The integrability problem consists in finding an explicit the class of functions that a first integral of a given planar polynomial differential system must belong to. In this paper we propose a ...method to find integrability via algebraic changes of variables in order to transform the planar polynomial differential into a simple system or equation to determine a first integral of it.
•A method to find integrability via algebraic changes of variables is proposed.•Looking for one transformation of planar polynomial differential system into a simple one.•The method is applied to several examples.
In this work the algebraic characterization of the Darboux, Liouville, and Puiseux first integrals of the vocal fold model is given. The multiple Puiseux solutions, also defined in Puiseux ...integrability, are studied for the first time.
•The Darboux polynomials of the vocal fold are provided.•The Darboux and Liouville integrability are studied.•The Puiseux integrability is studied.•The multiple Puiseux solutions are studied for the first time.
This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze ...scalar products of off-shell Bethe states and explore their connections to tau functions of integrable hierarchies. Furthermore, we discuss correlation functions within this formalism, examining their representations in terms of tau functions, as well as their Schur polynomial expansions.
This paper obtains optical soliton solutions to the Kundu–Eckhaus equation with general coefficients. The Riccati–Bernoulli's sub-ODE method as well as Kudryashov's scheme are employed to obtain ...soliton solutions to the model. This generalized earlier reported result of the model with specific coefficients. Subsequently, the multiplier approach was utilized to secure a conserved quantity with the bright soliton solution that was reported earlier.
Stochastic nonlinear Schrödinger equation is a mathematical model, used in the study of complex systems especially nonlinear optics and quantum mechanics. It explains the evolution of a quantum field ...with stochastic and nonlinear terms. A stochastic or random process is a mathematical representation that outlines the progression of a system in a way that incorporates probability. This is in contrast to a deterministic process, where the future states are strictly defined by initial conditions. In mathematical model, stochastic term is usually represent by variable or a function that involves random variable or a noise. In this paper, we study the generalized nonlinear Schrödinger equation by adding stochastic term for the first time to retrieve different types of optical solitons. To the best of our knowledge, no one work on this model with stochastic term. Our aim is to find optical solitons like bright and dark solitons, also solitary wave solutions for this model like bell type, hyperbolic functions, Jacobi Elliptic Function, Weierstrass Elliptic Function, periodic and rational solutions. These solutions will be obtained using the sub-ode method, subject to certain constraint conditions. Finally, we will use Jupyter as a machine learning tool and also mathematica to plot some obtained solutions in various dimensions like 3D, 2D and contour plots by using some libraries of PYTHON like numpy, scipy.integrate and matplotlib.pyplot. For the first time, we use this approach for this model and our results are new and novel 1.
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•Study the generalized nonlinear Schrödinger equation by adding a stochastic term.•Retrieve different types of optical solitons using sub-ode technique.•Obtain bright, dark, singular, Jacobi elliptic and other solitary wave solutions.•Use machine learning tools like Jupyter and Mathematica to sketch graphs.
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