Exploring Integrability and Advanced Analytical Methods for Various Higher-Order Partial Differential Equations Arising in Mathematical Physics

Abstract

Nonlinear partial differential equations with constant and variable coefficients play a vital role in modeling the difficult physical phenomena in the various fields of science and engineering. Finding the analytical solutions to these equations is again a very difficult task because of the complexity of these models. However, these analytical solutions help us in studying the various physical and natural characteristics of the models. Integrable property is the key feature for these nonlinear models in order to obtain the analytical solutions, because integrability ensures the availability of analytical solutions. In this dissertation, we adopted Painlevé analysis and Lax pairs methods to verify the integrability of governing model equations. However, Painlevé analysis method is one of the trustworthy and easiest approach, which is very easy to implement in computer algebra. Painlevé analysis method also helps to construct the auto-Bäcklund transformations. Furthermore, the auto-Bäcklund transformation method based on Painlevé analysis method is applied to generate the numerous analytical solutions to these nonlinear model equations. By employing this approach, new kind of wave patterns which include kink-soliton, anti-kink soliton, periodic wave, kink-antikink wave, bell-shaped wave, anti-bell shaped wave, solitary wave, bright-soliton, dark soliton etc. are recognized for the governing models. It has been noticed that this method is an easy approach to solve the variable-coefficients nonlinear equations analytically. The Bell polynomials method is applied to generate the bilinear form of nonlinear equations. These bilinear forms can generate multi-soliton solutions with the help of Hirota bilinear method. With the help of bilinear representation, the bilinear Bäcklund transformations and Lax pairs can be easily generated. The bilinear form is the key feature of the Hirota bilinear method, however, the bilinear forms are highly non-trivial. To overcome the difficulty of finding bilinear forms, the simplified Hirota method is developed which is used to obtain the multi-soliton solutions to these equations. This method is the generalized version of Hirota bilinear method. Furthermore, the Paul-Painlevé approach method is adopted to generate the different kinds of analytical solutions. All the obtained results are expressed graphically to understand the physical phenomena of the models.