Avinash Khare

• Savitribai Phule Pune University

We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form $\Psi(x,t) = \Phi(x) e^{-i \omega t}$ where the nonlinear interactions are a combination of vector-vector (V-V) and scalar-scalar (S-S) interactions with the interaction Lagrangian given by $L_I= \frac{g^2}{(\kappa+1)}(\bar{\psi} \psi)^{\kappa+1} -\frac{g^2}{p(\k...

We present a comprehensive study of the rational extension of the quantum anisotropic harmonic oscillator (QAHO) potentials with linear and/or quadratic perturbations. For the one-dimensional harmonic oscillator plus imaginary linear perturbation ($i\lambda x$), we show that the rational extension is possible not only for the even but also for the...

This paper presents the first-order supersymmetric rational extension of the quantum anisotropic harmonic oscillator (QAHO) in multiple dimensions, including full-line, half-line, and their combinations. The exact solutions are in terms of the exceptional orthogonal polynomials. The rationally extended potentials are isospectral to the conventional...

Motivated by the recent introduction of an integrable coupled massive Thirring model by Basu-Mallick et al, we introduce a new coupled Soler model. Further we generalize both the coupled massive Thirring and the coupled Soler model to arbitrary nonlinear parameter $\kappa$ and obtain exact solitary wave solutions in both cases. Remarkably, it turns...

In this work, we discuss an application of the “inverse problem” method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross–Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave functio...

We consider the question of the number of exactly solvable complex but PT-invariant reflectionless potentials with N bound states. By carefully considering the Xm rationally extended reflectionless potentials, we argue that the total number of exactly solvable complex PT-invariant reflectionless potentials are 2[(2N − 1)m + N].

We consider the question of the number of exactly solvable complex but PT-invariant reflectionless potentials with $N$ bound states. By carefully considering the $X_m$ rationally extended reflectionless potentials, we argue that the total number of exactly solvable complex PT-invariant reflectionless potentials are $2[(2N-1)m+N]$.

We consider one-dimensional Dirac equation with rationally extended scalar potentials corresponding to the radial oscillator, the trigonometric Scarf and the hyperbolic Pöschl–Teller potentials and obtain their solution in terms of exceptional orthogonal polynomials. Further, in the case of the trigonometric Scarf and the hyperbolic Pöschl–Teller c...

In this work, we study the existence and stability of constant density (flat-top) solutions to the Gross-Pitaevskii equation (GPE) in confining potentials. These are constructed by using the “inverse problem” approach which corresponds to the identification of confining potentials that make flat-top waveforms exact solutions to the GPE. In the one-...

In this work, we consider a ``reverse-engineering'' approach to construct confining potentials that support exact, constant density kovaton solutions to the classical Gross-Pitaevskii equation (GPE) also known as the nonlinear Schr\"odinger equation (NLSE). In the one-dimensional case, the exact solution is the sum of stationary kink and anti-kink...

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for “generic” initial data, how close are the integrable to the noninte...

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+ 1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase “mass” (i.e., the L2norm) and the nonlinearity parameter κ is explored. We observe both the...

We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the non...

We consider $1+1$-dimensional Dirac equation with rationally extended scalar potentials corresponding to the radial oscillator, the trigonometric Scarf and the hyperbolic Poschl-Teller potentials and obtain their solution in terms of exceptional orthogonal polynomials. Further, in the case of the trigonometric Scarf and the hyperbolic Poschl-Teller...

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable non-integrable one (the discrete nonlinear Schr\"odinger model). The question we ask is: for "generic" initial data, how close are the integrable to the non-i...

We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger...

In this paper, we obtain novel solutions of a coupled ϕ4, a coupled nonlinear Schrödinger equation and a coupled modified Korteweg de Vries equation which can be re-expressed as a linear superposition of either the sum or the difference of two hyperbolic pulse solutions or the sum of either a two-kink or a kink and an antikink solution. These resul...

