Amitava Choudhuri

We have studied a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological model with an additional assumption that the universe is filled with barotropic causal bulk viscous fluid. The bulk viscous coefficient of the fluid is related to the energy density $(\rho )$ and relaxation time $(\tau )$ by the relation $\zeta =c_{b}^{2}\tau \rho $. We have used a truncated version of the transport equation for the viscous pressure. The expansion rate of such a spatially flat FLRW universe is governed by the well-known modified Painlevé–Ince equation. We find that the Lie symmetry corresponding to scale invariance gives the power-law solution for this model equation from invariant curve condition. For the value of ${{c}_{b}}\sim 0.79$, our result satisfies the present experimental value $\sim -0.46$ of the deceleration parameter (q) for $N=\frac{1}{\tau H}=1$. The result obtained for the relaxation time $(\tau )$ which is of the order of the Hubble time supports the necessary condition for successful inflation. Our study not only shows the role of the bulk viscosity for the present accelerating expansion $(q\lt 0)$ but also predicts an age of around $\sim 25\;{\rm Gyr}$, which solves the age problem of the present universe.

We identify two alternative Lagrangian representations for the damped harmonic oscillator characterised by a frictional coefficient (gamma). The first one is explicitly time independent while the second one involves time parameter explicitly. With separate attention to both Lagrangians we make use of the Noether theorem to compute the variational symmetries and conservation laws in order to study how association between them changes as one goes from one representation to the other. In the case of time-independent representation squeezing symmetry leads to conservation of angular momentum for (gamma) = 0, while for the time-dependent Lagrangian the same conserved quantity results from rotational invariance. The Lie algebra (g) of the symmetry vectors that leaves the action corresponding to the time-independent Lagrangian invariant is semi-simple. On the other hand, g is only a simple Lie algebra for the action characterised by the time-dependent Lagrangian.

We have studied the modulational instability (MI) of the higher-order nonlinear Schrödinger (HNLS) equation with non-Kerr nonlinearities in an optical context and presented an analytical expression for MI gain to show the effects of non-Kerr nonlinearities and higher-order dispersions on MI gain spectra. In our study, we demonstrate that MI can exist not only for the anomalous group-velocity dispersion (GVD) regime, but also in the normal GVD regime in the HNLS equation in the presence of non-Kerr quintic nonlinearities. The non-Kerr quintic nonlinear effect reduces the maximum value of the MI gain and bandwidth and plays a sensitive role over the Kerr nonlinearity, which leads to continuous wave breaking into a number of stable wave trains of ultrashort optical pulses that can be used to generate the stable supercontinuum white-light coherent sources.

We analytically solve the higher-order nonlinear Schrödinger (HNLS) equation with non-Kerr nonlinearity under some parametric conditions and obtain results for bright and dark solitary wave solutions. The functional form of these solutions are different from the traditional sech and tanh bright and dark solitons. Periodic wave solutions are also presented. Going over to the traveling coordinate, we reduce the complicated HNLS equation to the Hamiltonian form and treat the resulting equations by the dynamical systems theory. The results of our study demonstrate that the equation can, in general, support both soliton (bright and dark) and periodic solutions. We estimate the size of the derivative non-Kerr nonlinear coefficients. The results are in good agreement with those of the waveguide made of highly nonlinear optical materials. Our calculated values can be used as model parameters for sub-10 fs pulse propagation.

We introduced complexly coupled modified KdV (ccmKdV) equations, which could be derived from a two-layer fluid model [Yang and Mao, Chin. Phys. Lett. 25, 1527 (2008); Hu, J. Phys. A: Math. Theor. 43, 185207 (2009)], and used the Miura transformation to construct expressions for their alternative Lax pair representations. We derived a Lagrangian-based approach to study the Hamiltonian structures of the ccmKdV equations and observed that the complexly coupled mKdV equations have an additional analytic structure. The coupled equations were characterized by two alternative Lagrangians not connected by a gauge term. We examined how the alternative Lagrangian descriptions of the system affect the bi-Hamiltonian structures.

We show that the complex modified KdV (cmKdV) equation and generalized nonlinear Schr\"odinger (GNLS) equation belong to the Ablowitz, Kaup, Newell and Segur or so-called AKNS hierarchy. Both equations do not follow from the action principle and are nonintegrable. By introducing some auxiliary fields we obtain the variational principle for them and study their canonical structures. We make use of a coupled amplitude-phase method to solve the equations analytically and derive conditions under which they can support bright and dark solitary wave solutions.

