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For a second-order real linear partial differential equation in two variables, an invertible local change of variables simplifies its principal part. At regular points, the hyperbolic case has mixed-derivative principal part $U_{\xi \eta }$, the elliptic case has principal part $U_{\xi \xi }+U_{\eta \eta }$, and the parabolic case has rank-one principal part $U_{\xi \xi }$, after multiplication by nonzero factors. Lower-order terms and forcing must also be transformed with the chain rule; simplifying only the principal part does not in general give the full canonical equation. In the hyperbolic partial differential equation case the new variables can be chosen constant along the two characteristic families.

This partial differential equation has discriminant $y$, so it is a hyperbolic partial differential equation above $y=0$ and an elliptic partial differential equation below it. In its hyperbolic region the displayed characteristic coordinates have Jacobian determinant $18\sqrt{y}$. The chain rule givesThe first derivative terms arise from differentiating the coordinate coefficients and cannot be discarded. The coordinate change degenerates at the line where the two characteristic directions merge.

For a scalar second-order partial differential equation in two variables, a tangent direction along which Cauchy data fail to determine the highest transverse derivative through the equation. For principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, a tangent $(dx,dy)$ is characteristic exactly when $A(dy{)}^2-2Bdxdy+C(dx{)}^2=0$. Equivalently its conormal annihilates the principal symbol. A hyperbolic partial differential equation has two distinct real characteristic directions at each point.

For a hyperbolic partial differential equation in two variables, two independent characteristic coordinates remove both pure second partial derivatives, leaving a nonzero mixed second partial derivative. A nonzero Jacobian determinant is essential: two characteristic invariants that become dependent do not give a valid coordinate transformation.

For the canonical hyperbolic partial differential equation $V_{\xi \eta }=0$, every solution is $V=g(\xi )+h(\eta )$. Data on the characteristic $\eta ={\eta }_0$ determine $g$ up to an additive constant, but only determine $h$ at one point. If a transverse derivative is also prescribed, it must satisfy $V_{\eta }(\xi ,{\eta }_0)=h^{\prime }({\eta }_0)$, independent of $\xi$. A varying prescribed value is incompatible; a constant compatible value still leaves the rest of $h$ arbitrary. Thus characteristic Cauchy data can produce nonexistence or nonuniqueness. A derivative in original coordinates must first be transformed by the chain rule before applying this compatibility test.

For a second-order hyperbolic partial differential equation, the usual reduction to a mixed derivative uses twice continuously differentiable characteristic coordinates. Continuity of the principal coefficients alone does not ensure this regularity, even locally. Consider $R=0$, $S=1/2$ and $T=|x|$ near the origin; the discriminant $S^2-RT$ is the positive constant $1/4$. A characteristic coordinate must satisfyThe two characteristic families have first integrals $x$ and $y-x|x|/2$. Locally, any coordinate transverse to the first family has the form $\eta =h(y-x|x|/2)$ with $h^{\prime }\ne 0$. If this were twice continuously differentiable, its restriction at $x=0$ would make $h$ twice continuously differentiable. But its second $x$ derivative approaches $-h^{\prime }(y)$ from the right and $h^{\prime }(y)$ from the left, a contradiction. Independent characteristic coordinates cannot both choose the other family. Thus no twice continuously differentiable characteristic coordinate system exists near the origin.

With sufficiently smooth coefficients, the two distinct characteristic direction fields do yield smooth local coordinates. If $J$ is their Jacobian determinant and $B$ is the transformed mixed coefficient, then $B^2=J^2(S^2-RT)>0$. Division by $2B$ gives the mixed-derivative canonical form, including all lower-order terms produced by the chain rule. This is a local result; global intersections of characteristic curves can create separate obstructions.

Coauthor with Randall J. LeVeque and Mauricio J. del Razo of an interactive book on Riemann problems and approximate solvers for hyperbolic partial differential equations. Computational notebooks let readers vary initial states and compare wave patterns.^[ref]

The hyperbolic Laplacian is the Laplace-Beltrami operator for the metric of hyperbolic space. In upper-half-space coordinates $(x_1,\dots ,x_{n-1},y)$ with $y>0$,Using the nonnegative convention $\Delta =-\mathrm{div}\mathrm{grad}$ givesIndeed, insert $g^{ij}=y^2{\delta }^{ij}$ and $\sqrt{\det g}=y^{-n}$ into the coordinate formula $-{\left(\det g\right)}^{-1/2}{\partial }_i(\sqrt{\det g}{g}^{ij}{\partial }_j)$. The choice of overall sign must be specified when comparing spectral formulas.^[ref]

In dimension two the first-derivative term disappears. For $f(y)=y^s$, direct differentiation gives $\Delta f=s(n-1-s)f$, and in particular $s(1-s)f$ in the hyperbolic plane. This is the eigenvalue expression underlying many Maass form conventions.

