debdeep sinha - Academia.edu (original) (raw)

Papers by debdeep sinha

Journal of Physics A, Nov 19, 2019

The quantization of many-body systems with balanced loss and gain is investigated. Two types of m... more The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A partial set of integrals of motion are constructed for each type of model. Specific examples for the translational invariant systems include Calogero-type many-body systems with balanced loss and gain, where each particle is interacting with other particles via four-body inverse-square potential plus pair-wise two-body harmonic terms. A many-body system interacting via short range four-body plus six-body inverse square potential with pair-wise two-body harmonic terms in presence of balanced loss and gain is also considered. In general, the eigen values of these two models contain quantized as well as continuous spectra. A completely quantized spectra and bound states involving all the particles may be obtained by employing box-normalization on the particles having continuous spectra. The normalization of the ground state wave functions in appropriate Stoke wedges is discussed. The exact n-particle correlation functions of these two models are obtained through a mapping of the relevant integrals to known results in random matrix theory. It is shown that a rotationally symmetric system with generic many-body potential does not have entirely real spectra, leading to unstable quantum modes. The eigenvalue problem of a Hamiltonian system with balanced loss and gain and admitting dynamical O(2, 1) symmetry is also considered.

The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-li... more The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-linearity are considered in presence of a linear term characterized, in general, by a non-hermitian matrix which under certain condition incorporates balanced loss and gain and a linear coupling between the complex fields of the governing non-linear equations. It is shown that the systems posses a Lax pair and an infinite number of conserved quantities and hence integrable. Apart from the particular form of the local and non-local reductions, the systems are integrable when the matrix representing the linear term is pseudo hermitian with respect to the hermitian matrix comprising the generic cubic non-linearity. The inverse scattering transformation method is employed to find exact soliton solutions for both the local and non-local cases. The presence of the linear term restricts the possible form of the norming constants and hence the polarization vector. It is shown that for integrable v...

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using t... more We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for the abundance using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the abundances of k-chains. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. ... more A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo- Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion is constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the dissipation parameter has u...

A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potentia... more A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self-induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non-autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a d + 1 dimensional generalization of it, admits all the symmetries of the d + 1 dimensional Schrödinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the "pseudo-width" of the wave-packet for the special case when the system admits a O(2, 1) conformal symmetry.

The European Physical Journal Plus, 2021

Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic har... more Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic harmonic oscillator. Exact solutions for the system are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp's shift to the noncommuting degrees of freedom. Employing Simon's separability criterion, we find that the states of the system are entangled provided a unique function of the (mass and frequency) parameters obeys an inequality. Entanglement of Formation for this system is also computed and its relation to the degree of anisotropy is discussed. It is worth mentioning that, even in a noncommutative space, entanglement is generated only if the harmonic oscillator is anisotropic. Interestingly, the Entanglement of Formation saturates for higher values of the deformation parameter θ, that quantifies spatial noncommutativity.

A two body rational Calogero model with balanced loss and gain is investigated. The system yields... more A two body rational Calogero model with balanced loss and gain is investigated. The system yields a Hamiltonian which is symmetric under the combined operation of parity (P) and time reversal (T ) symmetry. It is shown that the system is integrable and exact, stable classical solutions are obtained for particular ranges of the parameters. The corresponding quantum system admits bound state solutions for exactly the same ranges of the parameters for which the classical solutions are stable. The eigen spectra of the system is presented with a discussion on the normalization of the wave functions in proper Stokes wedges. Finally, the Calogero model with balanced loss and gain is studied classically, when the pair-wise harmonic interaction term is replaced by a common confining harmonic potential. The system admits stable solutions for particular ranges of the parameters. However, the integrability and/or exact solvability of the system is obscure due to the presence of the loss and gai...

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using t... more We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance C k of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for C k using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the C k. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and... more A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of co-ordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient.The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A ...

