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For an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients, real-analytic Cauchy data on a real-analytic non-characteristic hypersurface determine a unique real-analytic solution in a neighbourhood of each point of that hypersurface.

For the first-order quasilinear partial differential equation $P(x,y,z)z_x+Q(x,y,z)z_y=R(x,y,z)$, the displayed ordinary differential equations trace curves tangent to every classical solution surface. Indeed, if $z=u(x,y)$ solves the equation and $x_s=P$, $y_s=Q$, then the chain rule gives $z_s=u_{x}P+u_{y}Q=R$. Conversely a surface swept out by these curves solves the equation wherever its projection to $(x,y)$ is invertible and hence represents a differentiable graph. The coefficients can depend on the unknown $z$, which distinguishes these equations from characteristic equations for a prescribed linear transport equation.

This scalar quasilinear partial differential equation results by imposing $h=Q/u$ in a frictional river momentum equation and measuring $X$ upstream. It retains a relaxation toward $u_0=Q^{1/3}$ and carries information at speed $c(u)=Q/(F^2{u}^2)-u$. Near equilibrium the relaxation time is $\epsilon F^2{u}_0/3$. A slowly moving downstream boundary can generate an attached boundary layer when $c(u)$ exceeds its speed throughout the relaxation. The model retains momentum dynamics only: exactly constant flux in both original Saint-Venant equations would instead force time-independent depth.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

The smooth ideal-MHD system couples mass conservation, momentum, adiabatic pressure transport and the ideal magnetohydrodynamic induction equation, subject to the solenoidal magnetic-field constraint. Without gravity it is a quasilinear partial differential equation system. Across shocks, conservative total energy replaces smooth adiabatic pressure transport.

The method of characteristics converts a first-order partial differential equation into ordinary differential equations along characteristic curves. Forsolve $\dot{X}=b(t,X)$, $X(0)=\alpha$, and $\dot{U}=h(t,X,U)$, $U(0)=g(\alpha )$. The chain rule identifies $U(t,\alpha )=u(t,X(t,\alpha ))$. If the characteristic flow map is invertible, it reconstructs the solution by $u(t,x)=U(t,X_t^{-1}(x))$. This turns spatial propagation and the evolution of the carried value into separate calculations.^[ref]

For $u_t+t{u}_x=0$, the characteristic curves are $X(t,\alpha )=\alpha +t^2/2$ and $U=g(\alpha )$, so $u(t,x)=g(x-t^2/2)$. Substitution verifies both the partial differential equation and the initial condition. More generally, a quasilinear partial differential equation allows $b$ to depend on $U$, so the two ordinary differential equations become coupled. Noncharacteristic Cauchy data and a nonzero projection Jacobian determinant give local reconstruction; characteristic crossing can destroy it even while the lifted trajectories remain smooth.

For a quasilinear partial differential equation, the characteristic equations for a quasilinear equation are the ordinary differential equationsA solution surface is locally a differentiable graph $z=u(x,y)$ whose tangent plane contains the vector $(P,Q,R)$ at every point. Its normal is $(-u_x,-u_y,1)$, so the tangency condition is precisely $P{u}_x+Q{u}_y=R$. Thus every characteristic lying on that graph follows the vector field without leaving the graph.

Parameterize the given initial curve by $\gamma (t)=(x_0(t),y_0(t),z_0(t))$. For each $t$, solve the characteristic equations for a quasilinear equation with initial value $\gamma (t)$ at $s=0$. The resulting map $(s,t)\mapsto (x(s,t),y(s,t),z(s,t))$ sweeps out a candidate solution surface. If the coefficients and initial data are continuously differentiable, the Picard-Lindelöf theorem supplies this local family. To make it a graph we also need the projection $(s,t)\mapsto (x,y)$ to be locally invertible. At $s=0$ its Jacobian determinant isThe Noncharacteristic Cauchy data condition is $J(0,t)\ne 0$. The inverse function theorem then gives $s=s(x,y)$ and $t=t(x,y)$, and hence $u(x,y)=z(s(x,y),t(x,y))$. Its derivatives obey the PDE by the chain rule. The same characteristic initial-value uniqueness gives local uniqueness of this solution wherever the projection remains invertible and the relevant characteristic meets the prescribed data.

