Weakly nonlinear high frequency waves. (English) Zbl 0547.35070

Consider a strictly hyperbolic system of quasilinear partial differential equations of the first order
\[ \sum^n_{i=1} A^{(i)}(x,u)u_{x_i}+B(x,u)=0, \ x\in\mathbb R^n, \tag{1} \]
where \(u\) and \(B\) are \(N\)-vectors, the \(A^{(i)}\) are \(N\times N\) matrices, and \(B\) and \(A^{(i)}\) are of class \(C^1\) with respect to \(u\).
Let \(u^{(0)}\) be a solution of (1); the authors seek a solution \(u\) which differs from \(u^{(0)}\) by \(mm\) small amplitude high frequency waves, i.e.:
\[ u=u^{(0)}+\varepsilon v(x,\theta,\varepsilon), \tag{2} \]
with \(\theta =(\theta_1,\theta_ 2,\ldots,\theta_m)\), \(\theta_j=\phi^{(j)}(x)/\varepsilon\). Assuming the asymptotic expansion for \(v\)
\[ v(x,\theta,\varepsilon)=v^{(0)}(x,\varepsilon)+v^{(1)}(x,\varepsilon)+O(\varepsilon^2), \tag{3} \]
they show that
\[ u(x,\varepsilon) = u^{(0)}(x) + \varepsilon\tilde v(x) + \varepsilon\sum^m_1 a^{(j)}(x,\phi^{(j)}(x)/\varepsilon)R^{(j) }(x) + O(\varepsilon^2). \tag{4} \]
Here, \(\phi^{(j)}\) satisfies the eikonal equation
\[ \det \left(\sum^n_1\phi_{x_i}^{(j)} A^{(i)}(x,u^{(0)})\right) = 0, \tag{5} \]
which is solved by the method of the characteristics. The scalar function \(a^{(j)}\) solves a transport equation along the characteristics of (5).
This leads to an implicit system which may have several solutions, so that shocks have to be considered. If (1) is in conservation form, the shocks satisfy a Rankine-Hugoniot condition, and the authors give an asymptotic to the position of the shock, which allows them to conclude that the speed of a weak shock is the average of the speeds on the two sides of the shock. These techniques reproduce the results of the nonlinearization techniques of Landau and Witham.
As an application, the authors find an asymptotic solution of the system of gas dynamics in three dimensional space; this solution is the sum of a constant state and of \(j\) high frequency sound waves and \(k\) high frequency vorticity waves.


35L60 First-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
76N15 Gas dynamics (general theory)
35C20 Asymptotic expansions of solutions to PDEs
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