Nonlinear waves in physical systems are wave phenomena governed by equations in which the restoring forces or constitutive relations depend nonlinearly on the field amplitudes, leading to amplitude-dependent propagation characteristics and interactions. Unlike linear waves, they can exhibit self-steepening, shock formation, soliton generation, modulational instability, and harmonic generation. Mathematically, they arise from nonlinear partial differential equations such as the Korteweg–de Vries, nonlinear Schrödinger, sine-Gordon, and nonlinear elastic or hydrodynamic equations. Nonlinear waves play a central role in fluid dynamics, plasma physics, nonlinear optics, condensed matter, and elastic media, where they mediate energy transport, pattern formation, and coherent structure dynamics.
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