Fractal Dimension in Nonlinear Wave Dynamics Governed by a Nonlinear Partial Differential Equation

TL;DR

This paper investigates the fractal geometry of solutions to a nonlinear wave system, revealing multiscale, self-affine patterns that connect analytical solutions with fractal analysis, relevant to fluid and plasma turbulence.

Contribution

It introduces a new class of exact solutions for the (2+1)-dimensional BLP system and applies fractal dimension analysis to characterize their complex geometrical features.

Findings

Solutions exhibit non-integer fractal dimensions

Patterns are self-affine and multiscale

Fractal analysis links solutions to turbulence phenomena

Abstract

This work presents a detailed analytical and geometrical investigation of the (2+1)-dimensional Boiti-Leon-Pempinelli system, a nonlinear dispersive model arising in the context of fluid and plasma dynamics. By employing a projective Riccati-based ansatz, a new class of exact solutions is systematically derived. These solutions, when visualized, exhibit intricate geometrical features that evolve across multiple spatial scales. To quantify this complexity, a voxel-based box-counting dimension analysis is conducted on the corresponding surface profiles. The analysis reveals non-integer fractal dimensions that vary with magnification, confirming the self-affine nature of the patterns and highlighting the multiscale structure inherent in the system. Such fractal character is not only of theoretical interest but also reflects real-world behaviors in turbulent plasma flows and fine-scale fluid instabilities. The study thus bridges exact analytical solutions with computational fractal geometry, providing a deeper understanding of the BLP system and its relevance in describing natural phenomena characterized by spatial complexity and multiscale interactions.