arXiv subjects

Avinash Khare

Publications and source records attributed to Avinash Khare.

At least 73 records · Page 4Linked to original sources

arXiv2017

A systematic construction of parity-time ($\cal PT$)-symmetric and non-$\cal PT$-symmetric complex potentials from the solutions of various real nonlinear evolution equations

We systematically construct a distinct class of complex potentials including parity-time ($\cal PT$) symmetric potentials for the stationary Schrödinger equation by using the soliton and periodic solutions of the four integrable real nonlinear evolution equations (NLEEs) namely the sine-Gordon (sG) equation, the modified Korteweg-de Vries (mKdV) equation, combined mKdV-sG equation and the Gardner equation. These potentials comprise of kink, breather, bion, elliptic bion, periodic and soliton potentials which are explicitly constructed from the various respective solutions of the NLEEs. We demonstrate the relevance between the identified complex potentials and the potential of the graphene model from an application point of view.

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arXiv2017

A class of exactly solvable rationally extended non-central potentials in Two and Three Dimensions

We start from a seven parameters (six continuous and one discrete) family of non-central exactly solvable potential in three dimensions and construct a wide class of ten parameters (six continuous and four discrete) family of rationally extended exactly solvable non-central real as well as PT symmetric complex potentials. The energy eigenvalues and the eigenfunctions of these extended non-central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these po- tentials are either associated with the exceptional orthogonal polynomials (EOPs) or some type of new polynomials which can be further re-expressed in terms of the corresponding classical orthogonal polynomials. Similarly, we also construct a wide class of rationally extended exactly solvable non-central real as well as complex PT-invariant potentials in two dimensions.

quant-ph↗

arXiv2017

Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry

We discuss the effect of small perturbation on nodeless solutions of the nonlinear \Schrodinger\ equation in 1+1 dimensions in an external complex potential derivable from a parity-time symmetric superpotential that was considered earlier [Phys.~Rev.~E 92, 042901 (2015)]. In particular we consider the nonlinear partial differential equation $\{ \, \rmi \, \partial_t + \partial_x^2 + g |ψ(x,t)|^2 - V^{+}(x) \, \} \, ψ(x,t) = 0$, where $V^{+}(x) = \qty( -b^2 - m^2 + 1/4 ) \, \sech^2(x) - 2 i \, m \, b \, \sech(x) \, \tanh(x)$ represents the complex potential. Here we study the perturbations as a function of $b$ and $m$ using a variational approximation based on a dissipation functional formalism. We compare the result of this variational approach with direct numerical simulation of the equations. We find that the variational approximation works quite well at small and moderate values of the parameter $b m$ which controls the strength of the imaginary part of the potential. We also show that the dissipation functional formalism is equivalent to the generalized traveling wave method for this type of dissipation.

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arXiv2017

Stability of new exact solutions of the nonlinear Schrodinger equation in a Poschl-Teller external potential

We discuss the stability properties of the solutions of the general nonlinear \Schrodinger\ equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time ($\PT$) symmetric superpotential $W(x)$ that we considered earlier \cite{PhysRevE.92.042901}. In particular we consider the nonlinear partial differential equation $ \{ i \, \partial_t + \partial_x^2 - V(x) + g | ψ(x,t) |^{2κ} \} \, ψ(x,t) = 0 \>, $ for arbitrary nonlinearity parameter $κ$, where $g= \pm1$ and $V$ is the well known P{ö}schl-Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and $κ$. For the repulsive potential (and $g=+1$) we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1\% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential (and $g=+1$), changing the initial conditions by 1 \% modifies the domain of stability only slightly. For the case of the attractive potential and negative $g$ perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation.

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arXiv2017

A short note on "Group theoretic approach to rationally extended shape invariant potentials" [Annals of Physics 359, 46 (2015)]

It is proved the equivalence of the compatibility condition of [A. Ramos, J. Phys. A 44 (2011) 342001, Phys. Lett. A 376 (2012) 3499] with a condition found in [Yadav et al., Ann. Phys. 359 (2015) 46]. The link of Shape Invariance with the existence of a Potential Algebra is reinforced for the rationally extended Shape Invariant potentials. Some examples on X1 and Xl Jacobi and Laguerre cases are given.

math-ph↗

arXiv2017

A class of exactly solvable rationally extended Calogero-Wolfes type 3-body problems

In this work, we start from the well known Calogero-Wolfes type 3-body problems on a line and construct the corresponding exactly solvable rationally extended 3-body potentials. In particular, we obtain the corresponding energy eigenvalues and eigenfunctions which are in terms of the product of Xm Laguerre and Xp Jacobi exceptional orthogonal polynomials where both m,p = 1,2,3,....

quant-ph↗

arXiv2016

Truncated Calogero-Sutherland Models on a Circle

We investigate a quantum many-body system with particles moving on a circle and subject to two-body and three-body potentials. In this new class of models, that extrapolates from the celebrated Calogero-Sutherland model and a system with interactions among nearest and next-to-nearest neighbors, the interactions can be tuned as a function of range. We determine the exact ground state energy and wavefunction and obtain a part of the excitation spectrum.

quant-ph↗

arXiv2016

Variational Approach to studying solitary waves in the nonlinear Schrodinger equation with Complex Potentials

We discuss the behavior of solitary wave solutions of the nonlinear Schr{ö}dinger equation (NLSE) as they interact with complex potentials, using a four parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope of $dp(t)/dv(t)$ becomes negative when plotted parametrically as a function of time, where $p(t)$ is the momentum of the solitary wave and $v(t)$ the velocity.