Despite its name, this is an elliptic differential operator on $y>0$: its leading symbol in the chosen sign convention is $y^2|\xi {|}^2$. The adjective hyperbolic refers to the geometry, not to the classification of a hyperbolic partial differential equation.

A second-order partial differential equation is of mixed type when the type of its principal part changes across its domain. For $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}=0$, the sign of $B^2-AC$ distinguishes a hyperbolic partial differential equation, parabolic type and an elliptic partial differential equation. For example, $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ is elliptic above $y=0$, hyperbolic below it, and degenerate on the interface. On its hyperbolic side, the change $\xi =x+y$, $\eta =x-y$ reduces it to $u_{\xi \eta }=0$; above it is the Laplace equation. Type classification alone supplies no matching rule for the normal derivative.

For a second-order linear partial differential equation written $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\cdots =0$, the discriminant is $B^2-AC$. HereIt is positive exactly when $x\ne 0$, proving that the equation is a hyperbolic partial differential equation on each half-plane. At $x=0$ its principal part degenerates and it is not hyperbolic.

For a second-order linear partial differential equation with principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, the discriminant is $B^2-AC$. Here $A=1$, $B=0$, $C=-x^2$, soThe equation is therefore a hyperbolic partial differential equation throughout its domain. The lower-order term $-u_x/x$ does not affect this classification.

The characteristic curves satisfyIntegrating gives the two families $y\mp x^2/2=\text{constant}$. Choose coordinates constant along these families:Their Jacobian determinant is $8x\ne 0$, so they define an invertible change of variables on $x>0$, with image $\xi +\eta >0$. Write $u(x,y)=U(\xi ,\eta )$. The chain rule givesBoth the pure second partial derivatives and the first partial derivatives cancel in the original equation, leavingThis is the full canonical form of a second-order partial differential equation, including the effect of the lower-order term. Integrating shows $U_{\xi }$ is a function of $\xi$ alone, hencewith arbitrary twice differentiable functions $f,g$. The representation is global on $x>0$: for each fixed $\xi$, the allowed $\eta >-\xi$ form an interval, so $U_{\xi }$ is independent of $\eta$ throughout that interval; after subtracting its antiderivative, the same reasoning in the other coordinate leaves only $g(\eta )$. These are quadratic characteristic coordinates for a variable-speed wave equation.

For the data on $x=1$, set $s=1-2y$, so the other argument is $2-s$. The two initial conditions becomeDifferentiating the first equation with respect to $s$ also gives $f^{\prime }(s)-g^{\prime }(2-s)=0$. Solving these two linear equations yields $f^{\prime }(s)=g^{\prime }(2-s)=1/2$. Since $s$ ranges over all real numbers,ConsequentlyIndeed $u_{xx}=2$, $u_{yy}=0$, $u_x=2x$, so the equation and both initial conditions hold directly.

Use the convention $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}=0$, with discriminant $B^2-AC$. Here $A=1$, $B=0$, $C=\mathrm{sign}(y)$.

For $y>0$, the discriminant is $-1<0$, so the equation is an elliptic partial differential equation, already in the canonical Laplace equation form $u_{xx}+u_{yy}=0$.

For $y<0$, the discriminant is $1>0$, so it is a hyperbolic partial differential equation, the wave equation $u_{xx}-u_{yy}=0$. Introduce characteristic coordinatesThen ${\partial }_x={\partial }_{\xi }+{\partial }_{\eta }$ and ${\partial }_y={\partial }_{\xi }-{\partial }_{\eta }$, givingOn $y=0$, the discriminant vanishes and the principal part has rank one; the degenerate parabolic type is $u_{xx}=0$. This is an interface line, rather than an open parabolic subdomain. ThusThis is a mixed-type partial differential equation. The classification concerns the principal part and remains meaningful even though the coefficient jumps at the interface.

First consider the usual local reduction with sufficiently smooth coefficients. The hyperbolic partial differential equation has two distinct characteristic directions because $S^2-RT>0$. Choose independent local characteristic coordinates $\xi ,\eta$ whose gradients satisfyLet $J={\xi }_x{\eta }_y-{\xi }_y{\eta }_x\ne 0$ andThe determinant of the transformed principal coefficient matrix gives $B^2=J^2(S^2-RT)>0$. By the chain rule, the left side of the original partial differential equation becomeswith the analogous expression for $A_{\eta }$. The original first derivatives are $z_x=u_{\xi }{\xi }_x+u_{\eta }{\eta }_x$ and $z_y=u_{\xi }{\xi }_y+u_{\eta }{\eta }_y$. Substitution and division by $2B$ yield the requested mixed-derivative canonical form. The construction is local, in a neighborhood where the two characteristic foliations give a diffeomorphism.

