[Show abstract][Hide abstract] ABSTRACT: For a large number of real nonlinear equations, either continuous or
discrete, integrable or nonintegrable, we show that whenever a real nonlinear
equation admits a solution in terms of $\sech x$, it also admits solutions in
terms of the PT-invariant combinations $\sech x \pm i \tanh x$. Further, for a
number of real nonlinear equations we show that whenever a nonlinear equation
admits a solution in terms $\sech^2 x$, it also admits solutions in terms of
the PT-invariant combinations $\sech^2 x \pm i \sech x \tanh x$. Besides, we
show that similar results are also true in the periodic case involving Jacobi
elliptic functions.
[Show abstract][Hide abstract] ABSTRACT: In the present work, we consider a prototypical example of a PT-symmetric
Dirac model. We discuss the underlying linear limit of the model and identify
the threshold of the PT-phase transition in an analytical form. We then focus
on the examination of the nonlinear model. We consider the continuation in the
PT-symmetric model of the solutions of the corresponding Hamiltonian model and
find that the solutions can be continued robustly as stable ones all the way up
to the PT-transition threshold. In the latter, they degenerate into linear
waves. We also examine the dynamics of the model. Given the stability of the
waveforms in the PT-exact phase we consider them as initial conditions for
parameters outside of that phase. We find that both oscillatory dynamics and
exponential growth may arise, depending on the size of the corresponding
"quench". The former can be characterized by an interesting form of
bi-frequency solutions that have been predicted on the basis of the SU(1,1)
symmetry. Finally, we explore some special, analytically tractable, but not
PT-symmetric solutions in the massless limit of the model.
[Show abstract][Hide abstract] ABSTRACT: In the present work, we combine the notion of $\mathcal{PT}$-symmetry with
that of super-symmetry (SUSY) for a prototypical case example with a complex
potential that is related by SUSY to the so-called P{\"o}schl-Teller potential
which is real. Not only are we able to identify and numerically confirm the
eigenvalues of the relevant problem, but we also show that the corresponding
nonlinear problem, in the presence of an arbitrary power law nonlinearity, has
an exact bright soliton solution that can be analytically identified and has
intriguing stability properties, such as an oscillatory instability, which the
corresponding solution of the regular nonlinear Schr{\"o}dinger equation with
arbitrary power law nonlinearity does not possess. The spectral properties and
dynamical implications of this instability are examined. We believe that these
findings may pave the way towards initiating a fruitful interplay between the
notions of $\mathcal{PT}$-symmetry, super-symmetric partner potentials and
nonlinear interactions.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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 · 2.10 Impact Factor
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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 · 1.18 Impact Factor
[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; 378(42). DOI:10.1016/j.physleta.2014.09.006 · 1.68 Impact Factor
[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 08/2014; 90(2-1):023208. · 2.29 Impact Factor
[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.
[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.
[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.
Physical Review E 02/2014; 90(2). DOI:10.1103/PhysRevE.90.023208 · 2.29 Impact Factor
[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. DOI:10.1016/j.physleta.2013.08.015 · 1.68 Impact Factor
[Show abstract][Hide abstract] 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.
[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; 379(3). DOI:10.1016/j.physleta.2014.11.009 · 1.68 Impact Factor
[Show abstract][Hide abstract] 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 · 2.81 Impact Factor
[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. DOI:10.1103/PhysRevE.87.044901 · 2.29 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\"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 · 6.13 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.