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arXiv2016

Stability of exact solutions of the nonlinear Schroedinger equation in an external potential having supersymmetry and parity-time symmetry

We discuss the stability properties of the solutions of the general nonlinear Schroedinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time (PT) symmetric superpotential $W(x)$ that we considered earlier [Kevrekedis et al Phys. Rev. E 92, 042901 (2015)]. In particular we consider the nonlinear partial differential equation $\{ i \partial_t + \partial_x^2 - V^{-}(x) +| ψ(x,t) |^{2κ} \} \, ψ(x,t) = 0$, for arbitrary nonlinearity parameter $κ$. We study the bound state solutions when $V^{-}(x) = (1/4- b^2)$ sech$^2(x)$, which can be derived from two different superpotentials $W(x)$, one of which is complex and $PT$ symmetric. Using Derrick's theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth $b^2$ of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov-Kolokolov (V-K) stability criterion. The numerical results of applying the V-K condition give the same answer for the domain of stability as the analytic result obtained from applying Derrick's theorem. Our main result is that for $κ>2$ a new regime of stability for the exact solutions appears as long as $b > b_{crit}$, where $b_{crit}$ is a function of the nonlinearity parameter $κ$. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for $κ>2$.

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arXiv2016

Scattering amplitudes for the rationally extended P T symmetric complex potentials

In this paper, we consider the rational extensions of two different P T symmetric complex potentials namely the asymptotically vanishing Scarf II and asymptotically non-vanishing Rosen-Morse II [ RM-II] potentials and obtain bound state eigenfunc- tions in terms of newly found exceptional Xm Jacobi polynomials and also some new type of orthogonal polynomials respectively. By considering the asymptotic behaviour of the exceptional polynomials, we obtain the reflection and transmission amplitudes for them and discuss the various novel properties of the corresponding amplitudes.

quant-ph↗

arXiv2016

Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} ψ)^2 + \frac{ g_2^2}{2} ( {\bphi} ϕ)^2 + g_3^2 ({\bpsi} ψ) ( {\bphi} ϕ)$ as well as vector-vector interactions of the form $\frac{g_1^2 }{2} (\bpsi γ_μ ψ)(\bpsi γ^μ ψ)+ \frac{g_2^2 }{2} (\bphi γ_μ ϕ)(\bphi γ^μ ϕ) + g_3^2 (\bpsi γ_μ ψ)(\bphi γ^μ ϕ). $ Writing the two components of the assumed solitary wave solution of these equations in the form $ψ= e^{-i ω_1 t} \{R_1 \cos θ, R_1 \sin θ\}$, $ϕ= e^{-i ω_2 t} \{R_2 \cos η, R_2\sin η\}$, and assuming that $ θ(x),η(x)$ have the {\it same} functional form they had when $g_3$=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for $R_i(x)$ which are valid for small values of $g_3^2/ g_2^2 $ and $g_3^2/ g_1^2$. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at $x \rightarrow \pm \infty$.

nlin.PS↗

arXiv2016

Novel PT-invariant Kink and Pulse Solutions For a Large Number of Real Nonlinear Equations

For a large number of real nonlinear equations, either continuous or discrete, integrable or nonintegrable, uncoupled or coupled, we show that whenever a real nonlinear equation admits kink solutions in terms of $\tanh βx$, where $β$ is the inverse of the kink width, it also admits solutions in terms of the PT-invariant combinations $\tanh 2βx \pm i \sech 2 βx$, i.e. the kink width is reduced by half to that of the real kink solution. We show that both the kink and the PT-invariant kink are linearly stable and obtain expressions for the zero mode in the case of several PT-invariant kink solutions. Further, for a number of real nonlinear equations we show that whenever a nonlinear equation admits periodic kink solutions in terms of $\sn(x,m)$, it also admits periodic solutions in terms of the PT-invariant combinations $\sn(x,m) \pm i \cn(x,m)$ as well as $\sn(x,m)\pm i \dn(x,m)$. Finally, for coupled equations we show that one cannot only have complex PT-invariant solutions with PT eigenvalue $+1$ or $-1$ in both the fields but one can also have solutions with PT eigenvalue $+1$ in one field and $-1$ in the other field.

nlin.PS↗

arXiv2015

Reply to the Comment by Zhang and Li [J. Math. Phys. 56, 084101, 2015] on the Paper [J. Math. Phys. {\bf 56}, 032104, 2015]}

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.

nlin.PS↗

arXiv2015

Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations

We obtain a class of elliptic wave solutions of coupled nonlinear Helmholtz (CNLH) equations describing nonparaxial ultra-broad beam propagation in nonlinear Kerr-like media, in terms of the Jacobi elliptic functions and also discuss their limiting forms (hyperbolic solutions). Especially, we show the existence of non-trivial 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 phase-unlocking behaviour due to the presence of nonparaxial parameter.

nlin.PS↗

arXiv2015

Parametric symmetries in exactly solvable real and P T symmetric complex potentials

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 Scarf-I and P T symmetric Scarf-II 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.

quant-ph↗

arXiv2015

Integrable and Superintegrable Klein-Gordon and Schrödinger Type Dimers

A $PT$-symmetric dimer is a two-site nonlinear oscillator or a nonlinear Schrödinger 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 Schrödinger type dimers where again the Hamiltonian is of arbitrary even order.

nlin.SI↗

arXiv2015

Solitary waves of a PT-symmetric Nonlinear Dirac equation

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.

nlin.PS↗

arXiv2015

Novel PT-invariant Solutions For a Large Number of Real Nonlinear Equations

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.

nlin.SI↗