arXiv subjects

M. Senthilvelan

Publications and source records attributed to M. Senthilvelan.

At least 37 records · Page 2Linked to original sources

arXiv2022

Similarity reductions of peakon equations: integrable cubic equations

We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa-Holm (mCH) equation and Novikov's equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov's equation to a negative mKdV flow and a negative Sawada-Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second order and second degree, while for Novikov's equation the reduction is a particular case of Painlevé V. Furthermore, we show that each of these two different Painlevé-type equations is related to the particular cases of Painlevé III that arise from analogous similarity reductions of the Camassa-Holm and the Degasperis-Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlevé III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.

nlin.SI↗

arXiv2022

Nth order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

In this paper, we investigate smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger (GNLS) equation which contains higher order nonlinear effects. With the help of generalized Darboux transformation (GDT) method we construct $N$th order smooth positon solutions of GNLS equation. We study the effect of higher order nonlinear terms on these solutions. Our investigations show that the positon solutions are highly compressed by higher order nonlinear effects. The direction of positons are also get changed. We also derive $N$th order breather-positon (B-P) solution with the help of GDT. We show that these B-Ps are well compressed by the effect of higher order nonlinear terms but the period of B-P solution is not affected as in the breather solution case.

nlin.SI↗

arXiv2022

Data driven soliton solution of the nonlinear Schrödinger equation with certain $\mathcal{PT}$-symmetric potentials via deep learning

We investigate the physics informed neural network method, a deep learning approach, to approximate soliton solution of the nonlinear Schrödinger equation with parity time symmetric potentials. We consider three different parity time symmetric potentials, namely Gaussian, periodic and Rosen-Morse potentials. We use physics informed neural network to solve the considered nonlinear partial differential equation with the above three potentials. We compare the predicted result with actual result and analyze the ability of deep learning in solving the considered partial differential equation. We check the ability of deep learning in approximating the soliton solution by taking squared error between real and predicted values. {Further, we examine the factors that affect the performance of the considered deep learning method with different activation functions, namely ReLU, sigmoid and tanh. We also use a new activation function, namely sech which is not used in the field of deep learning and analyze whether this new activation function is suitable for the prediction of soliton solution of nonlinear Schrödinger equation for the aforementioned parity time symmetric potentials. In addition to the above, we present how the network's structure and the size of the training data influence the performance of the physics informed neural network. Our results show that the constructed deep learning model successfully approximates the soliton solution of the considered equation with high accuracy.

nlin.PS↗

arXiv2022

A Class of Isochronous and Non-Isochronous Nonlinear Oscillators

In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent frequency of oscillations or amplitude dependent frequency of oscillations from the integrals of the simple harmonic oscillator equation. To achieve this, we consider the case where the integrals are in the same form both for the linear and the nonlinear oscillators in either of the cases. We also discuss the method of deriving the associated integrals and the general solution in harmonic form for both the types. We demonstrate the applicability of this method up to 2N coupled first order nonlinear ODEs in both the cases. Further, we illustrate the theory with an example in each case.

nlin.SI↗

arXiv2022

Prediction of chaotic attractors in quasiperiodically forced logistic map using deep learning

We forecast two different chaotic dynamics of the quasiperiodically forced logistic map using the well-known deep learning framework Long Short-Term Memory. We generate two data sets and use one in the training process and the other in the testing process. The predicted values are evaluated using the metric called Root Mean Square Error and visualized using the scatter plots. The robustness of the Long Short-Term Memory model is evaluated using the number of units in the layers of the model. We also make multi-step forecasting of the considered system. We show that the considered Long Short-Term Memory model performs well in predicting chaotic attractors upto three steps.

cs.LG↗

arXiv2022

Exotic states induced by co-evolving connection weights and phases in complex networks

We consider an adaptive network, whose connection weights co-evolve in congruence with the dynamical states of the local nodes that are under the influence of an external stimulus. The adaptive dynamical system mimics the adaptive synaptic connections common in neuronal networks. The adaptive network under external forcing displays exotic dynamical states such as itinerant chimeras whose population density of coherent and incoherent domains co-evolves with the synaptic connection, bump states and bump frequency cluster states, which do not exist in adaptive networks without forcing. In addition the adaptive network also exhibits partial synchronization patterns such as phase and frequency clusters, forced entrained, and incoherent states. We introduce two measures for the strength of incoherence based on the standard deviation of the temporally averaged (mean) frequency and on the mean frequency in order to classify the emergent dynamical states as well as their transitions. We provide a two-parameter phase diagram showing the wealth of dynamical states. We additionally deduce the stability condition for the frequency-entrained state. We use the paradigmatic Kuramoto model of phase oscillators which is a simple generic model that has been widely employed in unraveling a plethora of cooperative phenomena in natural and man-made systems.

