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M. Lakshmanan

Publications and source records attributed to M. Lakshmanan.

At least 253 records · Page 14Linked to original sources
arXiv2003

Analytic calculation of nonadiabatic transition probabilities from monodromy of differential equations

The nonadiabatic transition probabilities in the two-level systems are calculated analytically by using the monodromy matrix determining the global feature of the underlying differential equation. We study the time-dependent 2x2 Hamiltonian with the tanh-type plus sech-type energy difference and with constant off-diagonal elements as an example to show the efficiency of the monodromy approach. The application of this method to multi-level systems is also discussed.

quant-ph↗
arXiv2003

Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape changing collisions, logic gates and partially coherent solitons

The novel dynamical features underlying soliton interactions in coupled nonlinear Schr{ö}dinger equations, which model multimode wave propagation under varied physical situations in nonlinear optics, are studied. In this paper, by explicitly constructing multisoliton solutions (upto four-soliton solutions) for two coupled and arbitrary $N$-coupled nonlinear Schr{ö}dinger equations using the Hirota bilinearization method, we bring out clearly the various features underlying the fascinating shape changing (intensity redistribution) collisions of solitons, including changes in amplitudes, phases and relative separation distances, and the very many possibilities of energy redistributions among the modes of solitons. However in this multisoliton collision process the pair-wise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons. Detailed asymptotic analysis also shows that when solitons undergo multiple collisions, there exists the exciting possibility of shape restoration of atleast one soliton during interactions of more than two solitons represented by three and higher order soliton solutions. From application point of view, we have shown from the asymptotic expressions how the amplitude (intensity) redistribution can be written as a generalized linear fractional transformation for the $N$-component case. Also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters, leading to possible multistate logic. In addition, we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape changing collisions, thereby explaining their variable profile and shape variation in collision process.

nlin.SI↗
arXiv2002

Effect of Phase Shift in Shape Changing Collision of Solitons in Coupled Nonlinear Schroedinger Equations

Soliton interactions in systems modelled by coupled nonlinear Schroedinger (CNLS) equations and encountered in phenomena such as wave propagation in optical fibers and photorefractive media possess unusual features : shape changing intensity redistributions, amplitude dependent phase shifts and relative separation distances. We demonstrate these properties in the case of integrable 2-CNLS equations. As a simple example, we consider the stationary two-soliton solution which is equivalent to the so-called partially coherent soliton (PCS) solution discussed much in the recent literature.

nlin.SI↗
arXiv2001

Shape-changing Collisions of Coupled Bright Solitons in Birefringent Optical Fibers

Wecritically review the recent progress in understanding soliton propagation in birefringent optical fibers.By constructing the most general bright two-soliton solution of the integrable coupled nonlinear Schroedinger equation (Manakov model) we point out that solitons in birefringent fibers can in general change their shape after interaction due to a change in the intensity distribution among the modes even though the total energy is conserved. However, the standard shape-preserving collision (elastic collision) property of the (1+1)-dimensional solitons is recovered when restrictions are imposed on some of the soliton parameters. As a consequence the following further properties can be deduced using this shape-changing collision. (i) The exciting possibility of switching of solitons between orthogonally polarized modes of the birefringent fiber exists. (ii) When additional effects due to periodic rotation of birefringence axes are considered, the shape changing collision can be used as a switch to suppress or to enhance the periodic intensity exchange between the orthogonally polarized modes. (iii) For ultra short optical soliton pulse propagation in non-Kerr media, from the governing equation an integrable system of coupled nonlinear Schroedinger equation with cubic-quintic terms is identified. It admits a nonlocal Poisson bracket structure. (iv) If we take the higher-order terms in the coupled nonlinear Schroedinger equation into account then their effect on the shape-changing collision of solitons, during optical pulse propagation, can be studied by using a direct perturbational approach.

nlin.SI↗
arXiv2001

Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schroedinger equations

We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schroedinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general $N$-coupled nonlinear Schroedinger equations (N-CNLS). It is pointed out that the underlying solitons undergo inelastic (shape changing) collisions due to intensity redistribution among the modes. We also analyse the various possibilities and conditions for such collisions to occur. Further, we report the significant fact that the various partial coherent solitons (PCS) discussed in the literature are special cases of the higher order bright soliton solutions of the N-CNLS equations.

