Lie symmetry analysis is one of the powerful tools to analyse nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, 𝜆-symmetries, adjoint symmetries and telescopic vector fields of a secondorder ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between

1. symmetries and integrating factors and

2. symmetries and integrals are also discussed and illustrated through the same example.

The interconnections between some of the above symmetries, i.e.,

1. Lie point symmetries and 𝜆-symmetries and

2. exponential nonlocal symmetries and 𝜆-symmetries are also discussed.

The order reduction procedure is invoked to derive the general solution of the second-order equation.

In this paper, we briefly present an overview of the recent developments made in identifying/generating systems of Liénard-type nonlinear oscillators exhibiting isochronous properties, including linear, quadratic and mixed cases and their higher-order generalizations. There exists several procedures/methods in the literature to identify/generate isochronous systems. The application of local as well as nonlocal transformations and 𝛺-modified Hamiltonian method in identifying and generating systems exhibiting isochronous properties of arbitrary dimensions is also discussed in detail. The identified oscillators include singular and nonsingular Hamiltonian systems and PT-symmetric systems.

Systematic analytic methods of deriving integrable mappings from integrable nonlinear ordinary differential, differential-difference and lattice equations are presented. More specifically, we explain how to derive integrable mappings through four different techniques namely,

1. dis-cretization technique,

2. Lax pair approach,

3. periodic reduction of integrable nonlinear partial difference equations and

4. construction of sufficient number of integrals of motion.

The applicability of methods have been illustrated through Ricatti equation, a scalar second-order nonlinear ordinary differential equation with cubic nonlinearity, 2- and 3-coupled second-order nonlinear ordinary differential equations with cubic nonlinearity, lattice equations of Korteweg–de Vries, modified Korteweg–deVries and sine-Gordon types.

In this short review, we present some applications and historical facts about the integrability detectors: Painlevé analysis, singularity confinement and algebraic entropy.

A systematic method is given to derive Lie point symmetries of time-fractional partial differential equation with Riemann–Liouville fractional derivative and its applicability illustrated through

1. time-fractional diffusive equation and

2. time-fractional cylindrical Korteweg–de Vries equation.

Using the Lie point symmetries obtained, we show that each of them has been transformed into ordinary differential equation of fractional order with a new independent variable. We also explain how exact or invariant solutions can be derived from the obtained point symmetries.

A detailed review of the invention of Cole–Hopf transformations for the Burgers equation and all the subsequent works which include generalizations of the Burgers equation and the corresponding developments in Cole–Hopf transformations are documented.

In this paper we show that the Darboux transformation for a large class of nonlinear evolution equations arises due to factorization and commutation. The factorization and commutation has been pointed out earlier for Schrödinger operator. We show that it extends to a large class of nonlinear differential equations which admit Lax pairs including Boussinesq, Davey–Stewartson, Bogoyavlensky–Schiff and 𝑛-wave interaction equation.

In this paper, we discuss the fascinating energy sharing collisions of multicomponent solitons in certain incoherently coupled and coherently coupled nonlinear Schrödinger-type equations arising in the context of nonlinear optics.

Reviewing briefly the concept of classical and quantum integrable systems, we propose an alternative Lax operator approach, leading to quasi-higher-dimensional integrable model, unravelling some hidden dimensions in integrable systems. As an example, we construct a novel integrable quasi-two-dimensional NLS equation at the classical and the quantum levels with intriguing application in rogue wave modelling.

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry, i.e., one of them has gain and the other an equal and opposite amount of loss. We first discuss various symmetries of the model. We show that both the linear system as well as a special case of the nonlinear system can be derived from a Hamiltonian, whose structure is similar to the Pais–Uhlenbeck Hamiltonian. Exact solutions are obtained in a few special cases. We show that the system is a superintegrable system within the rotating wave approximation (RWA). We also obtain several exact solutions of these RWA equations. Further, we point out a novel superposition in the context of periodic solutions in terms of Jacobi elliptic functions that we obtain in this problem. Finally, we briefly mention numerical results about the stability of some of the solutions.

Over the past twelve years, ideas and methods from nonlinear dynamics system theory, in particular, group theoretical methods in bifurcation theory, have been used to study, design, and fabricate novel engineering technologies. For instance, the existence and stability of heteroclinic cycles in coupled bistable systems has been exploited to develop and deploy highly sensitive, lowpower, magnetic and electric field sensors. Also, patterns of behaviour in networks of oscillators with certain symmetry groups have been extensively studied and the results have been applied to conceptualize a multifrequency up/down converter, a channelizer to lock into incoming signals, and a microwave signal generator at the nanoscale. In this manuscript, a review of the most recent work on modelling and analysis of two seemingly different systems, an array of gyroscopes and an array of energy harvesters, is presented. Empirical values of operational parameters suggest that damping and external forcing occur at a lower scale compared to other parameters, so that the individual units can be treated as Hamiltonian systems. Casting the governing equations in Hamiltonian form leads to a common approach to study both arrays. More importantly, the approach yields analytical expressions for the onset of bifurcations to synchronized oscillations. The expressions are valid for arrays of any size and the ensuing synchronized oscillations are critical to enhance performance.

