Harnessing Weakly Nonlinear Waves for Precision Ultrasound Applications

Abstract

The work focuses on the nonlinear evolution of acoustic waves in weakly diffracting and weakly nonlinear media, modeled by the (2+1)-dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation. This equation captures how nonlinearity, diffraction, and dissipation interact during the propagation of finite-amplitude acoustic beams. This model is a fundamental model in nonlinear acoustics describing the evolution of finite-amplitude sound beams in thermoviscous fluids. It captures the combined effects of nonlinearity, diffraction, and absorption under the paraxial approximation. Using the Khater II analytical scheme together with an improved Kudryashov method, a new class of exact and physically meaningful solutions is obtained. The correctness and convergence of these results are checked through He’s variational iteration method. Linear perturbation and spectral analyses are then used to explore the stability of the derived wave forms and to locate the limits between stable and unstable regimes. The bifurcation study points out how small parameter variations lead to distinct transitions among periodic, solitary, and blow-up type waves. Physically, these solutions describe the evolution of nonlinear acoustic beams, the onset of shock profiles, and the localized transport of acoustic energy in fluids and biological media. The combined Khater II–Kudryashov framework proves more effective than existing analytical approaches for handling multidimensional nonlinear models. The findings add to current understanding of nonlinear acoustic wave behavior and provide a useful basis for designing and interpreting acoustic systems governed by nonlinear effects.