nlin.SI↗
arXiv2000

Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method I. General Theory

The present paper concerns the derivation of phase-integral quantization conditions for the two-centre Coulomb problem under the assumption that the two Coulomb centres are fixed. With this restriction we treat the general two-centre Coulomb problem according to the phase-integral method, in which one uses an {\it a priori} unspecified {\it base function}. We consider base functions containing three unspecified parameters $C, \tilde C$ and $Λ$. When the absolute value of the magnetic quantum number $m$ is not too small, it is most appropriate to choose $Λ=|m|\ne 0$. When, on the other hand, $|m|$ is sufficiently small, it is most appropriate to choose $Λ= 0$. Arbitrary-order phase-integral quantization conditions are obtained for these choices of $Λ$. The parameters $C$ and $\tilde C$ are determined from the requirement that the results of the first and the third order of the phase-integral approximation coincide, which makes the first-order approximation as good as possible. In order to make the paper to some extent self-contained, a short review of the phase-integral method is given in the Appendix.

nlin.SI↗
arXiv2000

Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method III. Quantization Conditions in the General Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration

In this paper we take up the quantal two-centre problem where the Coulomb centres have arbitrary positive charges. In analogy with the symmetric case, treated in the second paper of this series of papers, we use the knowledge on the quasiclassical dynamics to express the contour integrals in the first- and third-order approximations of the phase-integral quantization conditions, given in the first paper of this series of papers, in terms of complete elliptic integrals. For various values of the distance between these charges the accuracy of the formulas obtained is illustrated by comparison with available numerically exact results.

nlin.SI↗
arXiv2000

Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method II. Quantization Conditions in the Symmetric Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration

The contour integrals, occurring in the arbitrary-order phase-integral quantization conditions given in a previous paper, are in the first- and third-order approximations expressed in terms of complete elliptic integrals in the case that the charges of the Coulomb centres are equal. The evaluation of the integrals is facilitated by the knowledge of quasiclassical dynamics. The resulting quantization conditions involving complete elliptic integrals are solved numerically to obtain the energy eigenvalues and the separation constants of the $1sσ$ and $2pσ$ states of the hydrogen molecule ion for various values of the internuclear distance. The accuracy of the formulas obtained is illustrated by comparison with available numerically exact results.

nlin.SI↗
arXiv2000

Rich Variety of Bifurcations and Chaos in a Variant of Murali-Lakshmanan-Chua Circuit

A very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element, exhibiting a rich variety of dynamical features, is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua(MLC). By constructing a two-parameter phase diagram in the $(F-ω)$ plane, corresponding to the forcing amplitude (F) and frequency $(ω)$, we identify, besides the familiar period-doubling scenario to chaos, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, Farey sequences, and so on. The chaotic dynamics is verified by both experimental as well as computer simulation studies including PSPICE.

nlin.CD↗
arXiv2000

Intermittency transitions to strange nonchaotic attractors in a quasiperiodically driven Duffing oscillator

Different mechanisms for the creation of strange nonchaotic attractors (SNAs) are studied in a two-frequency parametrically driven Duffing oscillator. We focus on intermittency transitions in particular, and show that SNAs in this system are created through quasiperiodic saddle-node bifurcations (Type-I intermittency) as well as through a quasiperiodic subharmonic bifurcation (Type-III intermittency). The intermittent attractors are characterized via a number of Lyapunov measures including the behavior of the largest nontrivial Lyapunov exponent and its variance as well as through distributions of finite-time Lyapunov exponents. These attractors are ubiquitous in quasiperiodically driven systems; the regions of occurrence of various SNAs are identified in a phase diagram of the Duffing system.

nlin.CD↗
arXiv2000

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map : Mechanisms and their characterizations

A simple quasiperiodically forced one-dimensional cubic map is shown to exhibit very many types of routes to chaos via strange nonchaotic attractors (SNAs) with reference to a two-parameter $(A-f)$ space. The routes include transitions to chaos via SNAs from both one frequency torus and period doubled torus. In the former case, we identify the fractalization and type I intermittency routes. In the latter case, we point out that atleast four distinct routes through which the truncation of torus doubling bifurcation and the birth of SNAs take place in this model. In particular, the formation of SNAs through Heagy-Hammel, fractalization and type--III intermittent mechanisms are described. In addition, it has been found that in this system there are some regions in the parameter space where a novel dynamics involving a sudden expansion of the attractor which tames the growth of period-doubling bifurcation takes place, giving birth to SNA. The SNAs created through different mechanisms are characterized by the behaviour of the Lyapunov exponents and their variance, by the estimation of phase sensitivity exponent as well as through the distribution of finite-time Lyapunov exponents.

nlin.CD↗
arXiv2000

On the conditions for synchronization in unidirectionally coupled chaotic oscillators