The impact of current on static and kinetic depinning fields of a domain wall in a onedimensional ferromagnetic nanostrip is investigated analytically and numerically by solving the Landau–Lifshitz–Gilbert equation with adiabatic and non-adiabatic spin-transfer torques. The results show that in the absence of current, the static depinning field is greater than the kinetic depinning field. Both the depinning fields decrease by increasing the current applied in a direction opposite to the direction of the applied field. Both the depinning fields can also be tuned by the current to make them equal.

We discuss about the theory of nonlinear localized excitations, such as soliton and compactons in the gap of the linear spectrum of the nonlinear systems. We show how the gap originates in the linear spectrum using examples of a few systems, such as nonlinear lattices, Bose–Einstein condensates in optical lattice and systems represented by coupled nonlinear evolution equations. We then analytically show the excitation of solitons and compacton-like solutions in the gap of the linear spectrum of a system of coupled Korteweg–de Vries (KdV) equations with linear and nonlinear dispersions. Finally, we discuss about the theory of Feshbach resonance management and dispersion management of the soliton solutions.

Some important properties of photorefractive spatial solitons and their applications have been reviewed in the present paper. Using band transport model, the governing principle of photorefractive nonlinearity has been addressed and nonlinear dynamical equations of spatial solitons owing to this nonlinearity have been discussed. Mechanisms of formation of screening and photovoltaic solitons of three different configurations, i.e., bright, dark and grey varieties have been examined. Incoherently coupled vector solitons due to single and two-photon photorefractive phenomena have been highlighted. Modulation instability of a broad quasicontinuous optical beam has also been discussed. Finally possible applications have been highlighted.

We present a practical design of novel photonic crystal fibre (PCF) to investigate the nonlinear propagation of femtosecond pulses for the application of optical coherence tomography (OCT) based on supercontinuum generation (SCG) process. In addition, this paper contains a brief introduction of the physical phenomena of soliton and SCG. Typically, here we discuss how the ultrabroadband radiation in PCF can be generated by SCG through various nonlinear effects of the fibre. To accomplish the proposed aim, we put forth liquid core PCF (LCPCF) structure filled with chloroform for OCT measurements of the eye. From the proposed design, we observe that proposed LCPCFs with liquid material exhibit significant broadened wavelength spectrum with low input pulse energy over small propagation distances for the OCT application.

It is well known that solitons in integrable systems recover their original profiles after their mutual collisions. This is not true in the case of optical fibre arrays, governed by a set of integrable coupled nonlinear Schrödinger (CNLS) equations. We consider the Manakov- and mixed-type `two-component' CNLS systems. The most important characteristics of these systems are:

1. The polarizations of the two-component solitons are changed through their mutual collisions (Manakov system) and

2. the energy (intensity) switching occurs through the head-on collision (mixed system).

By placing the above solitons on the primary star graph (PSG), we see that soliton collisions give rise to interesting phase changes in PSG:

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These results will be applicable to network protocols using optical fibre arrays.

It is shown that in (2+1)-dimensional condensed matter systems, induced gravitational Chern–Simons (CS) action can play a crucial role for coherent spin transport in a finite geometry, provided zero-curvature condition is satisfied on the boundary. The role of the resultant KdV solitons is explicated. The fact that KdV solitons can pass through each other without interference, represent `resistanceless' spin transport.

We present a unified formulation to investigate solitons for all background densities in the Bose–Einstein condensate of a system of hard-core bosons with nearest-neighbour attractive interactions, using an extended Bose–Hubbard lattice model. We derive in detail the characteristics of the solitons supported in the continuum version, for the various cases possible. In general, two species of solitons appear: A nonpersistent (NP) type that fully delocalizes at its maximum speed and a persistent (P) type that survives even at its maximum speed. When the background condensate density is nonzero, both species coexist, the soliton is associated with a constant intrinsic frequency, and its maximum speed is the speed of sound. In contrast, when the background condensate density is zero, the system has neither a fixed frequency, nor a speed of sound. Here, the maximum soliton speed depends on the frequency, which can be tuned to lead to a cross-over between the NP-type and the P-type at a certain critical frequency, determined by the energy parameters of the system. We provide a single functional form for the soliton profile, from which diverse characteristics for various background densities can be obtained. Using mapping to spin systems enables us to characterize, in a unified fashion, the corresponding class of magnetic solitons in Heisenberg spin chains with different types of anisotropy.