The European Physical Journal Plus, Nov 1, 2017

A two body rational Calogero model with balanced loss and gain is investigated. The system yields... more A two body rational Calogero model with balanced loss and gain is investigated. The system yields a Hamiltonian which is symmetric under the combined operation of parity (P) and time reversal (T) symmetry. It is shown that the system is integrable and exact, stable classical solutions are obtained for particular ranges of the parameters. The corresponding quantum system admits bound state solutions for exactly the same ranges of the parameters for which the classical solutions are stable. The eigen spectra of the system is presented with a discussion on the normalization of the wave functions in proper Stokes wedges. Finally, the Calogero model with balanced loss and gain is studied classically, when the pair-wise harmonic interaction term is replaced by a common confining harmonic potential. The system admits stable solutions for particular ranges of the parameters. However, the integrability and/or exact solvability of the system is obscure due to the presence of the loss and gain terms. The perturbative solutions are obtained and are compared with the numerical results.

Journal of Physics A: Mathematical and Theoretical

Annals of Physics

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and... more A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of coordinates , the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain coefficient .The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A few exact solutions for both the cases are presented for specific choices of the potential and spacedependent gain/loss coefficients , which include periodic stable solutions. Quantization of the system is discussed with the construction of the integrals of motion for specific choices of the potential and gain-loss coefficients. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges are presented.

Annals of Physics

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. ... more A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also shown that with the choice of a suitable coordinate , the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion is constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

Physical Review E, 2015

A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potentia... more A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self-induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non-autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a d + 1 dimensional generalization of it, admits all the symmetries of the d + 1 dimensional Schrödinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the "pseudo-width" of the wave-packet for the special case when the system admits a O(2, 1) conformal symmetry.

Physics Letters A, 2017

A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-... more A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-induced PT symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of these charges being non-hermitian. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is also mentioned. An inhomogeneous version of this VNLSE with space-time modulated nonlinear interaction term is also considered and a mapping of this Eq. with standard VNLSE through similarity transformation is used to generate its solutions.

Physical Review D, 2013

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using t... more We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance C k of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for C k using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the C k. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

Journal of Physics A, Nov 19, 2019

The quantization of many-body systems with balanced loss and gain is investigated. Two types of m... more The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A partial set of integrals of motion are constructed for each type of model. Specific examples for the translational invariant systems include Calogero-type many-body systems with balanced loss and gain, where each particle is interacting with other particles via four-body inverse-square potential plus pair-wise two-body harmonic terms. A many-body system interacting via short range four-body plus six-body inverse square potential with pair-wise two-body harmonic terms in presence of balanced loss and gain is also considered. In general, the eigen values of these two models contain quantized as well as continuous spectra. A completely quantized spectra and bound states involving all the particles may be obtained by employing box-normalization on the particles having continuous spectra. The normalization of the ground state wave functions in appropriate Stoke wedges is discussed. The exact n-particle correlation functions of these two models are obtained through a mapping of the relevant integrals to known results in random matrix theory. It is shown that a rotationally symmetric system with generic many-body potential does not have entirely real spectra, leading to unstable quantum modes. The eigenvalue problem of a Hamiltonian system with balanced loss and gain and admitting dynamical O(2, 1) symmetry is also considered.

The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-li... more The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-linearity are considered in presence of a linear term characterized, in general, by a non-hermitian matrix which under certain condition incorporates balanced loss and gain and a linear coupling between the complex fields of the governing non-linear equations. It is shown that the systems posses a Lax pair and an infinite number of conserved quantities and hence integrable. Apart from the particular form of the local and non-local reductions, the systems are integrable when the matrix representing the linear term is pseudo hermitian with respect to the hermitian matrix comprising the generic cubic non-linearity. The inverse scattering transformation method is employed to find exact soliton solutions for both the local and non-local cases. The presence of the linear term restricts the possible form of the norming constants and hence the polarization vector. It is shown that for integrable v...

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using t... more We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for the abundance using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the abundances of k-chains. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. ... more A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo- Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion is constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the dissipation parameter has u...