cond-mat.dis-nn↗

arXiv2021

Prediction of Occurrence of Extreme Events using Machine Learning

Machine learning models play a vital role in the prediction task in several fields of study. In this work, we utilize the ability of machine learning algorithms to predict the occurrence of extreme events in a nonlinear mechanical system. Extreme events are rare events that occur ubiquitously in nature. We consider four machine learning models, namely Logistic Regression, Support Vector Machine, Random Forest and Multi-Layer Perceptron in our prediction task. We train these four machine learning models using training set data and compute the performance of each model using the test set data. We show that the Multi-Layer Perceptron model performs better among the four models in the prediction of extreme events in the considered system. The persistent behaviour of the considered machine learning models is cross-checked with randomly shuffled training set and test set data.

cs.LG↗

arXiv2021

High-order synchronization in a system of nonlinearly coupled Stuart-Landau oscillators

The high-order synchronization was studied in systems driven by external force and in autonomous systems with proper frequency mismatch. Differing from the literature, in this article, we demonstrate the occurrence of high-order (1:2) synchronization in an autonomous nonlinearly coupled (Stuart-Landau) oscillators which admit a particular form of rotational symmetry. Interestingly, the observed 1:2 synchronization happens not only for a particular choice of natural frequencies but for all possible choices of frequencies. We have observed such a behaviour in the case of 1:1 synchronization, where we have seen a variety of couplings in the literature that forces the oscillators to have almost equal frequencies and makes the system to oscillate with a common frequency independent of whether the oscillators are identical or non-identical. Similarly, in this article we observe that, whether the initial choice of frequencies is of the ratio 1:2 or not, the given nonlinear coupling forces the system to oscillate with the frequency ratio 1:2. Further, we present synchronization and other dynamical behaviours of the system by considering different choices of natural frequencies.

nlin.CD↗

arXiv2021

Manifestation of strange nonchaotic attractors in extended systems: A study through out-of-time-ordered correlators

We study the spatial spread of out-of-time-ordered correlators (OTOCs) in coupled map lattices (CMLs) of quasiperiodically forced nonlinear maps. We use instantaneous speed (IS) and finite-time Lyapunov exponents (FTLEs) to investigate the role of strange non-chaotic attractors (SNAs) on the spatial spread of the OTOC. We find that these CMLs exhibit a characteristic on and off type of spread of the OTOC for SNA. Further, we provide a broad spectrum of the various dynamical regimes in a two-parameter phase diagram using IS and FTLEs. We substantiate our results by confirming the presence of SNA using established tools and measures, namely the distribution of finite-time Lyapunov exponents, phase sensitivity, spectrum of partial Fourier sums, and $0-1$ test.

cond-mat.stat-mech↗

arXiv2021

Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation

We construct rogue wave solutions of a fifth-order nonlinear Schrödinger equation on the Jacobian elliptic function background. By combining Darboux transformation and the nonlinearization of spectral problem, we generate rogue wave solution on two different periodic wave backgrounds. We analyze the obtained solutions for different values of system parameter and point out certain novel features of our results. We also compute instability growth rate of both $dn$ and $cn$ periodic background waves for the considered system through spectral stability problem. We show that instability growth rate decreases (increases) for $dn$-$(cn)$ periodic waves when we vary the value of the elliptic modulus parameter.

nlin.PS↗

arXiv2021

Constant Bias and Weak Second Periodic Forcing : Tools to Mitigate Extreme Events

We propose two potentially viable non-feedback methods, namely (i) constant bias and (ii) weak second periodic forcing as tools to mitigate extreme events. We demonstrate the effectiveness of these two tools in suppressing extreme events in two well-known nonlinear dynamical systems, namely (i) Liénard system and (ii) a non-polynomial mechanical system with velocity dependent potential. As far as the constant bias is concerned, in the Liénard system, the suppression occurs due to the decrease in large amplitude oscillations and in the non-polynomial mechanical system the suppression occurs due to the destruction of chaos into a periodic orbit. As far as the second periodic forcing is concerned, in both the examples, extreme events get suppressed due to the increase in large amplitude oscillations. We also demonstrate that by introducing a phase in the second periodic forcing one can decrease the probability of occurrence of extreme events even further in the non-polynomial system. To provide a support to the complete suppression of extreme events, we present the two parameter probability plot for all the cases. In addition to the above, we examine the feasibility of the aforementioned tools in a parametrically driven version of the non-polynomial mechanical system. Finally, we investigate how these two methods influence the multistability nature in the Liénard system.

math.DS↗

arXiv2021

Model-free prediction of emergence of extreme events in a parametrically driven nonlinear dynamical system by Deep Learning