There is a regularity qualification in the printed hypotheses: merely continuous coefficients need not give twice continuously differentiable characteristic coordinates. Such coordinates are needed for the second-derivative chain rule above. For example, take $R=0$, $S=1/2$ and $T=|x|$ near the origin. The characteristic equation factors as ${\xi }_y({\xi }_x+|x|{\xi }_y)=0$. One family is $x=\text{constant}$ and the other has first integrals $\eta =h(y-x|x|/2)$. For an independent coordinate $h^{\prime }\ne 0$, but ${\eta }_{xx}$ has opposite one-sided limits at $x=0$. Hence no twice continuously differentiable characteristic coordinate system exists there. This illustrates that continuous hyperbolic coefficients need not admit smooth characteristic coordinates. The formal general argument requires additional smoothness, for example twice continuously differentiable coefficients. The concrete coefficients below are smooth, so this issue does not affect their solution.

For the concrete partial differential equation, putThe Jacobian determinant is $1/y^2\ne 0$. Direct use of the chain rule givesThereforeTo impose the data efficiently, let $v=z_x$. Then $2x{v}_x+y{v}_y=0$, whose characteristic curves preserve $x/y^2$, so $v=H(x/y^2)$. On the initial line $x=1$, the value $v(1,y)=y^2$ gives $H(\xi )=1/\xi$ for every $\xi >0$. Thus $z_x=y^2/x$, and integration in $x$ gives $z=y^2\log x+C(y)$. The remaining initial condition gives $C(y)=0$:Every point in this region is reached from the data by a characteristic with positive $x/y^2$. The line $x=0$ separates these characteristics from those with $x<0$, and the logarithmic divergence prevents a smooth extension of this solution through $x=0$. The original condition restricts us to $y>0$.

For a nonzero second-order principal part, the discriminant is $Q^2-PR$: positive gives a hyperbolic partial differential equation, negative an elliptic partial differential equation, and zero the parabolic type when the principal part has rank one. If all three coefficients vanish, the equation is instead degenerate to lower order.

Here $P=x$, $2Q=x^2$, $R=0$, so $Q^2-PR=x^4/4>0$ for $x>0$. The characteristic equation isThus chooseTheir Jacobian determinant with respect to $(x,y)$ is $x\ne 0$, so they are valid coordinates on the specified half-plane. The level sets are horizontal lines and the right-hand halves of parabolas $y=x^2/2+\eta$.

Writing $z(x,y)=Z(\xi ,\eta )$, the chain rule givesThe original equation reduces to $-x{Z}_{\eta }-x^3{Z}_{\xi \eta }=-x{Z}_{\eta }+x^3$, hence the canonical form is $\boxed{Z_{\xi \eta }=-1}$. Integrating once in each characteristic variable givesorOn $x=1$, differentiation gives $z_x=y-g^{\prime }(y-1/2)$. The specified derivative therefore forces $g^{\prime }=0$ for every argument, so $g$ is a constant. The specified value then gives $f(y)+g=1+y^2-y/2$. ConsequentlyIts derivatives are $z_x=xy$, $z_{xx}=y$ and $z_{xy}=x$, which verify both the equation and the data directly.

The principal coefficients are $A=-xy$, $B=0$, $C=1$, in the convention $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$. Their discriminant is $B^2-AC=xy$. Thus it is a hyperbolic partial differential equation where $xy>0$ and an elliptic partial differential equation where $xy<0$. Its principal part has parabolic type where $xy=0$; however the full equation contains $u_y/y$ and is undefined on $y=0$. On its actual domain $y\ne 0$, the parabolic locus is $x=0$.

The characteristic equation of a partial differential equation isIn a hyperbolic region this gives two distinct real characteristic directions. The singular lower-order coefficient should not be mistaken for an ordinary parabolic point of the full equation on $y=0$.

For $\beta =0$, the equation for the characteristic curves isThe two families are therefore $y=\text{constant}$ and the characteristic projections from part (a), $y=K{x}^2$, together with the vertical characteristic $x=0$. On either region $y>0$ or $y<0$, the hyperbolic partial differential equation admits the characteristic coordinatesIndeed $\phi$ is constant along the first family, while $(x{\partial }_x+2y{\partial }_y)\psi =0$. Their Jacobian determinant isThe coefficients of the two pure second partial derivatives in these coordinates vanish, and the coefficient of the mixed partial derivative is $2y/\sqrt{|y|}\ne 0$. Hence they give the hyperbolic characteristic normal form. They also remain regular at $x=0$, provided $y\ne 0$.

