A nonlinear scattering length formula in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg" display="inline" id="d1e23"><mml:mi>p</mml:mi></mml:math>-Laplacian framework

This paper investigates nonlinear diffusion processes within the framework of the p -Laplacian operator, focusing on the interplay between path integrals, cumulative potential, and scattering length. We propose a novel nonlinear scattering length definition, suitable for multi-scale, multi-state systems with non-uniform diffusion. Furthermore, we derive analytical results that reveal connections between the first eigenvalue, energy functionals, and scattering properties, offering new insights into statistical models and nonlinear dynamics. To better capture the dynamics governed by the p -Laplacian operator, we employ nonlinearity-driven stochastic processes as a conceptual framework instead of relying on Brownian-like motion models. • Innovative Definition : Introduces a scattering length for multi-scale systems in p-Laplacian nonlinear diffusion. • Theoretical Connections : Links scattering length with eigenvalues and energy in nonlinear systems. • Application Potential : Supports nonlinear diffusion models in statistics and physics, enabling further studies.