Ateneo roses

Posted 2024-06-15
Updated 2025-07-07

goal

• Develop a novel extension of the tanh method, producing tunable families of exact solutions for the forced Boussinesq equation and other nonlinear systems
• Draft comprehensive literature review and conduct preliminary theoretical investigation on quantum computing applications using photonic integrated circuits

output

R. Torres and B.B. Dingel, New exact solution families for forced Boussinesq equation via an extension of generalized tanh-function method, Proc. Samahang Pisika ng Pilipinas 43, SPP-2025-PB-16 (2025), poster, open copy

In this paper, we present a novel generalization of the tanh method for solving nonlinear partial differential equations, and a subsequent extension. This approach features a tunable parameter p, allowing the obtainable solution families to be tweaked. We derived new tunable families of soliton, non-soliton traveling wave, and plane periodic solutions after applying our method to the forced classical Boussinesq equation, all of which reduce to standard tanh method solutions when p=1. While our study was limited to 0<=p<=1, future research should explore solutions beyond this range and investigate the applicability of this generalization to other nonlinear systems, particularly those where finding exact solutions is challenging.

R. Torres, B.B. Dingel, and C. Bennett, Novel exact solutions for forced Boussinesq equation via extended generalized tanh-function method, thesis, Ateneo de Manila University (2025), talk, proposal, manuscript

This thesis work introduces a novel extended generalized tanh-function method for deriving exact solutions to nonlinear partial differential equations. Central to this approach is an ansatz Y_p incorporating a tunable parameter p which provides significant flexibility in the characteristics of the resulting solution families. The method is applied to the classical Boussinesq equation. Application of this extended method yields 8 unique families of exact, tunable solutions, including solitons, non-soliton traveling waves, and plane periodic solutions. Critically, for p!=1, these solutions pertain to a forced Boussinesq equation with the forcing term F(Y_p) explicitly dependent on p. Solutions to the original, unforced Boussinesq equation, as obtained through standard tanh method, are recovered when p=1 is set and where F(Y_p) vanishes. The parameter p is found to greatly influence solution characteristics. For 0<=p<=1, localized waves generally widen and flatten. For p>1, they narrow and heighten. A fundamental transformation to trigonometric forms and oscillatory behavior occurs for p<0, where the wave number becomes imaginary, potentially introducing singularities. This work significantly expands the analytical solution space for Boussinesq-type equations, demonstrating the method's capacity to generate a diverse spectrum of wave behaviors. The study underscores the importance of the tunable parameter p and the associated forcing function, opening new avenues for theoretical modeling and understanding nonlinear wave phenomena. Future research includes applying the method to other nonlinear systems and further exploring the parameter space and physical implications.

Keywords: nonlinear partial differential equations, Boussinesq equation, generalized tanh method, extended generalized tanh method, tunable solutions, solitary waves, solitons, periodic waves, forcing function