ible.

We note that the approach presented above generalizes the results obtained in [@GrundlandZelazny:1983] where the simple states have been constructed with wave vectors $\lambda$ of constant direction for hyperbolic inhomogeneous systems (\[eq:SW:1\]).

Simple mode solutions for inhomogeneous quasilinear system. {#sec:6}
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We now generalize the concept of simple wave solutions for inhomogeneous, quasilinear, systems of form (\[eq:SW:1\]). As in the case of the simple wave, the system can be either hyperbolic or elliptic. We look for a real solution, in terms of a Riemann invariant $r$ and its complex conjugate $\bar{r}$, of the form

\[eq:ms:1\] u=f(r,|[r]{}),r(u,x)=\_i(u)x\^i,|[r]{}(u,x)=|\_i(u)x\^i,i=1,…, p,

where $\lambda(u)={\left( \lambda_1(u),\ldots,\lambda_p(u) \right)}$ is a complex wave vector and $\bar{\lambda}(u)$ its complex conjugate. The Jacobian matrix takes the form

\[eq:ms:2\] u=\^[-1]{}[( + | )]{}\^[qp]{},

where we assume that the matrix

\[eq:ms:3\] =[( I\_q-- )]{}\^[qq]{}.

is invertible. Replacing the Jacobian matrix (\[eq:ms:2\]) into the system (\[eq:SW:1\]), we obtain

\[eq:ms:4\] \^[i]{}\^[-1]{}[( + | )]{}=b,\^i=[( A\^[i]{}\_)]{}\^[qq]{}.

We introduce a rotation matrix $L=L(x,u)\in SO(q,{\mathbb{C}})$ and its complex conjuguate $\bar{L}=\bar{L}(x,u)\in SO(q,{\mathbb{C}})$ such that the relations

\[eq:ms:5\] \^[-1]{}=L b + ,\^[-1]{}=| |[L]{} b+ |,

hold, where $\Omega(x,u)$ and its conjugate $\bar{\Omega}$ are scalar complex functions, while $\tau(x,u)$ and its complex conjugate vector $\bar{\tau}(x,u)$ satisfy

\[eq:ms:7\] \^i\_i +\^i|\_i |=0.

The vectors $\tau$ and $\bar{\tau}$ can be seen as characteristic vectors of the homogeneous part of equation (\[eq:ms:4\]). We eliminate the vectors $\Phi^{-1}({\partial}f/{\partial}r)$ and $\Phi^{-1}({\partial}f/{\partial}\bar{r})$ from equation (\[eq:ms:4\]) using equations (\[eq:ms:5\]). Considering the equation (\[eq:ms:7\]), we obtain, as a condition on functions $\Omega$,