The conditions for synchronization in unidirectionally coupled chaotic oscillators are revisited. We demonstrate with typical examples that the conditional Lyapunov exponents (CLEs) play an important role in distinguishing between intermittent and permanent synchronizations, when the analytic conditions for chaos synchronization are not uniformly obeyed. We show that intermittent synchronization can occur when CLEs are very small positive or negative values close to zero while permanent synchronization occurs when CLEs take sufficiently large negative values. There is also strong evidence for the fact that for permanent synchronization the time of synchronization is relatively low while it is high for intermittent synchronization.

nlin.CD↗
arXiv2000

Spatiotemporal Patterns in Arrays of Coupled Nonlinear Oscillators

Nonlinear reaction-diffusion systems admit a wide variety of spatiotemporal patterns or structures. In this lecture, we point out that there is certain advantage in studying discrete arrays, namely cellular neural/nonlinear networks (CNNs), over continuous systems. Then, to illustrate these ideas, the dynamics of diffusively coupled one and two dimensional cellular nonlinear networks (CNNs), involving Murali-Lakshmanan-Chua circuit as the basic element, is considered. Propagation failure in the case of uniform diffusion and propagation blocking in the case of defects are pointed out. The mechanism behind these phenomena in terms of loss of stability is explained. Various spatiotemporal patterns arising from diffusion driven instability such as hexagons, rhombous and rolls are considered when external forces are absent. Existence of penta-hepta defects and removal of them due to external forcing is discussed. The transition from hexagonal to roll structure and breathing oscillations in the presence of external forcing is also demonstrated. Further spatiotemporal chaos, synchronization and size instability in the coupled chaotic systems are elucidated.

nlin.PS↗
arXiv2000

Secure Digital Signal Transmission by Multistep Parameter Modulation and Alternative Driving of Transmitter Variables

The idea of secure communication of digital signals via chaos synchronization has been plagued by the possibility of attractor reconstruction by eavesdroppers as pointed out by Perez and Cerdeira. In this Letter, we wish to present a very simple mechanism by which this problem can be overcome, wherein the signal is transmitted via a multistep parameter modulation combined with alternative driving of different transmitter variables, which makes the attractor reconstruction impossible. The method is illustrated by means of the Lorenz system and Chua's circuit as examples.

nlin.CD↗
arXiv1999

Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media

We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair, conserved quantities and exact soliton solutions for the proposed integrable model are given. Explicit form of two-solitons are used to study soliton interaction showing many intriguing features including inelastic (shape changing) scattering. Another novel system of coupled equations with fifth-degree nonlinearity is derived, which represents vector generalization of the known chiral-soliton bearing system.

solv-int↗
arXiv1999

Suppression and Enhancement of Soliton Switching During Interaction in Periodically Twisted Birefringent Fiber

Soliton interaction in periodically twisted birefringent optical fibers has been analysed analytically with refernce to soliton switching. For this purpose we construct the exact general two-soliton solution of the associated coupled system and investigate its asymptotic behaviour. Using the results of our analytical approach we point out that the interaction can be used as a switch to suppress or to enhance soliton switching dynamics, if one injects multi-soliton as an input pulse in the periodically twisted birefringent fiber.

solv-int↗
arXiv1998

Different routes to chaos via strange nonchaotic attractor in a quasiperiodically forced system

This paper focusses attention on the strange nonchaotic attractors (SNA) of a quasiperiodically forced dynamical system. Several routes, including the standard ones by which the appearance of strange nonchaotic attractors takes place, are shown to be realizable in the same model over a two parameters ($f-ε$) domain of the system. In particular, the transition through torus doubling to chaos via SNA, torus breaking to chaos via SNA and period doubling bifurcations of fractal torus are demonstrated with the aid of the two parameter ($f-ε$) phase diagram. More interestingly, in order to approach the strange nonchaotic attractor, the existence of several new bifurcations on the torus corresponding to the novel phenomenon of torus bubbling are described. Particularly, we point out the new routes to chaos, namely, (1) two frequency quasiperiodicity $\to$ torus doubling $\to$ torus merging followed by the gradual fractalization of torus to chaos, (2) two frequency quasiperiodicity $\to$ torus doubling $\to$ wrinkling $\to$ SNA $\to$ chaos $\to$ SNA $\to$ wrinkling $\to$ inverse torus doubling $\to$ torus $\to$ torus bubbles followed by the onset of torus breaking to chaos via SNA or followed by the onset of torus doubling route to chaos via SNA. The existence of the strange nonchaotic attractor is confirmed by calculating several characterizing quantities such as Lyapunov exponents, winding numbers, power spectral measures and dimensions. The mechanism behind the various bifurcations are also briefly discussed.

chao-dyn↗