A method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. In addition to the above we introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equation. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a twodimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere S 2 and hyperbolic plane H 2 , is discussed.MSC Classification: 37J35, 34A34, 34C15, 70H06 a)

A Liénard type nonlinear oscillator of the form x+kxx+(k2/9)x3+lambda1x=0, which may also be considered as a generalized Emden-type equation, is shown to possess unusual nonlinear dynamical properties. It is shown to admit explicit nonisolated periodic orbits of conservative Hamiltonian type for lambda1>0. These periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator. This is completely contrary to the standard characteristic property of nonlinear oscillators. Interestingly, the system though appears deceptively a dissipative type for lambda1< or =0 does admit a conserved Hamiltonian description, where the characteristic decay time is also independent of the amplitude. The results also show that the criterion for conservative Hamiltonian system in terms of divergence of flow function needs to be generalized.

In this paper, we consider a generalized second order nonlinear ordinary differential equation of the formẍ + (k 1 x q + k 2 )ẋ + k 3 x 2q+1 + k 4 x q+1 + λ 1 x = 0, where k i 's, i = 1, 2, 3, 4, λ 1 and q are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden type equation and its hierarchy, generalized Duffing-van der Pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent q, q ∈ R. The q = 1 and q = 2 cases are analyzed in detail and the results are generalized to arbitrary q. Our results show that many classical integrable nonlinear oscillators can be derived as sub-cases of our results and significantly enlarge the list of integrable equations that exist in the contemporary literature. To explore the above underlying results we use the recently introduced generalized extended Prelle-Singer procedure applicable to second order ODEs. As an added advantage of the method we not only identify integrable regimes but also construct integrating factors, integrals of motion and general solutions for the integrable cases, wherever possible, and bring out the mathematical structures associated with each of the integrable cases.

In this work, we establish a connection between the extended Prelle-Singer procedure 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 the modified Emden equation as a suitable example.

In this paper, we demonstrate that the modified Emden type equation (MEE),ẍ + αxẋ + βx 3 = 0, is integrable either explicitly or by quadrature for any value of α and β. We also prove that the MEE possesses appropriate time-independent Hamiltonian function for the full range of parameters α and β. In addition, we show that the MEE is intimately connected with two well known nonlinear models, namely the force-free Duffing type oscillator equation and the two dimensional Lotka-Volterra (LV) equation and thus the complete integrability of the latter two models can also be understood in terms of the MEE.

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.

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. a Corresponding author E-mail: lakshman@cnld.bdu.ac.in 1 Discovery of solitons has created a new pathway to understand the wave propagation in many physical systems with nonlinearity [1]. In particular, the existence of optical solitons in nonlinear Kerr media [2] provoked the investigation on solitons from different perspectives, particularly from applications point of view. By generalizing the waves propagating in an isotropic medium[3] to an anisotropic medium, a pair of coupled equations for orthogonally polarized waves has been obtained by Manakov [4,5] aswhere q j , j = 1, 2, describe orthogonally polarized complex waves. Here the subscripts z and t represent normalized distance and retarded time, respectively. Equation (1) also appears in many physical situations such as single optical field propagation in birefringent fibers [6], self trapped incoherent light beam propagation in photorefractive medium [7-9] and so on. Generalization of Eq. (1) to arbitrary N-waves is useful to model optical pulse propagation in multi-mode fibers [10]. It has been identified [4] that the polarization vectors of the solitons change when orthogonally polarized waves nonlinearly interact with each other leading to energy exchange interaction between the modes [11]. Experimental observation of the latter has been demonstrated in [12-14]. The shape changing collision property of such waves, which we designate here as degenerate polarized soliton propagating with identical velocity and wave number in the two modes, gave rise to the possibility of constructing logic gates leading to all optical computing atleast in a theoretical sense [15-17]. Energy sharing collisions among the optical vector solitons has been explored [16]by constructing multi-soliton solutions explicitly to the multi-component nonlinear Schrödinger equations. Further, it has been shown that the multi-soliton interaction process satisfies Yang-Baxter relation [18]. It is clear from these studies that the shape changing collision that occurs among the solitons with identical wave numbers in all the modes has been well understood. However, to our knowledge, studies on solitons with non-identical wave numbers in all the modes have not been considered so far. Consequently one would like to...