In this work, we consider the nonlinear Schr\"odinger equation (NLSE) in $2+1$ dimensions with arbitrary nonlinearity exponent $\kappa$ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability over their "mass" (i.e., the $L^2$ norm) and the parameter $\kappa$ is explored. We observe bot...

Here, we present and analyze the nature of a wide class of exact nonlinear wave solutions for a bimodal optical fiber doped with a three level lambda ( $\Lambda$ ) type system and delineate the parameter regimes for their occurrence. Solutions that are non-resonant with the transition frequencies of the doped system are found to undergo stable prop...

This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrödinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [ | ψ ( x , t ) | 2 + | ψ ( − x , t ) | 2 ] κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e. fully int...

We present several one-parameter families of higher-order field theory models some of which admit explicit kink solutions with an exponential tail while others admit explicit kink solutions with a power-law tail. Various properties of these families of kink solutions are examined in detail. Further, by applying the recent Manton formalism, we provi...

We start from a given one dimensional rationally extended shape invariant potential associated with Xm exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter (λ) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three...

This work focuses on the study of solitary wave solutions to a nonlocal, nonlinear Schr\"odinger system in $1$+$1$ dimensions with arbitrary nonlinearity parameter $\kappa$. Although the system we study here was first reported by Yang (Phys. Rev. E, 98 (2018), 042202) for the fully integrable case $\kappa=1$, we extend its considerations and offer...

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of...

We discuss the response of both moving and trapped solitary wave solutions of a two-component nonlinear Schrödinger system in 1 + 1 dimensions to an odd- PT external periodic complex potential. The dynamical behavior of perturbed solitary waves is explored by conducting numerical simulations of the nonlinear system and using a collective coordinate...

We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature. We obtain for various curves in two and three dimensions the transforma...

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (non-integrable) discrete nonlinear Schr{\"o}dinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechani...

We present trapped solitary wave solutions of a coupled nonlinear Schrödinger (NLS) system in 1 + 1 dimensions in the presence of an external, supersymmetric and complex PT -symmetric potential. The Schrödinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated by means of...

We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature and then we can obtain the transformation for the 1+1 dimensional Dirac...

We discuss the response of both moving and trapped solitary wave solutions of a nonlinear two-component nonlinear Schr\"odinger system in 1+1 dimensions to an anti-$\mathcal{PT}$ external periodic complex potential. The dynamical behavior of perturbed solitary waves is explored by conducting numerical simulations of the nonlinear system and using a...

We present a wide class of potentials which admit kinks and corresponding mirror kinks with either a power law or an exponential tail at the two extreme ends and a power-tower form of tails at the two neighbouring ends. We analyse kink stability equation in all these cases and show that there is no gap between the zero mode and the beginning of the...

This review is intended for readers who want to have a quick understanding on the theoretical underpinnings of coherent states and squeezed states which are conventionally generated from the prototype harmonic oscillator but not always restricting to it. Noting that the treatments of building up such states have a long history, we collected the imp...

We consider a novel one dimensional model of a logarithmic potential which has super-super-exponential kink profiles as well as kink tails. We provide analytic kink solutions of the model—it has 3 kinks, 3 mirror kinks and the corresponding antikinks. While some of the kink tails are super-super-exponential, some others are super-exponential wherea...

We start from a given one dimensional rationally extended potential associated with $X_m$ exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter ($\lambda$) family of rationally extended strictly isospectral potentials whose solutions are also associated with Xm exceptional or...

We present trapped solitary wave solutions of a coupled nonlinear Schr\"odinger system in $1$+$1$ dimensions in the presence of an external, supersymmetric and complex $\mathcal{PT}$-symmetric potential. The Schr\"odinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated...

This review is intended for readers who want to have a quick understanding on the theoretical underpinnings of coherent states and squeezed states which are conventionally generated from the prototype harmonic oscillator but not always restricting to it. Noting that the treatments of building up such states have a long history, we collected the imp...

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satis...

