arXiv subjects

Avinash Khare

Publications and source records attributed to Avinash Khare.

At least 91 records · Page 5Linked to original sources

arXiv2015

Nonlinear Dirac equation solitary waves in the presence of external driving forces

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 μγ^0 Ψ$, where both $f$ and $Ψ$ 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.

nlin.PS↗

arXiv2015

Group theoretic approach to rationally extended shape invariant potentials

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.

quant-ph↗

arXiv2014

Rationally extended shape invariant potentials in arbitrary D-dimensions associated with exceptional $X_m$ polynomials

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.

quant-ph↗

arXiv2014

Elliptic Waves in Two Component Long Wave--Short Wave Resonance Interaction System in One and Two dimensions

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 Lamé 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.

nlin.SI↗

arXiv2014

Successive phase transitions and kink solutions in $ϕ^{8}$, $ϕ^{10}$, and $ϕ^{12}$ field theories

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 co-existence 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 quasi-exactly solvable (QES) 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.

math-ph↗

arXiv2014

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar self interaction $ \frac{g^2}{κ+1} ({\bar Ψ} Ψ)^{κ+1}$ and with mass $m$. Using the exact analytic form for rest frame solitary waves of the form $Ψ(x,t) = ψ(x) e^{-i ωt}$ for arbitrary $ κ$, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem, the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed 4th-order operator splitting integration method. For different ranges of $κ$ we map out the stability regimes in $ω$. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time $t_c$, it takes for the instability to set in, is an exponentially increasing function of $ω$ and $t_c$ decreases monotonically with increasing $κ$.

nlin.PS↗

arXiv2014

Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrödinger 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 Lamé polynomials of order 1, but it also admits solutions in terms of Lamé 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.

nlin.PS↗

arXiv2014

Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations

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 Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, $λϕ^4$, 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.

math-ph↗

arXiv2013

The Scattering amplitude for Rationally extended shape invariant Eckart potentials

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.

hep-th↗

arXiv2013

Nonlinear Dirac equation solitary waves in external fields

We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{κ+1} ({\bPsi} Ψ)^{κ+1}$ 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{ö}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.

nlin.PS↗

arXiv2013

Statistical Mechanics of a Discrete Schrödinger Equation with Saturable Nonlinearity

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. {\bf 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 non-trivial, 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.

nlin.PS↗

arXiv2013

Novel symmetries in N = 2 supersymmetric quantum mechanical models

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.

hep-th↗

arXiv2013

The Scattering amplitude for one parameter family of shape invariant potentials related to Xm Jacobi polynomials

We consider the recently discovered, one parameter family of exactly solvable shape invariant potentials which are isospectral to the generalized Pöschl-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.

quant-ph↗

arXiv2013

Linear Superposition for a Large Number of Nonlinear Equations

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.

nlin.SI↗

arXiv2012

The Scattering amplitude for Newly found exactly solvable Potential

The scattering amplitude for the recently discovered exactly solvable shape invariant potential, which is isospectral to the generalized Pöschl-Taylor potential, is calculated explicitly by considering the asymptotic behavior of the $X_{1}$ Jacobi exceptional polynomials associated with this system.

math-ph↗

arXiv2011

New QES Hermitian as well as non-Hermitian PT invariant Potentials

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.

math-ph↗

arXiv2011

Solutions of Several Coupled Discrete Models in terms of Lame Polynomials of Arbitrary Order

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 Lamé polynomials of arbitrary order. The models discussed are (i) coupled Salerno model, (ii) coupled Ablowitz-Ladik model, (iii) coupled $ϕ^4$ model, and (iv) coupled $ϕ^6$ model. In all these cases we show that the coefficients of the Lamé polynomials are such that the Lamé polynomials can be reexpressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.

math-ph↗

arXiv2011

Forced Nonlinear Schroedinger Equation with Arbitrary Nonlinearity

We consider the nonlinear Schr{ö}dinger equation (NLSE) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{κ+1} (ψ^\star ψ)^{κ+1}$ in the presence of the external forcing terms of the form $r e^{-i(kx + θ)} -δψ$. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where $v_k=2 k$. These new exact solutions reduce to the constant phase solutions of the unforced problem when $r \rightarrow 0.$ 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)/d \dot{q} (t) < 0$, where $p(t)$ is the normalized canonical momentum $p(t) = \frac{1}{M(t)} \frac {\partial L}{\partial {\dot q}}$, and $\dot{q}(t)$ is the solitary wave velocity. Here $M(t) = \int dx ψ^\star(x,t) ψ(x,t)$. 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.

nlin.PS↗