arXiv subjects

M. Lakshmanan

Publications and source records attributed to M. Lakshmanan.

At least 145 records · Page 8Linked to original sources

arXiv2015

On certain analytical methods in finding integrable systems and their interconnections

In this paper, to begin with, we review six different analytical methods which are widely used to derive symmetries, integrating factors, multipliers, Darboux polynomials and integrals of second order nonlinear ordinary differential equations. We illustrate the algorithm behind each method by considering a nonlinear oscillator equation as an example. In the second part of this paper, we examine the interconnections between these methods. We establish a road map between extended Prelle-Singer procedure with all other methods cited above and thereby demonstrate the interplay between Lie point symmetries, $λ$-symmetries, adjoint symmetries, null forms, integrating factors, Darboux polynomials, and Jacobi multipliers of second order integrable ODEs. The interconnections are illustrated with the same example finally.

nlin.SI↗

arXiv2015

Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ordinary differentiable equations (ODEs). We establish an important interconnection between extended Prelle-Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

nlin.SI↗

arXiv2014

Factorization technique and isochronous condition for coupled quadratic and mixed Liénard-type nonlinear systems

In this paper, we discuss a systematic and self consistent procedure to factorize a rather general class of coupled nonlinear ordinary differential equations (ODEs), namely coupled quadratic and mixed Liénard type equations, which include various physical and mathematical models. The procedure is broadly divided into two parts. In the first part, we consider a general factorized form for the equation under consideration in terms of some unknown functions and identify the determining equations for them. In the second part, we systematically solve the determining equations and identify the compatible factorizing form for this class of equations. In addition, we also discuss the problem of identification of isochronous dynamical systems belonging to the above class of equations. In particular, we deduce an isochronocity condition for the coupled quadratic Liénard equation. We also present specific examples of physical interest.

nlin.SI↗

arXiv2014

Enhanced synchronization in an array of spin torque nano oscillators in the presence of oscillating external magnetic field

We demonstrate that the synchronization of an array of electrically coupled spin torque nano-oscillators (STNO) modelled by Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation can be enhanced appreciably in the presence of a common external microwave magnetic field. The applied microwave magnetic field stabilizes and enhances the regions of synchronization in the parameter space of our analysis, where the oscillators are exhibiting synchronized oscillations thereby emitting improved microwave power. To characterize the synchronized oscillations we have calculated the locking range in the domain of external source frequency.

nlin.CD↗

arXiv2014

Multicomponent long-wave--short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons

We consider a general multicomponent (2+1)-dimensional long-wave--short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long-wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multi-soliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.

nlin.SI↗

arXiv2014

Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Liénard type nonlinear oscillators

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schr$ö$dinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering we show that the class of systems can also be exactly solvable and are also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Li$é$nard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

quant-ph↗

arXiv2014

Manipulating matter-rogue waves and breathers in Bose-Einstein condensates

We construct higher order rogue wave solutions and breather profiles for the quasi-one-dimensional Gross-Pitaevskii equation with a time-dependent interatomic interaction and external trap through the similarity transformation technique. We consider three different forms of traps, namely (i) time-independent expulsive trap, (ii) time-dependent monotonous trap and (iii) time-dependent periodic trap. Our results show that when we change a parameter appearing in the time-independent or time-dependent trap the second and third-order rogue waves transform into the first-order like rogue waves. We also analyze the density profiles of breather solutions. Here also we show that the shapes of the breathers change when we tune the strength of trap parameter. Our results may help to manage rogue waves experimentally in a BEC system.

nlin.SI↗

arXiv2014

Mechanism for intensity induced chimera states in globally coupled oscillators

We identify the mechanism behind the existence of intensity induced chimera states in globally coupled oscillators. We find that the effect of intensity in the system is to cause multistability by increasing the number of fixed points. This in turn increases the number of multistable attractors and we find that their stability is determined by the strength of coupling . This causes the coexistence of different collective states in the system depending upon the initial state. We demonstrate that intensity induced chimera is generic to both periodic and chaotic systems. We have discussed possible applications of our results to real world systems like the brain and spin torque nano oscillators.

