Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation

This manuscript focuses on the application of the (m + 1/G’)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted. © 2020 by the authors

New wave behaviors of the (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation

In this study, the (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation that indicated the propagation of nonlinear dispersive waves in inhomogeneous media is given for consideration. The generalized exponential rational function method is used to seek some new exact solutions for the considered equation. The three-dimensional surfaces and two-dimensional graphs of the obtained solutions are plotted by choosing the appropriate values of the involving free parameters

Bright and Dark Soliton Solutions of the (2 + 1)-Dimensional Evolution Equations

In this paper, we obtained the 1-soliton solutions of the (2+1)-dimensional Boussinesq equation and the Camassa–Holm–KP equation. By using a solitary wave ansatz in the form of sechp function, we obtain exact bright soliton solutions and another wave ansatz in the form of tanhp function we obtain exact dark soliton solutions for these equations. The physical parameters in the soliton solutions are obtained nonlinear equations with constant coefficients

Alçak geçiren elektrik iletim hatlarının kesirli mertebeden lineer olmayan modelinin ilerleyen dalga çözümlerinin oluşturulması

In this article, the structure of the improved tan(φ/2)-expansion method and the simplest equation method are applied. The fractional nonlinear model of the low-pass electrical transmission lines via Atangana-Baleanu derivative operator is taken into consideration and exact solutions have been constructed of this equation using proposed methods. This article explores the applicability and effectiveness of these methods on fractional nonlinear evolution equations.Bu makalede, geliştirilmiş tan(φ/2)-açılım yöntemi ve en basit denklem yöntemi uygulanmıştır. Alçak geçiren elektrik iletim hatlarının Atangana-Baleanu türev operatörü aracılığıyla kesirli mertebeden lineer olmayan modeli dikkate alınmış ve önerilen yöntemler kullanılarak bu denklemin tam çözümleri oluşturulmuştur. Bu makale, bu yöntemlerin kesirli doğrusal olmayan evrim denklemleri üzerindeki uygulanabilirliğini ve etkinliğini araştırmaktadır

New classifications of nonlinear Schrödinger model with group velocity dispersion via new extended method

This work investigates the nonlinear Schrödinger equation (NLSE) with group velocity dispersion and second order spatiotemporal dispersion coefficients. The governing model is reduced into the classical nonlinear ordinary differential equation. Extended direct algebraic method (EDAM) is implemented to construct many novel mixed dark, and complex optical solutions. As a result, some important analytical solutions such as travelling mixed dark, and complex travelling wave solutions for the model are extracted.</p

Integrability, rational solitons and symmetries for nonlinear systems in Biology and Materials Physics

[ES] Los sistemas no lineales constituyen un tema de investigación de creciente interés en las últimas décadas dada su versatilidad en la descripción de fenómenos físicos en diversos campos de estudio. Generalmente, dichos fenómenos vienen modelizados por ecuaciones diferenciales no lineales, cuya estructura matemática ha demostrado ser sumamente rica, aunque de gran complejidad respecto a su análisis. Dentro del conjunto de los sistemas no lineales, cabe destacar un reducido grupo, pero a la vez selecto, que se distingue por las propiedades extraordinarias que presenta: los denominados sistemas integrables. La presente tesis doctoral se centra en el estudio de algunas de las propiedades más relevantes observadas para los sistemas integrables. En esta tesis se pretende proporcionar un marco teórico unificado que permita abordar ecuaciones diferenciales no lineales que potencialmente puedan ser consideradas como integrables. En particular, el análisis de integralidad de dichas ecuaciones se realiza a través de técnicas basadas en la Propiedad de Painlevé, en combinación con la subsiguiente búsqueda de los problemas espectrales asociados y la identificación de soluciones analíticas de naturaleza solitónica. El método de la variedad singular junto con las transformaciones de auto-Bäcklund y de Darboux jugarán un papel fundamental en este estudio. Además, también se lleva a cabo un análisis complementario basado en las simetrías de Lie y reducciones de similaridad, que nos permitirán estudiar desde esta nueva perspectiva los problemas espectrales asociados. Partiendo de la archiconocida ecuación de Schrödinger no lineal, se han investigado diferentes generalizaciones integrables de dicha ecuación con numerosas aplicaciones en diversos campos científicos, como la Física Matemática, Física de Materiales o Biología.[EN] Nonlinear systems emerge as an active research topic of growing interest during the last decades due to their versatility when it comes to describing physical phenomena. Such scenarios are typically modelled by nonlinear differential equations, whose mathematical structure has proved to be incredibly rich, but highly nontrivial to treat. In particular, a narrow but surprisingly special group of this kind stands out: the so-called integrable systems. The present doctoral thesis focuses on the study of some of the extraordinary properties observed for integrable systems. The ultimate purpose of this dissertation lies in providing a unified theoretical framework that allows us to approach nonlinear differential equations that may potentially be considered as integrable. In particular, their integrability characterization is addressed by means of Painlevé analysis, in conjunction with the subsequent quest of the associated spectral problems and the identification of analytical solutions of solitonic nature. The singular manifold method together with auto-Bäckund and Darboux transformations play a critical role in this setting. In addition, a complementary methodology based on Lie symmetries and similarity reductions is proposed so as to analyze integrable systems by studying the symmetry properties of their associated spectral problems. Taking the ubiquitous nonlinear Schrödinger equation as the starting point, we have investigated several integrable generalizations of this equation that possess copious applications in distinct scientific fields, such as Mathematical Physics, Material Sciences and Biology

Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics

Integrable and non-integrable equations with peaked soliton solutions

This thesis explores a number of nonlinear PDEs that have peaked soliton solutions, to apply reductions to such PDEs and solve the resultant equations. Chapter 1 provides a brief history of peakon equations, where they come from and the different viewpoints of various authors. The rest of the chapter is then devoted to detailing the mathematical tools that will be used throughout the rest of the thesis. Chapter 2 concerns a coupling of two integrable peakon equations, namely the Popowicz system, which itself is not integrable. The 2-peakon dynamics are studied, and an explicit solution to the 2-peakon dynamics is given alongside some features of the interaction. In chapter 3 a reduction from two integrable peakon equations with quadratic nonlinearity to the third Painlev´e equation is given. B¨acklund transformations and solutions for the Painlev´e equations are expressed, and then used to find solutions of the original PDEs. A general peakon family, the b-family, is also explored, giving a more general result. Chapter 4 examines two peakon equations with cubic nonlinearity, and their reductions to Painlev´e equations. A link is shown between these cubic nonlinear peakon equations and the quadratic nonlinear equations in chapter 3. Chapter 5 has conclusions and outlook in the area