A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potentia... more A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self-induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non-autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a d + 1 dimensional generalization of it, admits all the symmetries of the d + 1 dimensional Schrödinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the "pseudo-width" of the wave-packet for the special case when the system admits a O(2, 1) conformal symmetry.

The European Physical Journal Plus, 2021

Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic har... more Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic harmonic oscillator. Exact solutions for the system are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp's shift to the noncommuting degrees of freedom. Employing Simon's separability criterion, we find that the states of the system are entangled provided a unique function of the (mass and frequency) parameters obeys an inequality. Entanglement of Formation for this system is also computed and its relation to the degree of anisotropy is discussed. It is worth mentioning that, even in a noncommutative space, entanglement is generated only if the harmonic oscillator is anisotropic. Interestingly, the Entanglement of Formation saturates for higher values of the deformation parameter θ, that quantifies spatial noncommutativity.

A two body rational Calogero model with balanced loss and gain is investigated. The system yields... more A two body rational Calogero model with balanced loss and gain is investigated. The system yields a Hamiltonian which is symmetric under the combined operation of parity (P) and time reversal (T ) symmetry. It is shown that the system is integrable and exact, stable classical solutions are obtained for particular ranges of the parameters. The corresponding quantum system admits bound state solutions for exactly the same ranges of the parameters for which the classical solutions are stable. The eigen spectra of the system is presented with a discussion on the normalization of the wave functions in proper Stokes wedges. Finally, the Calogero model with balanced loss and gain is studied classically, when the pair-wise harmonic interaction term is replaced by a common confining harmonic potential. The system admits stable solutions for particular ranges of the parameters. However, the integrability and/or exact solvability of the system is obscure due to the presence of the loss and gai...

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using t... more We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance C k of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for C k using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the C k. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and... more A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of co-ordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient.The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A ...

The European Physical Journal Plus, Nov 1, 2017

A two body rational Calogero model with balanced loss and gain is investigated. The system yields... more A two body rational Calogero model with balanced loss and gain is investigated. The system yields a Hamiltonian which is symmetric under the combined operation of parity (P) and time reversal (T) symmetry. It is shown that the system is integrable and exact, stable classical solutions are obtained for particular ranges of the parameters. The corresponding quantum system admits bound state solutions for exactly the same ranges of the parameters for which the classical solutions are stable. The eigen spectra of the system is presented with a discussion on the normalization of the wave functions in proper Stokes wedges. Finally, the Calogero model with balanced loss and gain is studied classically, when the pair-wise harmonic interaction term is replaced by a common confining harmonic potential. The system admits stable solutions for particular ranges of the parameters. However, the integrability and/or exact solvability of the system is obscure due to the presence of the loss and gain terms. The perturbative solutions are obtained and are compared with the numerical results.

Journal of Physics A: Mathematical and Theoretical

Annals of Physics

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and... more A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of coordinates , the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain coefficient .The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A few exact solutions for both the cases are presented for specific choices of the potential and spacedependent gain/loss coefficients , which include periodic stable solutions. Quantization of the system is discussed with the construction of the integrals of motion for specific choices of the potential and gain-loss coefficients. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges are presented.

Annals of Physics

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. ... more A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also shown that with the choice of a suitable coordinate , the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion is constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

Physical Review E, 2015

A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potentia... more A class of non-local non-linear Schrödinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self-induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non-autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a d + 1 dimensional generalization of it, admits all the symmetries of the d + 1 dimensional Schrödinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the "pseudo-width" of the wave-packet for the special case when the system admits a O(2, 1) conformal symmetry.

Physics Letters A, 2017

A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-... more A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-induced PT symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of these charges being non-hermitian. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is also mentioned. An inhomogeneous version of this VNLSE with space-time modulated nonlinear interaction term is also considered and a mapping of this Eq. with standard VNLSE through similarity transformation is used to generate its solutions.

Physical Review D, 2013

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using t... more We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance C k of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for C k using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the C k. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.