nlin.AO↗

arXiv2014

Mixed solitons in (2+1) dimensional multicomponent long-wave--short-wave system

We derive a (2+1)-dimensional multicomponent long-wave$-$short-wave resonance interaction (LSRI) system as the evolution equation for propagation of $N$-dispersive waves in weak Kerr type nonlinear medium in the small amplitude limit. The mixed (bright-dark) type soliton solutions of a particular (2+1)-dimensional multicomponent LSRI system, deduced from the general multicomponent higher dimensional LSRI system, are obtained by applying the Hirota's bilinearization method. Particularly, we show that the solitons in the LSRI system with two short-wave components behave like scalar solitons. We point out that for $N$-component LSRI system with $N>3$, if the bright solitons appear in atleast two components, interesting collision behaviour takes place resulting in energy exchange among the bright solitons. However the dark solitons undergo standard elastic collision accompanied by a position-shift and a phase-shift. Our analysis on the mixed bound solitons shows that the additional degree of freedom which arises due to the higher dimensional nature of the system results in a wide range of parameters for which the soliton collision can take place.

nlin.SI↗

arXiv2014

Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling

By developing the concepts of strength of incoherence and discontinuity measure, we show that a distinct quantitative characterization of chimera and multichimera states which occur in networks of coupled nonlinear dynamical systems admitting nonlocal interactions of finite radius can be made. These measures also clearly distinguish between chimera or multichimera states (both stable and breathing types) and coherent and incoherent as well as cluster states....

nlin.CD↗

arXiv2014

Dark solitons, Breathers and Rogue Wave Solutions of the Coupled Generalized Nonlinear Schrödinger Equations

We construct dark-dark solitons, general breather (GB), Akhmediev breather (AB), Ma soliton (MS) and rogue wave (RW) solutions of a coupled generalized nonlinear Schrödinger (CGNLS) equation. While the dark-dark solitons are captured in the defocusing regime of CGNLS system, the other solutions, namely GB, AB, MS and RW, are identified in the focusing regime. We also analyze the structures of GB, AB, MS and RW profiles with respect to the four-wave mixing parameter. We show that when we increase the value of the real part of the four wave mixing parameter, the number of peaks in the breather profile increases and the width of each peak gets shrunk. Interestingly the direction of this profile also changes due to this change. As far as the RW profile is concerned the width of the peak becomes very thin when we increase the value of this parameter. Further, we consider RW solution as the starting point and derive AB, MS and GB in the reverse direction and show that the solutions obtained in both the directions match with each other. In the course of the reverse analysis we also demonstrate how to capture the RW solutions directly from AB and MS.

nlin.SI↗

arXiv2014

Integrable Motion of Curves in Self-Consistent Potentials : Relation to Spin Systems and Soliton Equations

Motion of curves and surfaces in $\R^3$ lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that a more general situation in which the curves evolve in the presence of additional self consistent vector potentials can lead to interesting generalized spin systems with self consistent potentials or soliton equations with self consistent potentials. We obtain the general form of the evolution equations of underlying curves and report specific examples of generalized spin chains and soliton equations. These include principal chiral model and various Myrzakulov spin equations in (1+1) dimensions and their geometrically equivalent generalized nonlinear Schrödinger (NLS) family of equations, including Hirota-Maxwell-Bloch equations, all in the presence of self consistent potential fields. The associated gauge equivalent Lax pairs are also presented to confirm their integrability.

nlin.PS↗

arXiv2014

Intrinsic Localized Modes of a Classical Discrete Anisotropic Heisenberg Ferromagnetic Spin Chain

We report several exact intrinsic localized mode solutions of the classical spin evolution equation of a one-dimensional anisotropic Heisenberg ferromagnetic spin chain in terms of Jacobian elliptic functions. These include one, two and three spin excitations. All these solutions have smooth anticontinuum limits. Their linear stability and semiclassical quantization are also discussed briefly.

nlin.PS↗

arXiv2014

On the complete Lie point symmetries classification of the mixed quadratic-linear Li$\acute{\textbf{e}}$nard type equation $\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0$

