Avinash Khare

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

Are you Avinash Khare?

Claim your profile

Publications (164)218.9 Total impact

  • Source
    Avinash Khare, T. Kanna, K. Tamilselvan
    [Show abstract] [Hide abstract]
    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; · 1.63 Impact Factor
  • Avinash Khare, Ivan C Christov, Avadh Saxena
    [Show abstract] [Hide abstract]
    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.
    Physical review. E, Statistical, nonlinear, and soft matter physics. 08/2014; 90(2-1):023208.
  • Avinash Khare, Avadh Saxena
    [Show abstract] [Hide abstract]
    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.
    05/2014;
  • Source
    Avinash Khare, Avadh Saxena
    [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 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.
    03/2014; 55(3).
  • Source
    Avinash Khare, Ivan C. Christov, Avadh Saxena
    [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 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.
    02/2014;
  • Avinash Khare, Avadh Saxena
    [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):2761-2765. · 1.63 Impact Factor
  • Source
    [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.
    09/2013;
  • Source
    [Show abstract] [Hide abstract]
    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. · 2.31 Impact Factor
  • Source
    [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\"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). · 4.57 Impact Factor
  • Avinash Khare, Avadh Saxena
    [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;
  • Source
    [Show abstract] [Hide abstract]
    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. · 3.32 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider nonlinear Dirac equations (NLDE's) in the 1+1 dimension with scalar-scalar self-interaction 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 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, 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(4-2):046602. · 2.31 Impact Factor
  • Source
    R. P. Malik, Avinash Khare
    [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 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. · 3.32 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    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). · 0.56 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction 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 two-dimensional projections of its trajectory in the four-dimensional 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
  • Source
    Avinash Khare, Bhabani Prasad Mandal
    [Show abstract] [Hide abstract]
    ABSTRACT: We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex PT-invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as PT-invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender-Dunne polynomials.
    12/2011;
  • Source
    Avinash Khare, Avadh Saxena, Apoorva Khare
    [Show abstract] [Hide abstract]
    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 Ablowitz-Ladik 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
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We show that a two-dimensional 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
  • Source
    Avinash Khare, Avadh Saxena
    [Show abstract] [Hide abstract]
    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 Ablowitz-Ladik 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
  • [Show abstract] [Hide abstract]
    ABSTRACT: For a two-dimensional scalar discrete 4 model we obtain several exact static solutions in the form of the Jacobi elliptic functions (JEF) with arbitrary shift along the lattice. The Quispel–Roberts–Thompson-type quadratic maps are identified for the considered two-dimensional model by using a JEF solution. We also show that many of the static solutions can be constructed iteratively from these quadratic maps by starting from an admissible initial value. The kink solution, having the form of tanh , is numerically demonstrated to be generically stable.
    Journal of Physics A Mathematical and Theoretical 08/2011; 44(35):355207. · 1.77 Impact Factor

Publication Stats

3k Citations
218.90 Total Impact Points

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