The first Saint-Venant equation conserves water volume. The second balances material acceleration against downslope gravity, bed friction and the free-surface pressure gradient. With length scale $L$, depth scale $H$, slope $S$ and velocity scale $U$, the conventional gravity-friction scaling has $\epsilon =H/(SL)$ and Froude number $F=U/\sqrt{gH}$. The pressure-gradient/downslope-gravity ratio is $\epsilon$, and the inertia/downslope-gravity ratio is $\epsilon F^2$.

The uniform steady solution obeys $u^2=h$ and $hu=Q$, hence $\boxed{u_0=Q^{1/3},h_0=Q^{2/3}}$. In the constant-flux river momentum relaxation reduction, substitute $h=Q/u$ and $X=1-s$. Since $u_s=-u_X$ and $h_s=Q{u}_X/u^2$, the resulting quasilinear partial differential equation isA downstream boundary at $X=0$ supplies data into $X>0$ precisely when $c(u_1)>0$, equivalent to $\boxed{u_1<Q^{1/3}/F^{2/3}}$. Although water flows out there, the relevant characteristic curves carry information upstream in this subcritical regime.

Along a characteristic curve, $dX/dt=c(u)$ and $du/dt=(1-u^3/Q)/(\epsilon F^2)$. Positive $u$ relaxes toward $u_0$; near $u_0$, its displacement decays at rate $3/(\epsilon F^2{u}_0)$. Boundary-generated deviations therefore become small after a time of order $\epsilon$ and, while the characteristic speed stays bounded away from zero, within a distance of order $\epsilon$. The outer state is the initial uniform solution. A jump between initial and boundary values at their corner can produce a weak solution or a thin initial adjustment; a globally smooth corner solution requires compatible data.

For the moving boundary put $Y=X-vt$. The characteristic speed relative to it is $c(u)-v$, while the outer speed isFor the intended tidal slowing $0<u_1\le u_0$, $c$ is decreasing in $u$ and hence $c(u)\ge v_b$ throughout the relaxation. Thus $v<v_b$ leaves an incoming family and an attached thin boundary layer. The diagram shows numerical characteristics of this scalar reduction, with a slower boundary velocity. When $v>v_b$, the tidal boundary overtakes the unperturbed characteristic family; an attached smooth relaxation layer is no longer appropriate, and a tidal bore or shock wave can arise in a fuller river model.

There are two qualifications to the printed reduction. First, without the physical slowing assumption, $v<v_b$ alone need not make the boundary incoming: for $u_1>u_0$, also require $v<c(u_1)$. Second, imposing $hu=Q$ exactly everywhere in the original continuity equation gives $h_t=0$, and consequently $u_t=0$. The time-dependent characteristics above solve the requested momentum-only approximation, rather than the complete Saint-Venant equation system with an exactly constant flux.

For a first-order quasilinear partial differential equation, take $C^1$ coefficients $a_j,b$, $C^1$ prescribed data $u_0$ on a smooth $C^2$ hypersurface $S$, and a point $x_0\in S$. The data are non-characteristic there ifwhere $n$ is any nonzero normal to $S$. Then there is a neighborhood of $x_0$ on which a unique $C^1$ solution realizes the data. Uniqueness is local, after restricting to a neighborhood swept out by the nearby characteristics.

The method of characteristics solves $\dot{x}=a(x,u)$, $\dot{u}=b(x,u)$ from points of $S$. In local surface coordinates $\eta$, the map $(\eta ,s)\mapsto x(\eta ,s)$ has nonsingular derivative at $s=0$ exactly under the stated normal condition. The inverse function theorem makes these variables local coordinates, giving the asserted solution and uniqueness.

