Shock Wave in Resonant Dispersion Media

For nonlinear one-dimensional waves propagated in resonant dispersion media, i.e. which contains uniformly distributed identical resonators with eigenfrequency ω₀, the so-called Resonant Dispersion equation (RD-equation) has been obtained [1]: 1 https://www.w3.org/1998/Math/MathML"> 1 + ∂ 2 ∂ τ 2 Ψ τ + Ψ η + 2 Ψ Ψ η − σ 2 Ψ η = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn1b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where τ = ω₀t, η = ω₀x / c — dimensionless time and coordinate. We have to choose Ψ in (1) as a wave variable which will satisfy the equation https://www.w3.org/1998/Math/MathML"> Ψt+cΨx+2ΨΨx=0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/ieq1142.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if there are no any resonators in medium (σ = 0).