In this paper we develop a systematic and self consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter) and non-maximal (one and two parameter) symmetry groups associated with the mixed quadratic-linear Li$\acute{e}$nard type equation, $\ddot {x} + f(x){\dot {x}}^{2} + g(x)\dot{x}+h(x)= 0$, where $f(x),\,g(x)$ and $h(x)$ are arbitrary functions of $x$. With the help of this procedure we show that a symmetry function $b(t)$ is zero for non-maximal cases whereas it is not so for the maximal case. On the basis of this result the symmetry analysis gets divided into two cases, $(i)$ the maximal symmetry group $(b\neq0)$ and $(ii)$ non-maximal symmetry groups $(b=0)$. We then identify the most general form of the mixed-quadratic linear Li$\acute{e}$nard type equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable. We present a specific example of physical interest. In the case of non-maximal symmetry groups the identified equations are all integrable. The integrability of all the equations is proved either by providing the general solution or by constructing time independent Hamiltonians. We also analyse the underlying equivalence transformations.

nlin.SI↗

arXiv2014

Dynamic environment coupling induce synchronized states in coupled time-delayed electronic circuits

We experimentally demonstrate the occurrence of various synchronized states in coupled piece-wise linear time-delayed electronic circuits using dynamic environment coupling where the environment has its own intrinsic dynamics via feedback from the circuits. We carry out these experiments in two different coupling configurations, namely mutual and subsystem coupling configurations. Depending upon the coupling strength and the nature of feedback, we observe a transition from nonsynchronization to complete synchronization via phase synchronization and from nonsynchronization to inverse synchronization via inverse-phase synchronization between the circuits in hyperchaotic regime. Snapshots of the time evolution, phase projection plots and localized sets of the circuits as observed experimentally from the oscilloscope, along with supporting numerical simulations confirm the existence of different synchronized states. Further, the transition to different synchronized states can be verified from the changes in the largest Lyapunov exponents, Correlation of Probability of Recurrence and Correlation Coefficient as a function of the coupling strength. We present a detailed linear stability analysis and obtain conditions for different synchronized states.

nlin.CD↗

arXiv2014

Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second order differential equations

In this work, we establish a connection between the extended Prelle-Singer procedure (Chandrasekar \textit{et al.} Proc. R. Soc. A 2005) with five other analytical methods which are widely used to identify integrable systems in the contemporary literature, especially for second order nonlinear ordinary differential equations (ODEs). By synthesizing these methods we bring out the interplay between Lie point symmetries, $λ$-symmetries, adjoint symmetries, null-forms, Darboux polynomials, integrating factors and Jacobi last multiplier in identifying the integrable systems described by second order ODEs. We also give new perspectives to the extended Prelle-Singer procedure developed by us. We illustrate these subtle connections with modified Emden equation as a suitable example.

nlin.SI↗

arXiv2013

Adaptive coupling induced multi-stable states in complex networks

Adaptive coupling, where the coupling is dynamical and depends on the behaviour of the oscillators in a complex system, is one of the most crucial factors to control the dynamics and streamline various processes in complex networks. In this paper, we have demonstrated the occurrence of multi-stable states in a system of identical phase oscillators that are dynamically coupled. We find that the multi-stable state is comprised of a two cluster synchronization state where the clusters are in anti-phase relationship with each other and a desynchronization state. We also find that the phase relationship between the oscillators is asymptotically stable irrespective of whether there is synchronization or desynchronization in the system. The time scale of the coupling affects the size of the clusters in the two cluster state. We also investigate the effect of both the coupling asymmetry and plasticity asymmetry on the multi-stable states. In the absence of coupling asymmetry, increasing the plasticity asymmetry causes the system to go from a two clustered state to a desynchronization state and then to a two clustered state. Further, the coupling asymmetry, if present, also affects this transition. We also analytically find the occurrence of the above mentioned multi-stable-desynchronization-multi-stable state transition. A brief discussion on the phase evolution of nonidentical oscillators is also provided. Our analytical results are in good agreement with our numerical observations.

nlin.AO↗

arXiv2013

A nonlocal connection between certain linear and nonlinear ordinary differential equations - Part II: Complex nonlinear oscillators

In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations (ODEs) with complex linear ODEs, thereby proving the integrability of the former. We also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples.

nlin.SI↗