Contract the printed Riemann curvature tensor using $R_{ab}=R^{\mu }{}_{a\mu b}$. Only the derivative terms can contain second derivatives of the metric tensor:Substituting the Christoffel symbol formula, its second derivative part isSet $H_b=g_{b\lambda }{H}^{\lambda }=g^{\mu \nu }{\partial }_{\mu }{g}_{\nu b}-\frac{1}{2}{g}^{\mu \nu }{\partial }_b{g}_{\mu \nu }$. The first, second and fourth terms are the second derivative terms of $\frac{1}{2}({\partial }_a{H}_b+{\partial }_b{H}_a)$. Derivatives of the inverse metric tensor produce only products of first derivatives. Thuswhere $Q$ contains at most first metric derivatives. In harmonic coordinates, $H^{\lambda }$ vanishes throughout the chart, so $H_b$ and its derivatives vanish. The vacuum equations have the wave principal partThese are quasilinear partial differential equations with the principal part of a wave equation, because $g^{\mu \nu }$ has Lorentzian signature. This is the harmonic reduction of the vacuum Einstein equations.

Take the primitive state vector $\mathbf{U}=(\rho ,p,u_x,u_y,u_z,B_x,B_y,B_z{)}^T$, with $\rho >0$. In the absence of gravity, the ideal magnetohydrodynamic equations are first order: expanding the material derivatives, pressure gradient and Lorentz force makes every term linear in a spatial derivative of $\mathbf{U}$, with coefficients depending on $\mathbf{U}$. Dividing momentum balance by $\rho$ therefore gives a quasilinear partial differential equation system ${\partial }_t\mathbf{U}+A_i(\mathbf{U}){\partial }_i\mathbf{U}=0$ with eight-by-eight matrices. The solenoidal magnetic-field constraint is an additional constraint on initial data; the ideal magnetohydrodynamic induction equation preserves it because the divergence of a curl vanishes.

For a one-dimensional simple wave in magnetohydrodynamics, write $\mathbf{U}=\mathbf{U}(q(x,t))$ with a nonzero state-space tangent. Substitution gives ${\mathbf{U}}_q{q}_t+A_x{\mathbf{U}}_q{q}_x=0$. A nonconstant profile requires the tangent to be a right eigenvector of $A_x$:The chain rule then gives the wave-speed equation:This is the Inviscid Burgers equation, including the special case of constant $v$. Its characteristic curves are $x=a+v_0(a)t$, with $v=v_0(a)$. As long as the mapping remains invertible,Thus a region with $v_0^{\prime }<0$ steepens: faster characteristics catch slower ones, and the first gradient catastrophe occurs at $t_{\ast }=-1/{\min }_a{v}_0^{\prime }(a)$ when the minimum is negative. A genuinely nonlinear compressive simple wave therefore forms a shock wave. Rarefactive profiles can spread instead; steepening is not inevitable for every initial profile.

To continue past this time, use weak solutions of the conservative mass, momentum, fluid total-energy equation and ideal magnetohydrodynamic induction equation. The Rankine-Hugoniot conditions fix the jumps and shock velocity, while physical entropy production selects admissible magnetohydrodynamic shocks. Diffusive shock wave layers are replaced by moving discontinuities, so their microscopic structure need not be explicitly resolved. In particular, the smooth adiabatic pressure equation must be replaced by total-energy conservation when imposing the ideal magnetohydrodynamic shock conditions.

For the Alfvén waves, impose the one-dimensional solenoidal magnetic-field constraint, so $B_x$ is constant and $\delta B_x=0$. Put ${\mathbf{B}}_{\perp }=(B_y,B_z)$ and $c=v-u_x$. The transverse components of the linearized ideal magnetohydrodynamic equations giveThe longitudinal momentum equation additionally gives $\delta p+{\mathbf{B}}_{\perp }\cdot \delta {\mathbf{B}}_{\perp }/{\mu }_0=0$. The Alfvén eigenvectors have $\delta \rho =\delta p=\delta u_x=0$, hence their transverse polarization is perpendicular to ${\mathbf{B}}_{\perp }$. For $s=\pm 1$, their speed and explicit right Alfvén characteristic eigenvectors areFor ${\mathbf{B}}_{\perp }\ne 0$ one may choose $(h_y,h_z)=(B_z,-B_y)$, up to a nonzero scalar factor. If ${\mathbf{B}}_{\perp }=0$, either transverse polarization is allowed and this characteristic speed is degenerate. These expressions describe the two propagating Alfvén branches for $B_x\ne 0$; if $B_x=0$, they coalesce with advected, nonpropagating transverse disturbances.

