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For an autonomous ordinary differential equation, the tangent to a periodic orbit gives a unit Floquet multiplier. A hyperbolic attracting cycle has that multiplier simple and all transverse multipliers strictly inside the unit disk. Thus the periodic variational operator has a one-dimensional null space spanned by the tangent. Nonlinear asymptotic attraction without hyperbolicity does not suffice: $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ attracts to a unit-period circle but its radial variational equation is neutral.

Freeze the coefficients at a point. Characteristic normals $(\tau ,\xi )\ne (0,0)$ satisfy the homogeneous quadratic matrix pencil equation $\det (\tau A+\xi B)=0$. In the elementary characteristic classification, two distinct real characteristic directions give a hyperbolic system, no real directions an elliptic system, and a repeated real direction the parabolic case. An identically zero determinant polynomial is a further degeneracy, not covered by that nondegenerate classification.

When $A$ is invertible, write $C=A^{-1}B$. The finite characteristic slopes solve $\det (B-\lambda A)=0$, orIts discriminant $(\mathrm{tr}C{)}^2-4\det C$ is positive, negative, or zero in the three cases above. For the well-posed first-order evolution meaning of hyperbolicity, real eigenvalues must also admit a complete eigenbasis: a repeated but diagonalizable pair is non-strictly hyperbolic rather than a defective parabolic system. Distinct real roots suffice for strict hyperbolicity.

For the given shallow water equations, subtracting $u$ times mass conservation from momentum conservation and dividing by $h>0$ gives the primitive-variable systemThe characteristic polynomial is $(u-\lambda {)}^2-h$, soThey are real and distinct whenever $h>0$, proving strict hyperbolicity.

With $A$ invertible, write the system as ${\mathbf{u}}_x+M{\mathbf{u}}_y=\mathbf{d}$, $M=A^{-1}B$. Real distinct eigenvalues of $M$ give strict hyperbolicity; a real complete eigenbasis gives the usual diagonalizable hyperbolic form. The characteristic curves satisfy $dy/dx={\lambda }_i$. Left eigenvectors form characteristic dependent combinations $w_i={\ell }_i\mathbf{u}$; for constant coefficients they obey separate transport equations. Characteristic coordinates are functions constant on those curves, such as $y-{\lambda }_{i}x$ in the constant case. For variable coefficients extra derivatives of the eigenvectors can couple the combinations; local characteristic directions still follow the real matrix pencil. If $A$ is singular one first chooses a noncharacteristic coordinate direction with an invertible coefficient combination.

For the specified constants,Its speeds are $2$ and $-1$, with left eigenvectors $(2,1)$ and $(1,1)$. HenceThe transported Riemann invariants are $\boxed{R_{+}=2u+v-4x,R_{-}=u+v-x}$, constant on $y-2x=\text{constant}$ and $y+x=\text{constant}$, respectively. Initial data give $w_{+}(0,s)=s-\sin (s/2)$ and $w_{-}(0,s)=-e^s$, soInvert these two combinations:Both characteristic feet must lie on the supplied positive initial half-line. ThusOutside this sector at least one invariant needs unsupplied initial data. The closed expressions give one possible extension there but are not uniquely forced by the question's initial half-line.

Assume the ambient fluid has zero downslope velocity, no bed drag, and an entrainment coefficient $\alpha >0$. The entraining shallow-water layer receives volume at rate $\alpha |u|$, but neither excess buoyancy nor downslope momentum from that ambient fluid. Consequently volume conservation, conservation of excess buoyancy, and momentum conservation giveThe pressure contribution is the integral of the hydrostatic pressure from part (a). The last source is the downslope component of the gravitational force.

With $D={\partial }_t+u{\partial }_x$, the equivalent equations areThe last term is the momentum required to accelerate newly entrained fluid. For variables $(h,u,b)$, the matrix of the spatial derivatives isThus it is a strictly hyperbolic system for $h,b>0$ and $0\le \theta <\pi /2$. At $\theta =\pi /2$ all three characteristic speeds coincide and the matrix is not diagonalizable. The exclusion of that angle presumes the physical range with $\cos \theta \ge 0$: allowing an overhanging channel with $\cos \theta <0$ would also destroy hyperbolicity.

For the steady positive-velocity branch, conservation of buoyancy flux gives $bhu=F_0$. Substituting into the other two balances givesUse a power-law ansatz $h=C{x}^p$, $u=D{x}^q$. The entrainment balance forces $p=1$ and $C(1+q)=\alpha$. The momentum powers are $x^{2q}$, $x^{-q}$ and $x^{-q}$, so the nonzero inertial branch has $q=0$. Therefore the constant-speed entraining gravity current on a slope isIts positive speed requires $\tan \theta >\alpha /2$ for $\theta <\pi /2$. In the shallow, strictly hyperbolic range this meansThe same steady integral balances have a formal vertical-plume limit at $\theta =\pi /2$, although their time-dependent system then loses strict hyperbolicity. At the lower threshold the assumed positive-flux, finite-speed solution fails.

The line source is singular: $h,uh$ and the momentum flux tend to zero, but $b$ diverges as $x\downarrow 0$. Thus this is a downstream ideal-source solution, valid only far enough away that $b/g\ll 1$ and the Boussinesq approximation and top-hat plume model hold. It does not describe the immediate source region.

The shallow-water approximation requires a small depth-to-horizontal-scale ratio, slow streamwise variation of the channel and interface, small vertical acceleration, and rapid enough cross-sectional mixing for $u$ and $\varphi$ to be representative averages. Near the channel apex, or during the initial collapse at a sharp front, these conditions need not hold. The high Reynolds number permits neglect of leading viscous resistance; weak stirring mixes the particles without appreciable ambient fluid entrainment.

Put $\lambda =\beta /H$. At elevation $z$, the channel width is $2\lambda xz$, so the occupied area and interface width areVolume conservation and suspended-particle conservation giveThe excess hydrostatic pressure is ${\rho }_{0}d\varphi (h-z)$. Its streamwise gradient averaged over the triangular section gives acceleration $-d\varphi h_x-(dh/3){\varphi }_x$: the weighted mean of $h-z$ is $h/3$. Thus the triangular-channel shallow water equations areThe quasilinear coefficient matrix in $(h,u,\varphi )$ isIts three distinct real eigenvalues, for $h,d\varphi >0$, establish strict hyperbolicity:On the material characteristic curve, $d\varphi /dt=-2W_s\varphi /h$. On each of the other two characteristic curves, multiplying by the left eigenvector $(\pm 2c/h,1,\pm dh/(3c))$ gives the compatibility ordinary differential equationThe derivatives are along the indicated characteristic, with $g^{\prime }=d\varphi$. At vanishing concentration, the gravity-wave speeds coalesce and strict hyperbolicity is lost.

A first-order system $U_t+A(U)U_x=C(U)$ is strictly hyperbolic at a state when $A$ has distinct real eigenvalues. These separate its characteristic families and permit independent compatibility equations from left eigenvectors. At a repeated eigenvalue strict hyperbolicity fails; if the matrix is also defective, even hyperbolicity in the complete-eigenvector sense fails. For $A=\left(\begin{array}{cc}u& 2\\ 2v& u\end{array}\right)$ the eigenvalues $u\pm 2\sqrt{v}$ are distinct precisely when $v>0$.