Avinash Khare

Indian Institute of Science Education and Research, Pune, Poona, Maharashtra, India

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Publications (166)246.77 Total impact

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    ABSTRACT: We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction in the presence of external forces as well as damping of the form $ f(x,t) - i \mu \gamma^0 \Psi$, where both $f$ and $\Psi$ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also affected, because the soliton position oscillates around a mean trajectory. We then compare the results of the variational approximation with numerical simulations of the NLD equation, and find a good agreement, if we take into account a certain linear excitation with specific wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved. We also solve explicitly the CC equations of the variational approximation in the non-relativistic regime for a homogeneous external force and obtain excellent agreement with the numerical solution of the CC equations.
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    ABSTRACT: The exact bound state spectrum of rationally extended shape invariant real as well as $PT$ symmetric complex potentials are obtained by using potential group approach. The generators of the potential groups are modified by introducing a new operator $U (x, J_3 \pm 1/2 )$ to express the Hamiltonian corresponding to these extended potentials in terms of Casimir operators. Connection between the potential algebra and the shape invariance is elucidated.
    Annals of Physics 02/2015; 359. DOI:10.1016/j.aop.2015.04.002 · 3.07 Impact Factor
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    ABSTRACT: Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state 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.
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    ABSTRACT: In the present work, we explore the case of a general PT-symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schr\"odinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations.
    International Journal of Theoretical Physics 09/2014; DOI:10.1007/s10773-014-2429-6 · 1.19 Impact Factor
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    Avinash Khare, T. Kanna, K. Tamilselvan
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    ABSTRACT: We consider (2+1) and (1+1) dimensional long-wave short-wave 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; 378(42). DOI:10.1016/j.physleta.2014.09.006 · 1.63 Impact Factor
  • Avinash Khare, Ivan C Christov, Avadh Saxena
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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 second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order 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 higher-order field theories have kink solutions with algebraically decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order 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 higher-order 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.
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    Avinash Khare, Avadh Saxena
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    ABSTRACT: For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schr\"odinger equation (NLSE), nonlocal Ablowitz-Ladik (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.
    Journal of Mathematical Physics 05/2014; 56(3). DOI:10.1063/1.4914335 · 1.18 Impact Factor
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    Avinash Khare, Avadh Saxena
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    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 KdV-MKdV system, a mixed quadratic-cubic nonlinear Schr\"odinger equation, the Ablowitz-Ladik 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.
    Journal of Mathematical Physics 03/2014; 55(3). DOI:10.1063/1.4866781 · 1.18 Impact Factor
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    Avinash Khare, Ivan C. Christov, Avadh Saxena
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    ABSTRACT: We obtain exact solutions for kinks in $\phi^{8}$, $\phi^{10}$ and $\phi^{12}$ field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order 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 higher-order field theories have kink solutions with algebraically-decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order $\phi^4$ and $\phi^6$ theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the co-existence 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 higher-order field theories have specific cases in which only nonlinear phonons exist. For the $\phi^{10}$ field theory, which is a quasi-exactly 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.
  • Avinash Khare, Avadh Saxena
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    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):2761-2765. DOI:10.1016/j.physleta.2013.08.015 · 1.63 Impact Factor
  • Avinash Khare
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    ABSTRACT: In this article I shall present a brief review of the hundred-year young Bohr model of the atom. In particular, I will first introduce the Thomson and the Rutherford models of atoms, their shortcomings and then discuss in some detail the development of the atomic model by Niels Bohr. Further, I will mention its refinements at the hand of Sommerfeld and also its shortcomings. Finally, I will discuss the implication of this model in the development of quantum mechanics.
    Resonance 10/2013; 18(10). DOI:10.1007/s12045-013-0115-2
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    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; 379(3). DOI:10.1016/j.physleta.2014.11.009 · 1.63 Impact Factor
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    ABSTRACT: We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schr\"odinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.
    Physical Review A 07/2013; 88(3). DOI:10.1103/PhysRevA.88.032108 · 2.99 Impact Factor
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    ABSTRACT: We study the statistical mechanics of the one-dimensional 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 one-dimensional 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(4-1):044901. DOI:10.1103/PhysRevE.87.044901 · 2.33 Impact Factor
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    ABSTRACT: We consider the recently discovered, one parameter family of exactly solvable shape invariant potentials which are isospectral to the generalized P\"oschl-Teller 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). DOI:10.1016/j.physletb.2013.05.036 · 6.02 Impact Factor
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    Avinash Khare, Avadh Saxena
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    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.
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    ABSTRACT: The scattering amplitude for the recently discovered exactly solvable shape invariant potential, which is isospectral to the generalized P\"oschl-Taylor 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. DOI:10.1016/j.aop.2013.01.006 · 3.07 Impact Factor
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    ABSTRACT: We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{\kappa+1} ({\bPsi} \Psi)^{\kappa+1}$ in the presence of various external electromagnetic fields. Starting from the exact solutions for the unforced problem we study the behavior of solitary wave 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 time independent 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 similar to the nonlinear Schr{\"o}dinger equation (NLSE) that a sufficient dynamical condition for instability to arise is that $ dP(t)/d \dq(t) < 0$. Here $P(t)$ is the momentum of the solitary wave, and $\dq$ is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials we always have $ dP(t)/d \dq(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.
    Physical Review E 10/2012; 86(4-2):046602. DOI:10.1103/PhysRevE.86.046602 · 2.33 Impact Factor
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    R. P. Malik, Avinash Khare
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    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 X-Y plane under the influence of a magnetic field in the Z-direction. 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. DOI:10.1016/j.aop.2013.03.015 · 3.07 Impact Factor
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    ABSTRACT: Quantum Hamilton-Jacobi 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). DOI:10.1007/s12043-013-0558-8 · 0.72 Impact Factor

Publication Stats

4k Citations
246.77 Total Impact Points

Institutions

  • 2010–2015
    • Indian Institute of Science Education and Research, Pune
      Poona, Maharashtra, India
  • 1987–2014
    • Los Alamos National Laboratory
      • Center for Nonlinear Studies
      Los Alamos, California, United States
  • 2013
    • Banaras Hindu University
      • Department of Physics
      Vārānasi, Uttar Pradesh, India
  • 1990–2011
    • Institute of Physics, Bhubaneswar
      Bhubaneswar, Orissa, India
  • 1986–2002
    • University of Illinois at Chicago
      • Department of Physics
      Chicago, IL, United States
  • 1997
    • University of Calcutta
      • Department of Applied Mathematics
      Kolkata, Bengal, India
  • 1993
    • Université de Montréal
      Montréal, Quebec, Canada