Publications (192)326.34 Total impact
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ABSTRACT: In a recent paper, Zhang and Li [J. Math. Phys. 56, 084101 (2015)] have doubted our claim that whenever a nonlinear equation has solutions in terms of the Jacobi elliptic functions cn(x, m) and dn(x, m), then the same nonlinear equation will necessarily also have solutions in terms of dn(x,m)±mcn(x,m). We point out the flaw in their argument and show why our assertion is indeed valid.  [Show abstract] [Hide abstract]
ABSTRACT: We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PTsymmetry, i.e., one of them has gain and the other an equal and opposite amount of loss. We first discuss various symmetries of the model. We show that both the linear system as well as a special case of the nonlinear system can be derived from a Hamiltonian, whose structure is similar to the Pais–Uhlenbeck Hamiltonian. Exact solutions are obtained in a few special cases. We show that the system is a superintegrable system within the rotating wave approximation (RWA). We also obtain several exact solutions of these RWA equations. Further, we point out a novel superposition in the context of periodic solutions in terms of Jacobi elliptic functions that we obtain in this problem. Finally, we briefly mention numerical results about the stability of some of the solutions.  [Show abstract] [Hide abstract]
ABSTRACT: In a recent paper Zhang and Li have doubted our claim that whenever a nonlinear equation has solutions in terms of the Jacobi elliptic functions $\cn(x,m)$ and $\dn(x,m)$, then the same nonlinear equation will necessarily also have solutions in terms of $\dn(x,m) \pm \sqrt{m} \cn(x,m)$. We point out the flaw in their argument and show why our assertion is indeed valid.  [Show abstract] [Hide abstract]
ABSTRACT: We obtain a class of elliptic wave solutions of coupled nonlinear Helmholtz (CNLH) equations describing nonparaxial ultrabroad beam propagation in nonlinear Kerrlike media, in terms of the Jacobi elliptic functions and also discuss their limiting forms (hyperbolic solutions). Especially, we show the existence of nontrivial solitary wave profiles in the CNLH system. The effect of nonparaxiality on the speed, pulse width and amplitude of the nonlinear waves is analysed in detail. Particularly a mechanism for tuning the speed by altering the nonparaxial parameter is proposed. We also identify a novel phaseunlocking behaviour due to the presence of nonparaxial parameter.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex P T symmetric potentials. We focus our at tention on the conventional potentials such as the generalized Poschl Teller (GPT), o ScarfI and P T symmetric ScarfII which are invariant under certain parametric transformations. The resulting set of potentials are shown to yield a completely dif ferent behavior of the bound state solutions. Further the supersymmetric (SUSY) partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and P T symmetric complex potentials. These potentials are also observed to be shape invariant (SI) in nature. We subse quently take up a study of the newly discovered rationally extended SI Potentials, corresponding to the above mentioned conventional potentials, whose bound state solutions are associated with the exceptional orthogonal polynomials (EOPs). We discuss the transformations of the corresponding Casimir operator employing the properties of the so(2,1) algebra.  [Show abstract] [Hide abstract]
ABSTRACT: A $PT$symmetric dimer is a twosite nonlinear oscillator or a nonlinear Schr\"odinger dimer where one site loses and the other site gains energy at the same rate. We present a wide class of integrable oscillator type dimers whose Hamiltonian is of arbitrary even order. Further, we also present a wide class of integrable and superintegrable nonlinear Schr\"odinger type dimers where again the Hamiltonian is of arbitrary even order.  [Show abstract] [Hide abstract]
ABSTRACT: In the present work, we combine the notion of paritytime (PT) symmetry with that of supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the socalled PöschlTeller 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 powerlaw nonlinearity, has an exact bright soliton solution that can be analytically identified and has intriguing stability properties, such as an oscillatory instability, which is absent for the corresponding solution of the regular nonlinear Schrödinger equation with arbitrary powerlaw nonlinearity. The spectral properties and dynamical implications of this instability are examined. We believe that these findings may pave the way toward initiating a fruitful interplay between the notions of PT symmetry, supersymmetric partner potentials, and nonlinear interactions.  [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 PTinvariant 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 PTinvariant 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 PTsymmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PTphase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the PTsymmetric 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 PTtransition 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 PTexact 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 bifrequency solutions that have been predicted on the basis of the SU(1,1) symmetry. Finally, we explore some special, analytically tractable, but not PTsymmetric 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 supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the socalled P{\"o}schlTeller 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, supersymmetric partner potentials and nonlinear interactions.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalarscalar selfinteraction 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 twocomponent 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 nonrelativistic 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.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
ABSTRACT: In the present work, we explore the case of a general PTsymmetric 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 antisymmetric 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.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the nonlinear Dirac equation in 1+1 dimension with scalarscalar self interaction g2κ+1(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)eiωt 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 and the criterion of Bogolubsky as well as the VakhitovKolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourthorder 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 onehump profile, but not all onehump waves are stable, while all waves with two humps are unstable. We also find that the time tc, it takes for the instability to set in, is an exponentially increasing function of ω and tc decreases monotonically with increasing κ.  [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.  [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 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. 
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.  [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.  [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.
Publication Stats
5k  Citations  
326.34  Total Impact Points  
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Institutions

2015

Savirtibai Phule Pune University
 Department of Physics
Poona, Maharashtra, India


20102015

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


2013

Banaras Hindu University
 Department of Physics
Vārānasi, Uttar Pradesh, India


19842011

Institute of Physics, Bhubaneswar
Bhubaneswar, Orissa, India


19862002

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


19872001

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


1997

University of Calcutta
 Department of Applied Mathematics
Kolkata, Bengal, India