We predict the emergence of extreme events in a parametrically driven nonlinear dynamical system using three Deep Learning models, namely Multi-Layer Perceptron, Convolutional Neural Network and Long Short-Term Memory. The Deep Learning models are trained using the training set and are allowed to predict the test set data. After prediction, the time series of the actual and the predicted values are plotted one over the other in order to visualize the performance of the models. Upon evaluating the Root Mean Square Error value between predicted and the actual values of all three models, we find that the Long Short-Term Memory model can serve as the best model to forecast the chaotic time series and to predict the emergence of extreme events for the considered system.

cs.LG↗

arXiv2021

Rogue waves on an elliptic function background in complex modified Korteweg-de Vries equation

With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg-de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfunctions and eigenvalues. To begin we construct the elliptic function background. Then, on top of this background, we create a rogue wave. We demonstrate the outcome for three distinct elliptic modulus values. We find that when we increase the modulus value the amplitude of rogue waves on the dn-periodic background decreases whereas it increases in the case of cn-periodic background.

nlin.SI↗

arXiv2021

Emergence and mitigation of extreme events in a parametrically driven system with velocity-dependent potential

In this paper, we discuss the emergence of extreme events in a parametrically driven non-polynomial mechanical system with a velocity-dependent potential. We confirm the occurrence of extreme events from the probability distribution function of the peaks, which exhibits a long-tail. We also present the mechanism for the occurrence of extreme events. We found that the probability of occurrence of extreme events alternatively increase and decrease with a brief region where the probability is zero. At the point of highest probability of extreme events, when the system is driven externally, we find that the probability decreases to zero. Our investigation confirms that the external drive can be used as an useful tool to mitigate extreme events in this nonlinear dynamical system. Through two parameter diagrams, we also demonstrate the regions where extreme events gets suppressed. In addition to the above, we show that extreme events persits when the sytem is influenced by noise and even gets transformed to super-extreme events when the state variable is influenced by noise.

cond-mat.stat-mech↗

arXiv2021

Penrose Instabilities and the Emergence of Rogue waves in Sasa-Satsuma equation

In this paper, we calculate the region of emergence of rogue waves in the Sasa-Satsuma equation by performing Penrose stability analysis. We consider Wigner-transformed Sasa-Satsuma equation and separate out unstable solutions, namely Penrose instability modes. We superpose these modes in a small region. With the help of marginal property of the Wigner transform we identify the region in which rogue wave solution can emerge in the Sasa-Satsuma equation and calculate the amount of spatial localization. We also formulate a condition for the emergence of rogue wave solution in the Sasa-Satsuma equation.

nlin.PS↗

arXiv2021

Nonlinear tunneling of solitons in a variable coefficients nonlinear Schrödinger equation with $\mathcal{PT}$-symmetric Rosen-Morse potential

We construct soliton solution of a variable coefficients nonlinear Schrödinger equation in the presence of parity reflection - time reversal $(\mathcal{PT})-$ symmetric Rosen-Morse potential using similarity transformation technique. We transform the variable coefficients nonlinear Schrödinger equation into the nonlinear Schrödinger equation with $\mathcal{PT}-$symmetric potential with certain integrability conditions. We investigate in-detail the features of the obtained soliton solutions with two different forms of dispersion parameters. Further, we analyze the nonlinear tunneling effect of soliton profiles by considering two different forms of nonlinear barrier/well and dispersion barrier/well. Our results show that the soliton can tunnel through nonlinear barrier/well and dispersion barrier/well with enlarged and suppressed amplitudes depending on the sign of the height. Our theoretical findings are experimentally realizable and might help to model the optical devices.

nlin.PS↗

arXiv2021

Rogue waves on the double periodic background in Hirota equation

We construct rogue wave solutions on the double periodic background for the Hirota equation through one fold Darboux transformation formula. We consider two types of double periodic solutions as seed solutions. We identify the squared eigenfunctions and eigenvalues that appear in the one fold Darboux transformation formula through an algebraic method with two eigenvalues. We then construct the desired solution in two steps. In the first step, we create double periodic waves as the background. In the second step, we build rogue wave solution on the top of this double periodic solution. We present the localized structures for two different values of the arbitrary parameter each one with two different sets of eigenvalues.

nlin.PS↗

arXiv2019

Nondegenerate solitons in Manakov system

It is known that Manakov equation which describes wave propagation in two mode optical fibers, photorefractive materials, etc. can admit solitons which allow energy redistribution between the modes on collision that also leads to logical computing. In this paper, we point out that Manakov system can admit more general type of nondegenerate fundamental solitons corresponding to different wave numbers, which undergo collisions without any energy redistribution. The previously known class of solitons which allows energy redistribution among the modes turns out to be a special case corresponding to solitary waves with identical wave numbers in both the modes and travelling with the same velocity. We trace out the reason behind such a possibility and analyze the physical consequences.

nlin.SI↗