The Painlevé analysis and computational technique for new wave solutions with its numerical validation to the complex short pulse equation

This article seeks to provide novel varieties of exact solitonic wave solutions to the complex short pulse equation via impressive technique which is called the Ricatti-Bernolli sub-ODE method (RBSODM) that does not conform to the balance rule. For the first time, the resonance induced by the arbitrariness of the singular manifold is analyzed in the suggested model through the application of Painlevé analysis (PA). The complex short pulse (CSP) equation models the behaviour of ultra-short optical pulses in nonlinear media. It serves as a more accurate model than the non-linear Schrödinger equation when the pulse width approaches the optical cycle scale. The suggested model incorporates both dispersion and Kerr-type nonlinearity, capturing the essential features of femtosecond pulse dynamics. Diverse types of rogue wave soliton solutions have been extracted such as bright soliton, dark soliton, W-like soliton, M-like soliton, and higher-order breather soliton. Moreover, the numerical approximations for all achieved analytical traveling wave solutions have been implemented by using the Haar wavelet approach (HWA). A comparison of the obtained analytical and numerical solutions is introduced. The 2D and 3D graph simulations are introduced with the help of the Mathematica program. The graph simulation shows the novelty of the obtained results and helps in the explanation of the dynamical properties of the suggested model. • Painlevé analysis approach was used to test the integrability. • The Ricatti-Bernolli-Sub ODE method (RBSODM) was used in obtaining new forms of solitons. • Diverse types of rogue wave soliton solutions were extracted including bright soliton, dark soliton, W-like soliton, M-like soliton, and higher-order breather soliton. • The Haar wavelet approach was used to construct the numerical approximation, corresponding to obtained analytical results.