Quantum droplets are ultradilute liquid states that emerge from the competitive interplay of two Hamiltonian terms, the mean-field energy and beyond-mean-field correction, in a weakly interacting binary Bose gas. We relate the formation of droplets in symmetric and asymmetric two-component one-dimensional boson systems to the modulational instabili...

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter $\kappa$ is analyzed, when the external force is periodic in space and given by $f(x) =r\cos(K x)$, both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our vari...

A quantum droplet is a liquid state which emerges from the competitive interaction of two energy scales, the mean-field energy and the beyond-mean-field correction to the energy of a weakly interacting Bose gas. Here, we analyze both the stability and generating mechanisms of such droplets in a one-dimensional model of weakly interacting binary Bos...

We point out novel connections between complex PT-invariant solutions of several nonlinear equations such as , , sine-Gordon, hyperbolic sine-Gordon, double sine-Gordon, double hyperbolic sine-Gordon, mKdV, etc. We then use these connections to obtain several new complex PT-invariant periodic solutions of some of these equations.

We consider a novel one dimensional model of a logarithmic potential which has super-super-exponential kink profiles as well as kink tails. We provide analytic kink solutions of the model -- it has 3 kinks, 3 mirror kinks and the corresponding antikinks. While some of the kink tails are super-super-exponential, some others are super-exponential whe...

We present a wide class of potentials which admit kinks and corresponding mirror kinks with either a power law or an exponential tail at the two extreme ends and a power-tower form of tails at the two neighbouring ends, i.e. of the forms $ette$ or $pttp$ where $e, p$ and $t$ denote exponential, power law and power-tower tail, respectively. We analy...

We study a (1+1)-dimensional field theory based on $(\psi \ln \psi)^2$ potential. There are three degenerate minima at $\psi = 0$ and $\psi=\pm1$. There are novel, asymmetric kink solutions of the form $\psi = \mp\exp (-\exp(\pm x))$ connecting the minima at $\psi = 0$ and $\psi = \mp 1$. The domains with $\psi = 0$ repel the linear excitations, th...

We provide examples of a large class of one-dimensional higher-order field theories with kink solutions which asymptotically have a power law tail either at one end or at both ends. In particular, we provide examples of a family of potentials which admit a kink as well as a mirror kink solution where all four ends of the two kinks, or only two extr...

In this Letter, we address the long-range interaction between kinks and antikinks, as well as kinks and kinks, in φ2n+4 field theories for n>1. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the 2n/(n−1)th power of their separation,...

The \(\phi ^4\) model has been the “workhorse” of the classical Ginzburg–Landau phenomenological theory of phase transitions and, furthermore, the foundation for a large amount of the now-classical developments in nonlinear science. However, the \(\phi ^4\) model, in its usual variant (symmetric double-well potential), can only possess two equilibr...

In this Letter, we address the general long-range interaction between kinks and antikinks, as well as kinks and kinks, in $\varphi^{2n+4}$ field theories for $n>1$. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the $(\frac{2n}{n-1}...

We construct a rational extension of the truncated Calogero–Sutherland model by Pittman et al. The exact solution of this rationally extended model is obtained analytically and it is shown that while the energy eigenvalues remain unchanged, however the eigenfunctions are completely different and written in terms of exceptional X1 Laguerre orthogona...

We provide examples of a large class of one dimensional higher order field theories with kink solutions which asymptotically have a power-law tail either at one end or at both ends. We provide analytic solutions for the kinks in a few cases but mostly provide implicit solutions. We also provide examples of a family of potentials with two kinks, bot...

We obtain exact solitary wave solutions of a variant of the generalized derivative nonlinear Schrödinger equation in 1+1 dimensions with arbitrary values of the nonlinearity parameter κ in a Scarf-II potential. This variant of the usual derivative nonlinear Schrödinger equation has the properties that for real external potentials, the dynamics is d...

