Solutions of First-Order Quasilinear Systems Expressed in Riemann Invariants

Abstract

We present a new technique for constructing solutions of quasilinear systems of first-order partial differential equations, in particular inhomogeneous ones. A generalization of the Riemann invariants method to the case of inhomogeneous hyperbolic and elliptic systems is formulated. The algebraization of these systems enables us to construct certain classes of solutions for which the matrix of derivatives of the unknown functions is expressible in terms of special orthogonal matrices. These solutions can be interpreted as nonlinear superpositions of k waves (or k modes) in the case of hyperbolic (or elliptic) systems, respectively. Theoretical considerations are illustrated by several examples of inhomogeneous hydrodynamic-type equations which allow us to construct solitonlike solutions (bumps and kinks) and multiwave (mode) solutions.

Appendix: The Simple State Solutions [16]

A mapping $u:\mathbb{R}^{p}\rightarrow\mathbb{R}^{q}$ is called a simple state solution of the inhomogeneous system (1) if all first-order derivatives of u with respect to x ⁱ are decomposable in the following way

$$ du^\alpha(x)=\frac{\partial u^\alpha}{\partial x^i}dx^i=\gamma _0^\alpha (u)\lambda_i^0(u)dx^i, $$

(84)

where the real-valued functions $\lambda^{0}= ( \lambda _{1}^{0},\ldots,\lambda_{p}^{0} )\in E^{\ast}$ and $\gamma_{0}=(\gamma _{0}^{1},\ldots,\gamma _{0}^{q})\in T_{u}\mathcal{U}$ satisfy the algebraic relation

$$\mathcal{A}^{\mu i}_\alpha(u)\lambda_i^0 \gamma_{0}^\alpha=b^\mu(u). $$

In contrast to the condition

$$du^\alpha(x)=\xi\gamma^\alpha\lambda_idx^i, \qquad\mathcal{A}_\alpha^{\mu i}(u)\lambda_i \gamma^\alpha=0, $$

defining the simple wave solution for homogeneous systems, the expression (84) does not include a function ξ of x. Consequently, the compatibility conditions are not identically satisfied and they lead to the following conditions

$$ d(du)=d\gamma_0\wedge\lambda^0+ \gamma_0\otimes d\lambda^0=0, $$

(85)

whenever (84) hold, where

$$\begin{aligned} d\gamma_0 =& \biggl( \frac{\partial\gamma _0}{\partial u^\alpha} \biggr)du^\alpha =\gamma_{0,\gamma_0}\otimes\lambda^0,\quad \lambda^0= \lambda^0_i(u)dx^i, \\ d\lambda^0 =&du^\alpha\wedge\frac{\partial\lambda^0}{\partial u^\alpha }= \lambda^0\wedge\lambda^0_{,\gamma_0}. \end{aligned}$$

Here we have used the notation $x_{,y}=\frac{\partial x}{\partial u^{\alpha}}y^{\alpha}$. Hence the system (84) has a solution if

$$\lambda^0_{,\gamma_0}\wedge\lambda^0=0 $$

holds. This means that the direction of λ ⁰ is constant along the vector field γ ₀. Choosing a proper normalization for the wave vector λ ⁰→aλ ⁰ and the vector field γ ₀→a ⁻¹ γ ₀, where a=a(u), one can obtain that λ ₀ is constant along the vector field γ ₀, i.e.

$$\lambda_{0,\gamma_0}=0. $$

Thus the image of a solution is a curve u=f(r ⁰) tangent to the vector field γ ₀. So one can choose a parametrization of a solution u=f(r ⁰) such that the ordinary differential equations

$$\frac{df^\alpha}{dr^0}=\gamma_0^\alpha \bigl(f \bigl(r^0 \bigr) \bigr) $$

hold. The wave vector λ ⁰ has a constant direction and by choosing a proper length of λ ⁰ such that ∂λ ⁰/∂r ⁰=0, we may represent a simple state solution of the inhomogeneous system (1) in the form

$$u=f \bigl(r^0 \bigr),\quad r^0=\lambda^0_i x^i, $$

where

$$\mathcal{A}_\alpha^{\mu i}(u) \lambda_i \gamma_0^\alpha=b^\mu(u),\qquad\frac{\partial f^\alpha}{\partial r^0}= \gamma_0^\alpha \bigl(f \bigl(r^0 \bigr) \bigr). $$

This solution was introduced in analogy with a simple wave solution of a homogeneous system (1) which satisfies the relations (see e.g. [17, 25, 26]).

$$ \begin{aligned} &u=f(r),\qquad r=\lambda_i\bigl(f(r)\bigr)x^i, \\ &\mathcal{A}^{\mu i}_\alpha(u)\lambda^i\gamma^\alpha=0,\qquad\frac{\partial f^\alpha}{\partial r}=\gamma^\alpha\bigl(f(r)\bigr). \end{aligned} $$