Narrow Results

We apply Painlevé test to the most general variable coefficient nonlinear Schrödinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like group-theoretical approach and the Lax pairs technique of the soliton theory. We presented some exact solutions based on point transformations relating analytic solutions of VCNLS equations for specific choices of the coefficients to those of the standard integrable NLS equation.

We propose several different types of construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. In analogy to conformal and affine Toda theories based on root systems of semi-simple Lie algebras, also their Lorentzian extensions come about in conformal and massive variants. We carry out the Painlevé integrability test for the proposed theories, finding in general only one integer valued resonance corresponding to the energy-momentum tensor. Thus most of the Lorentzian Toda field theories are not integrable, as the remaining resonances, that grade the spins of the W-algebras in the semi-simple cases, are either non-integer or complex valued. We analyze in detail the classical mass spectra of several massive variants. Lorentzian Toda field theories may be viewed as perturbed versions of integrable theories equipped with an algebraic framework.

In this paper, we investigated a new form of nonlinear Schrödinger equation (NLSE), namely the Biswas–Arshed model (BAM) for the analysis of complete integrability with the help of Painlevé test ($P$-test). By applying this test, we analyze the singularity structure of the solutions of BAM, knowing the fact that the absence of specific sort of singularities like moveable branch points is a patent signal for the complete integrability of the discussed model. Passing the $P$-test is a powerful indicator that the studied model is resolvable by means of inverse scattering transformation (IST).

In this paper, our objective is to analyze integrability of three famous nonlinear models, namely unstable nonlinear Schrödinger equation (UNLSE), modified UNLSE (MUNLSE) as well as (2+1)-dimensional cubic NLSE (CNLSE) by utilizing Painlevé test ($P$-test). The non-appearance of some sort of singularities such as moveable branch points indicates a sound probability of complete integrability of the concerned NLSE. In case an NLSE passes the $P$-test, the studied model can be solved by implementing inverse scattering transformation (IST).

In this paper, two important pseudo-parabolic type equations are studied for extracting soliton solutions via the generalized projective riccati equations method. These equations are Oskolkov equation and Oskolkov–Benjamin–Bona–Mahony–Burgers equation. The proposed method extracts dark soliton and singular soliton. Furthermore, the Painlevé test (P-test) has also been employed on pseudo-parabolic type equations for investigating integrability. The proposed equations are proved to be integrable by P-test. The numerical simulations have also been carried out by 3D and 2D graphs of some of the obtained solutions.

We study a hierarchy of nonlinear autonomous systems of first ordinary differential equations. We show that it is completely integrable using the Painlevé test and finding the first integrals. We also show that it can be derived from the first integrals using Nambu mechanics. A corresponding dynamical system with chaotic behavior is also derived. A modified system with harmonic oscillators as first integrals is also considered. A connection with the Yang–Mills equation and the self-dual Yang–Mills equation is also discussed.

In this paper, we construct new exact solutions of the reaction–diffusion equation with time dependent variable coefficients by employing the mathematical computation via the Painlevé test. We describe the behaviors and their interactions of the obtained solutions under certain constraints and various variable coefficients.

In order to later find explicit analytic solutions, we investigate the singularity structure of a fundamental model of nonlinear optics, the four-wave mixing model in one space variable z. This structure is quite similar, and this is not a surprise, to that of the cubic complex Ginzburg-Landau equation. The main result is that, in order to be single valued, time-dependent solutions should depend on the space-time coordinates through the reduced variable , in which τ is the relaxation time.