Applications of Variational Integrators and Solitons Solutions for Evaluation Equations

Abstract

n extensive variety of fields, including biology, chemistry, physics, fiber optics, mechan- ics, atmospheric science, and electronics science, use nonlinear partial differential equa- tions (NLPDEs), which makes them important. NLPDEs of a particular type are the non- linear Schr¨odinger equations (NLSE). Every NLPDE that is integrable and nonlinear has a soliton solution. A specific type of solitary wave defined as a soliton has the ability to maintain its original structure even after interacting with another soliton. They are special wave packets which have the capacity to travel long distances without suffering any dis- tortion. Solitons are often used in communication due to they are able to transmit signals with no errors throughout long distances and contain an abundance of data. In general, A nonlinear partial differential equation can be resolved to generate a soliton in many appli- cations. Variational Integrators (VIs) is a numerical technique in which Lagrangian of the system is used in the action integral. VIs discretized the Lagrangian to obtain a discrete Euler Lagrange equation with the help of the Hamiltonian principle of stationary action. VIs are renowned for their capacity to preserve a distinct multi-symplectic structure while demonstrating desirable long-term energy characteristics. Finite-difference scheme (FDS) are a class of numerical techniques utilized by numerical analysis that approximate deriva- tives by employing finite differences in order to solve differential equations (DE). During the past two decades, research on nonlinear models has frequently focused on soliton, a vital description in the NLSE analysis. In particular, the establishment of optical solitons has given rise to a theoretical basis for nonlinear optics. In this thesis, Our objective is to study the nonlinear Klein-Gordon model (NLKGM) and the Stochastic Biswas-Milovic equation (SBME) with parabolic law nonlinearity using the VIs by the use of projection technique, forward, backward, and central difference schemes and the Sub-OdE method. Utilizing the projection technique, we study the VIs of our gov- erning model, NLKGM; Additionally, we investigate some of the numerical solutions by ix the use of central difference, forward, and backward techniques. The Sub-OdE approach is also utilized to obtain soliton solutions, that include avariety of solutions:three positive solitons, three Jacobian elliptic function solutions(JEFS), bright solitons, dark solitons, pe- riodic solitons, rational solitons, and hyperbolic function solutions. Bose-Einstein conden- sation, fiber optic sensors, plasma physics, optical communication, and other fields belong to the applications for these solitons.