We point out novel connections between complex PT-invariant solutions of several nonlinear equations such as $\phi^4$, $\phi^6$, sine-Gordon, hyperbolic sine-Gordon, double sine-Gordon, double hyperbolic sine-Gordon, mKdV, etc. We then use these connections to obtain several new complex PT-invariant periodic solutions of some of these equations.

We construct a rational extension of the truncated Calogero-Sutherland model by Pittman et al. The exact solution of this rationally extended model is obtained analytically and it is shown that while the energy eigenvalues remain unchanged, however the eigenfunctions are completely different and written in terms of exceptional X1 Laguerre orthogona...

The $\phi^4$ model has been the "workhorse" of the classical Ginzburg--Landau phenomenological theory of phase transitions and, furthermore, the foundation for a large amount of the now-classical developments in nonlinear science. However, the $\phi^4$ model, in its usual variant (symmetric double-well potential), can only possess two equilibria. M...

We obtain exact solitary wave solutions of a variant of the generalized derivative nonlinear Schrodinger\equation in 1+1 dimensions with arbitrary values of the nonlinearity parameter $\kappa$ in a Scarf-II potential. This variant of the usual derivative nonlinear Schrodinger equation has the properties that for real external potentials, the dynami...

For a large number of real nonlinear equations, either continuous or discrete, uncoupled or coupled, we show that whenever these equations admit real cnoidal wave solutions, they also admit complex PT-invariant cnoidal (and hyperbolic) wave solutions. Further, we show that for real coupled field theories, one cannot only have complex PT-invariant c...

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...

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 $\{ \,...

We systematically construct a distinct class of complex potentials including parity-time ($\cal PT$) symmetric potentials for the stationary Schr\"odinger 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)...

We systematically construct a distinct class of complex potentials including parity-time ($\cal PT$) symmetric potentials for the stationary Schr\"odinger 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)...

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...

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...

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 $\{ \,...

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...

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 ex...

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 ex...

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 polyn...

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 polyn...

A PT-symmetric dimer is a two-site nonlinear oscillator dimer or a two-site nonlinear Schrodinger 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 nonlinear Schrod...

We study different classes of solitary waves of the multicomponent nonlinear Helmholtz (mNLH) equations describing nonparaxial ultra broad beam propagation in planar waveguides. The mNLH equations are solved using Lam´e polynomials of order 1, 2 and 3 in the hyperbolic limit. The effect of nonparaxiality on the speed, pulse width, and amplitude of...

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 differen...

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 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 discuss the behavior of solitary wave solutions of the nonlinear Schr{\"o}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...

We discuss the behavior of solitary wave solutions of the nonlinear Schr{\"o}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...

DOI:http://dx.doi.org/10.1103/PhysRevE.93.059903

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 differen...

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 p...

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 p...

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} \psi)^2 + \frac{ g_2^2}{2} ( {\bphi} \phi)^2 + g_3^2 ({\bpsi} \psi) ( {\bphi} \phi)$ as well as vector-vector interactions of the form $\frac{g_1^2 }{2} (\bpsi \gamma_{\mu} \psi)(\bpsi \gamma^{\mu} \psi)+ \fr...

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} \psi)^2 + \frac{ g_2^2}{2} ( {\bphi} \phi)^2 + g_3^2 ({\bpsi} \psi) ( {\bphi} \phi)$ as well as vector-vector interactions of the form $\frac{g_1^2 }{2} (\bpsi \gamma_{\mu} \psi)(\bpsi \gamma^{\mu} \psi)+ \fr...

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 $\tanh \beta x$, where $\beta$ is the inverse of the kink width, it also admits solutions in terms of the PT-invariant combinations $\tanh...

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...

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 sh...

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. 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 Hamiltonia...

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 asser...

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...

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 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. T...

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. T...

In the present work, we combine the notion of parity-time (PT) symmetry with that of supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the so-called Pöschl-Teller potential which is real. Not only are we able to identify and numerically confirm the eigenvalues of the relevant problem, but we al...

A $PT$-symmetric dimer is a two-site 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 nonline...

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 tha...

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 th...

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...

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...