Prof M Lakshmanan

SYMMETRIES AND SINGULARITY STRUCTURES: Professor Lakshmanan and his students were the earliest to identify the connection between soliton equations, associated Lie symmetries and Painleve transcendental equations. Further detailed analysis of generalized Lie-Bäcklund symmetries have brought out their close connection with complete integrability of soliton systems along with identification of perturbed hierarchy of integrable soliton systems. Combining Painleve singularity structure analysis with Hirota bilinearization method, Prof. Lakshmanan and his students have successfully obtained multisoliton solutions for several (1+1) dimensional systems AND exponentially localized dromion solutions in (2+1) dimensions. Very recently they have also identified a novel method to solve (2+1) dimensional integrable nonlinear evolution equations based on these connections.

FINITE DIMENSIONAL INTEGRABLE SYSTEMS: Classification of integrable finite dimensional systems, including coupled nonlinear oscillators, based on Painleve singularity structure analysis and existence of generalized symmetries were made and new integrable systems identified by Professor Lakshmanan and his students. Very recently they have proposed a fundamentally new method of isolating, identifying and solving integrable nonlinear systems of finite order, in terms of what is known as generalized modified Prelle-Singer approach, and showed how a large class of new integrable systems can be identified, their integrals of motion and explicit solutions obtained. Generalized linearization procedures have also been proposed with exciting further potentialities. A nonlocal transformation method to identify integrable nonlinear systems have been proposed. Amplitude independent isochronous oscillations in nonlinear systems have been identified. Nonstandard Hamiltonian structures in dissipative dynamical systems have been pointed out, which includes the damped harmonic oscillator.

NONLINEAR EXCITATIONS IN MAGNETIC SPIN SYSTEMS: Starting from the above mentioned seminal contribution that the isotopic continuum ferromagnetic spin system is a completely integrable solitonic system, Professor Lakshmanan and coworkers have opened up new avenues of research in nonlinear magnetic spin dynamics. The radially symmetric Heisenberg spin chain was shown to be equivalent to a generalized nonlocal nonlinear Schrödinger equation and higher dimensional spin systems were shown to take an analysable form in terms of stereographic variables. The effect of Landau-Gilbert damping was shown to be formally a complex sealing of time of the undamped spin chain. Using such an approach localized magnetic excitations in higher spatial dimensions and spatio temporal patterns arising due to Suhl’s instability were identified. Spin torque effect in nanoferromagnets in terms of generalized Landau-Lifshitz-Gilbert equation was explained and dynamic and static excitations in classical discrete anisotropic Heisenberg spin chain were obtained.

OPTICAL SOLITON INTERACTIONS AND BEC SOLITONS: The dynamics of optical solitons in multimode fibers and photorefractive materials are governed by coupled nonlinear Schrodinger equations. Prof. Lakshmanan and his coworkers have brought out the remarkable fact that the underlying solitons undergo novel shape changing/intensity redistribution collisions corresponding to generalized linear fractional transformations. Such a possibility leads to the construction of various logic gates including the universal NAND gates, through purely light-light collisions leading to the exciting possibility of all optical computers atleast in a theoretical sense in homogeneous bulk media. Very recently, Prof. Lakshmanan and his coworkers have shown the remarkable fact that in multi component mixed signs of focusing and defocusing type nonlinear coefficients, signal amplification can be identified through exact shape changing soliton collisions which can become singular if certain conditions on initial conditions are violated. Higher dimensional (2+1) NLS equations of specific types also exhibit such properties. Very many interesting solitonic structures in Bose-Einstein condensates under different conditions have been brought out.

BIFURCATIONS AND CHAOS: Professor Lakshmanan and his students have brought out the detailed bifurcation routes and chaotic structures in a wide variety of nonlinear oscillators ranging from Bonhoeffer – van der Pol, double –well Duffing-van der Pol, Josephson junction and Fitz-Hugh-Nagumo oscillators, spin systems to velocity-dependent nonpolynomial oscillators, etc. Existence, characterization and classification of various types of strange non-chaotic attractors (SNAs) in quasi- periodically forced systems have been demonstrated in a number of physically interesting systems. New transitions and mechanisms involving different routes such as Torus bubbling, type III intermittency, and subharmonic instability bifurcations for SNAs have been identified in addition to the standard routes such as Heagy-Hammel, fractalization and type–I intermittency routes. Novel bifurcation routes with hyperchaotic attractors in time delay systems have also been identified. Experimental realizations of many of these bifurcations and chaos aspects have also been made. Dynamics of several time delay systems have been elucidated and also experimentally demonstrated.

NONLINEAR ELECTRONIC CIRCUITS: With Murali and Chua, Prof. Lakshmanan had introduced the simplest nonlinear dissipative chaotic circuit (MLC circuit) as a paradigmic model for non-autonomous nonlinear dynamical systems, which is a simple nonlinear generalization of the standard series RLC circuit. A simple variant MLC circuit which is a generalization of the parallel RLC circuit had also been introduced. These two circuits admit almost all known bifurcations routes to chaos and can be investigated experimentally, numerically and analytically. Various other analog circuits like modified canonical Chua circuit and negative conductance circuit have also been studied for their bifurcations and chaos scenario. These circuits have been playing a pivotal role in various studies of chaotic dynamics including bifurcations, chaos, controlling and synchronization.

CONTROLLING AND SYNCHRONIZATION OF CHAOS: Various algorithms for controlling of chaos by feedback and nonfeedback methods have been critically evaluated by Lakshmanan and his group and new algorithms such as application of constant and pulsed external forces and external noise have been suggested for controlling chaos. Very recently with his student Palaniyandi, Prof. Lakshmanan had proposed a potentially very effective technique for estimation of control parameters and flow functions from chaotic time series data using control algorithms. Along with Murali, he had introduced several effective techniques such as one way coupling, compounding chaotic signals, etc. to study synchronization of chaos, transmission of analog and digital chaotic signals, possible secure communication by multistep parameter modulation and so on. Very recently Senthilkumar and Lakshmanan have demonstrated the notion of different kinds of synchronization scenario such as complete, anticipatory, lag, oscillating, phase synchronizations in time delay systems. Along with Senthilkumar and Kurths, Prof. M. Lakshmanan had introduced the notion of phase in nonphase coherent hyperchaotic attractors modeled by time delay systems and studied phase synchronization in these systems. Prof. Lakshmanan and his students have also demonstrated size induced instability of synchronization manifolds in arrays of coupled nonlinear oscillators and electronic circuits and emergence of different kinds of spatio-temporal patterns in them as a function of system size. Time delay induced synchronization and chaotic phase synchronization in such systems have been classified. Notion of mass synchronization of pathological states, globally clustered chimeras and event related desynchronization/synchronization have been elucidated.

QUANTUM CHAOS : With K. Nakamura, Prof. Lakshmanan had shown that for quantum bound systems whose classical versions are nonintegrable, coupled dynamical equations for energy levels and eigenfunctions with the nonintegrability parameter taken as ‘time’ are equivalent to a completely integrable Calogero-Moser system in (1+1) dimensions with internal complex vector space. This then allows one to develop analytical formalism for energy level distributions associated with nonintegrable and chaotic systems. Quantum chaotic dynamics associated with hydrogenlike Rydberg atoms in external van der Waals type interaction have been extensively analysed by Lakshmanan and his student Ganesan to explain the behaviour of such systems. Extensive analysis of quassiclassical dynamics associated with two center Coulomb problems using generalized JWKB approximations had revealed the intricate properties of these systems jointly with Athavan (Ph.D. student), N. Fröman and P.O. Fröman.