We consider equations in the modified KdV (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV equations. We also show that the complex modified KdV (cmKdV) equation follows from the action principle to have a Lagrangian representation. This representation not only provides a basis to write the cmKdV equation in the canonical form endowed with an appropriate Poisson structure but also help us construct a semianalytical solution of it. The solution obtained by us may serve as a useful guide for purely numerical routines which are currently being used to solve the cmKdV eqution.

We present new type of Dark-in-the-Bright solution also called dipole soliton for the higher order nonlinear Schrödinger (HNLS) equation with non-Kerr nonlinearity under some parametric conditions and subject to constraint relation among the parameters in optical context. This equation could be a model equation of pulse propagation beyond ultrashort range in optical communication systems. The solitary wave solution is composed of the product of bright and dark solitary waves. This type of pulse shape to be formed both the group velocity dispersion and third-order dispersion must be compensated. We also investigated the stability of the solitary wave solution under some initial perturbation on the parametric conditions. We have shown that the shape of pulse remains unchanged up to 20 normalized lengths even under some very small violation in parametric conditions.

It is pointed out that the higher-order symmetries of the Camassa-Holm (CH) equation are nonlocal and nonlocality poses problems to obtain higher-order conserved densities for this integrable equation (J. Phys. A: Math. Gen. 2005, {\bf 38} 869-880). This difficulty is circumvented by defining a nolinear hierarchy for the CH equation and an explicit expression is constructed for the nth-order conserved density.

We have sought to work with an approach to Noether symmetry analysis which uses the properties of infinitesimal point transformations in the space-time (q, t) variable to establish the association between symmetries and conservation laws of a dynamical system. In this approach symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of two uncoupled Harmonic oscillators and two such oscillators coupled by an interaction. Both these systems can have alternative Lagrangian representations. We have studied in detail how the association between symmetries and conservation laws changes as one alters the analytic or Lagrangian representation. This analysis is carried out with a view to explicitly demonstrate that the correlation between symmetry transformation and corresponding invariant quantity depends crucially on the choice of the analytic representation.

We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as $K(n,\,m)$ equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the $K(2,\,2)$ and $K(3,\,3)$ cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the $K(3,\,2)$ equation for which the parameter can take only negative values. The $K(2,\,3)$ equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.

A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type recursion operator is then employed to construct a hierarchy of Lagrangian equations. It is explicitly demonstrated that the constructed system of equations has a Lax representation and two compatible Hamiltonian structures. The homogeneous balance method is used to derive analytic soliton solutions of the third- and fifth-order equations.

An expression for the transition probability or form factor in one-dimensional Rydberg atom irradiated by short half-cycle pulse was constructed. In applicative contexts, our expression was found to be more useful than the corresponding result given by Landau and Lifshitz. Using the new expression for the form factor, the motion of a localized quantum wave packet was studied with particular emphasis on its revival and super-revival properties. Closed form analytical expressions were derived for expectation values of the position and momentum operators that characterized the widths of the position and momentum distributions. Transient phase-space localization of the wave packet produced by the application of a single impulsive kick was explicitly demonstrated. The undulation of the uncertainty product as a function of time was studied in order to visualize how the motion of the wave packet in its classical trajectory spreads throughout the orbit and the system becomes nonclassical. The process, however, repeats itself such that the atom undergoes a free evolution from a classical, to a nonclassical, and back to a classical state.

Supersymmetrization of a nonlinear evolution equation in which the bosonic equation is independent of the fermionic variable and the system is linear in fermionic field goes by the name B-supersymmetrization. We provide B-supersymmetric extension of a number of quasilinear and fully nonlinear evolution equations and demonstrate that the supersymmetric system follows from the usual action principle. We observe that B-supersymmetrization can also be realized using a generalized Noetherian symmetry such that the resulting set of Lagrangian symmetries coincides with symmetries of the field equations. Following this viewpoint we derive conservation laws for the supersymmetric pair of equations. We attempt to realize the bosonic and fermionic fields in terms of bright and dark solitons. The interpretation sought by us has its origin in the classic work of Bateman who introduced a reverse-time system with negative friction to bring linear dissipative systems within the framework of variational principle.