Integrating along an Alfvén simple wave leaves $\rho ,p,u_x,B_x$ unchanged and gives $d{\mathbf{u}}_{\perp }=-sd{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0\rho }$ and ${\mathbf{B}}_{\perp }\cdot d{\mathbf{B}}_{\perp }=0$. Consequently the finite-amplitude nonlinear Alfvén wave relations areThe constant-magnitude condition is essential: otherwise a varying magnetic pressure would drive longitudinal compression. An arbitrary smooth phase profile gives an explicit family, $f_3=B_{\perp }\cos \Theta (x-v_{s}t)$ and $f_4=B_{\perp }\sin \Theta (x-v_{s}t)$, with $f_1,f_2$ as above. Direct substitution in the transverse momentum and ideal magnetohydrodynamic induction equation gives $(u_x-v_s){\mathbf{u}}_{\perp }^{\prime }=B_x{\mathbf{B}}_{\perp }^{\prime }/({\mu }_0\rho )$ and $(u_x-v_s){\mathbf{B}}_{\perp }^{\prime }=B_x{\mathbf{u}}_{\perp }^{\prime }$, confirming the solution without relying on the eigenvector argument. Longitudinal momentum holds because $p+B_{\perp }^2/(2{\mu }_0)$ is constant. The speed $v_s$ is independent of amplitude along this family: these linearly degenerate characteristic fields translate without steepening.

Consider an analytic quasilinear partial differential equationLet the analytic initial hypersurface be $\Gamma =\{\varphi =0\}$ and prescribe $u=u_0$ and one transverse derivative ${\partial }_{N}u=u_1$ on $\Gamma$. The tangential derivatives of $u_0$ together with $u_1$ determine the full first jet $Du$ on $\Gamma$. The hypersurface is non-characteristic at $x_0$ with respect to these data whenThis is precisely the principal symbol of a partial differential equation evaluated on the conormal $d\varphi (x_0)$.

The Cauchy-Kovalevskaya theorem then gives a unique real-analytic solution near $x_0$. In coordinates flattening $\Gamma$ to $x_n=0$, non-characteristicity lets the equation solve analytically for ${\partial }_n^{2}u$, after which the analytic equation and the two initial jets determine every higher Taylor coefficient.

The Cauchy-Kovalevskaya theorem says that an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients has a unique local real-analytic solution near each point of a real-analytic non-characteristic hypersurface, provided the prescribed Cauchy dataare real analytic there. The uniqueness is among local real-analytic solutions agreeing with all of those data.

For a scalar differential operator of order $k$, the principal symbol replaces each order-$k$ derivative ${\partial }^{\alpha }$ by ${\xi }^{\alpha }$ and discards lower-order terms. For a quasilinear partial differential equation, the coefficients are evaluated at the prescribed lower-order data.

A one-dimensional scalar conservation law describes the balance of a scalar density $u$ and its conservation law flux $f(u)$:The second identity follows from the first for smooth $u$: the quantity inside a fixed interval changes only through its boundary fluxes. For a classical solution, the chain rule also gives the quasilinear partial differential equation $u_t+f^{\prime }(u)u_x=0$, with characteristic speed $f^{\prime }(u)$.

The conservative form still defines a discontinuous weak solution. With initial condition $u_0$, it requiresfor every smooth compactly supported test function on $[0,\infty )\times \mathbb{R}$. For $f(u)=u^2/2$, a jump from $1$ to $0$ travels at speed $1/2$ by the Rankine-Hugoniot condition, between incoming characteristic speeds $1$ and $0$. Reversing those states still gives a conservative jump of speed $1/2$, but entropy solution admissibility selects a rarefaction wave. Thus conservation determines jump balance while an entropy condition selects the continuation after characteristic crossing.^[ref]