows the position, width, and phase of these waves to vary in time. We find that in this approximation the position q(t) of the center of the solitary wave obeys the usual behavior of a relativistic point particle in an external field. For timeindependent external fields, we find that the energy of the solitary wave is conserved but not the momentum, which becomes a function of time. We postulate that, similarly to the nonlinear Schrödinger equation (NLSE), a sufficient dynamical condition for instability to arise is that dP(t)/dq[over ̇](t)<0. Here P(t) is the momentum of the solitary wave, and q[over ̇] is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials that we always have dP(t)/dq[over ̇](t)>0, so, when instabilities do occur, they are due to a different source. We investigate the accuracy of our variational approximation using numerical simulations of the NLDE and find that, when the forcing term is small and we are in a regime where the solitary wave is stable, that the behavior of the solutions of the collective coordinate equations agrees very well with the numerical simulations. We found that the time evolution of the collective coordinates of the solitary wave in our numerical simulations, namely the position of the average charge density and the momentum of the solitary wave, provide good indicators for when the solitary wave first becomes unstable. When these variables stop being smooth functions of time (t), then the solitary wave starts to distort in shape.Physical Review E 10/2012; 86(42):046602. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We demonstrate the existence of a novel set of discrete symmetries in the context of N = 2 supersymmetric (SUSY) quantum mechanical model with a potential function f(x) that is a generalization of the potential of the 1D SUSY harmonic oscillator. We perform the same exercise for the motion of a charged particle in the XY plane under the influence of a magnetic field in the Zdirection. We derive the underlying algebra of the existing continuous symmetry transformations (and corresponding conserved charges) and establish its relevance to the algebraic structures of the de Rham cohomological operators of differential geometry. We show that the discrete symmetry transformations of our present general theories correspond to the Hodge duality operation. Ultimately, we conjecture that any arbitrary N = 2 SUSY quantum mechanical system can be shown to be a tractable model for the Hodge theory.Annals of Physics 08/2012; 334. · 3.32 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Quantum HamiltonJacobi formalism is used to give a proof for Gozzi's criterion that for eigenstates of the supersymmetric partners, corresponding to same energy, the difference in the number of nodes is equal to one when supersymmetry (SUSY) is unbroken and is zero when SUSY is broken. We show that this proof is also applicable to the case, where isospectral deformation is involved.Pramana 07/2012; 81(2). · 0.56 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalarscalar selfinteraction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(i(kx+θ))δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by twodimensional projections of its trajectory in the fourdimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.Physical Review E 04/2012; 85(4 Pt 2):046607. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We start with quasiexactly solvable (QES) Hermitian (and hence real) as well as complex PTinvariant, double sinhGordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using antiisospectral transformations, we obtain Hermitian as well as PTinvariant complex QES periodic potentials. We study in detail the various properties of the corresponding BenderDunne polynomials.12/2011; 
Article: Solutions of Several Coupled Discrete Models in terms of Lame Polynomials of Arbitrary Order
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ABSTRACT: Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lam\'e polynomials of arbitrary order. The models discussed are (i) coupled Salerno model, (ii) coupled AblowitzLadik model, (iii) coupled $\phi^4$ model, and (iv) coupled $\phi^6$ model. In all these cases we show that the coefficients of the Lam\'e polynomials are such that the Lam\'e polynomials can be reexpressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.Pramana 11/2011; 79(3). · 0.56 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We show that a twodimensional generalized cubic–quintic Ablowitz–Ladik lattice admits periodic solutions that can be expressed in analytical form. The framework for the stability analysis of these solutions is developed and applied to reveal the intricate stability behavior of this nonlinear system. We examine the stability of these solutions and find that staggering along one of the two dimensions is important for stability.Physica Scripta 11/2011; 84(6):065001. · 1.03 Impact Factor 
Article: Solutions of Several Coupled Discrete Models in terms of Lame Polynomials of Order One and Two
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ABSTRACT: Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lame polynomials of order one and two. Some of the models discussed are (i) coupled Salerno model, (ii) coupled AblowitzLadik model, (iii) coupled saturated discrete nonlinear Schrodinger equation, (iv) coupled phi4 model, and (v) coupled phi6 model. Furthermore, we show that most of these coupled models in fact also possess an even broader class of exact solutions.Pramana 10/2011; · 0.56 Impact Factor
Publication Stats
3k  Citations  
218.68  Total Impact Points  
Top Journals
Institutions

2010–2014

Indian Institute of Science Education and Research, Pune
Poona, Mahārāshtra, India 
Technical University of Denmark
 Department of Physics
København, Capital Region, Denmark


1987–2014

Los Alamos National Laboratory
 • Center for Nonlinear Studies
 • Theoretical Division
Los Alamos, California, United States


2012

University of Bayreuth
 Institute of Physics
Bayreuth, Bavaria, Germany


2010–2012

Santa Fe Institute
Santa Fe, New Mexico, United States


1990–2011

Institute of Physics, Bhubaneswar
Bhubaneswar, Orissa, India


2007–2010

Altai State Technical University
Barnaul, Altayskiy, Russia


1999–2002

Bhabha Atomic Research Centre
 Nuclear Physics Division
Mumbai, State of Maharashtra, India


1986–2002

University of Illinois at Chicago
 Department of Physics
Chicago, IL, United States


2000

S.N. Bose National Centre for Basic Sciences
Kolkata, Bengal, India


1987–1998

Visva Bharati University
 Department of Physics
Bolpur, Bengal, India


1993

Université de Montréal
Montréal, Quebec, Canada
