Riemann Invariant

Riemann Invariant

One-dimensional movements of a compressed medium

Published in G. I. Kanel′, Shock Waves in Solid State Physics, 2019

which are called Riemann invariants. The Riemann invariant is constant along the characteristic and is determined with an accuracy of a constant, which is found from initial or other additional conditions. A more visual and convenient form of writing equations (1.9) is the relations between the particle velocity and pressure along the characteristics in the form of the Riemann integrals: () up=u0−∫p0pdpρ0aalongC+; () up=u0+∫p0pdpρ0aalongC−;

Complex waves and their collisions of the breaking soliton model describing hydrodynamics

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Journal Information

Published in Waves in Random and Complex Media, 2022

Chao-Qing Dai, Yue-Yue Wang

The classical elementary waves in Riemann solutions comprise shock wave, rarefaction wave and contact discontinuity. In shock-bearing systems, the Riemann simple waves are important to comprehend shock dynamics in different fields because they admit the dynamical description according to a single evolving Riemann invariant [6]. (1 + 1)-dimensional travelling nonlinear waves in dispersiveless systems are called Riemann waves, which exist in diversified fields of science and engineering, such as hydrodynamics [7], optics [8] and plasmas [9]. In hydrodynamics, the breaking soliton model governs the interaction between a Riemann wave along the y-axis and a long wave along the x-axis. In homogeneous media, the Riemann waves continuously deform and transform to the shock waves. In these inhomogeneous cases, variable coefficient dynamical equations become the governing equations [10–15].

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One-dimensional movements of a compressed medium

Complex waves and their collisions of the breaking soliton model describing hydrodynamics