The alternative $\phi =y$, $\psi =y/x^2$ is valid on patches with $xy\ne 0$. On $y=0$, it is a parabolic partial differential equation away from the fully degenerate origin, so two independent hyperbolic characteristic coordinates are unavailable.

Take $z$ increasing upwards and write $s=\varphi /{\varphi }_{\max }$. The particles have downward settling velocity; their upward volume flux is $\varphi V$. Conservation of particle volume in any interval givesThus the kinematic sedimentation equation and its characteristic speed areFor $0<s<1$ this is a scalar hyperbolic partial differential equation: its one characteristic speed is real. The method of characteristics gives $ds/dt=0$ and $dz/dt=a(s)$. A concentration disturbance need not descend at the particle settling velocity; $a(s)$ is the derivative of the transported flux, not $V(s)$. In particular $a(s)$ can be positive even though every particle settles downwards. At $\alpha =0$, $a=-V_0$ and the equation is ordinary translation. At the packing endpoint $s=1$ the differentiability of the chosen constitutive law depends on $\alpha$; none of the initial values used below is at that endpoint.

Use the explicit printed width law $b(z)=\alpha z^2$. Its wetted area and hydrostatic pressure integral areAssuming hydrostatic pressure, a horizontal prismatic bed and cross-sectionally uniform velocity, volume conservation and momentum conservation give the prismatic-channel shallow water equationsEquivalently,The coefficient matrix in variables $(h,u)$ is $\left(\begin{array}{cc}u& h/3\\ g& u\end{array}\right)$. Its eigenvalues and the corresponding gravity-wave speed areFor $h>0$ these eigenvalues are distinct and real, establishing strict hyperbolicity in the hyperbolic partial differential equation sense. Since $dc/dh=c/(2h)$, substitution in the two equations showsFor example $({\partial }_t+(u+c){\partial }_x)(u+6c)=0$. This provides both the slopes and the conserved characteristic quantities. At a dry bed $h=0$, strict hyperbolicity degenerates and the front is interpreted as the limit from positive depth.

The descriptive word “parabolic” alone would normally suggest $b\propto \sqrt{z}$ and different coefficients. The explicit $z^2$ width in the PDF determines the calculation here; it must not be silently replaced by that other geometry.

The chain rule gives ${\partial }_x={\partial }_{\xi }+\beta {\partial }_{\eta }$ and ${\partial }_y=\alpha {\partial }_{\xi }+{\partial }_{\eta }$. The coefficients of $u_{\xi \xi }$ and $u_{\eta \eta }$ are $2+3{\alpha }^2-7\alpha$ and $2{\beta }^2+3-7\beta$. Their integer roots are $\alpha =2$ and $\beta =3$. Thus choose characteristic coordinates $\xi =x+2y$, $\eta =3x+y$, whose Jacobian determinant is $-5\ne 0$. The coefficient of $u_{\xi \eta }$ is $4\beta +6\alpha -7(1+\alpha \beta )=-25$, so the equation becomes $-25u_{\xi \eta }=0$, equivalently $u_{\xi \eta }=0$.

Integrating the transformed hyperbolic partial differential equation gives $u=F(\xi )+G(\eta )$. Along $x=-2y$ the coordinates are $(0,-5y)$, so the zero boundary condition gives $G(\eta )=-F(0)$ for every real $\eta$. Along $x=0$ they are $(2y,y)$, so $F(2y)-F(0)=4y^2$, hence $F(\xi )={\xi }^2+F(0)$. The constants cancel and the unique resulting solution isIt directly satisfies both boundary conditions and $2u_{xx}+3u_{yy}-7u_{xy}=4+24-28=0$.

Expanding the equation givesFor the hyperbolic partial differential equation discriminant, $A=1$, $B=\rho$, $C={\rho }^2-1$, soFor a curve $\rho =\rho (t)$, the characteristic hypersurface equation is $({\rho }^{\prime }{)}^2-2\rho {\rho }^{\prime }+{\rho }^2-1=0$, or ${\rho }^{\prime }=\rho \pm 1$. Therefore the two families of characteristic curves areThe values $C=0$ include the characteristic lines $\rho =-1$ and $\rho =1$. Every curve in the first family approaches $\rho =-1$ as $t\to -\infty$; every curve in the second approaches $\rho =1$. The two curves through $(\rho ,t)=(0,0)$ are $\rho =e^t-1$ and $\rho =1-e^t$, shown as the thick curves below. Equivalently, the characteristic coordinates are $e^{-t}(\rho +1)$ and $e^{-t}(\rho -1)$.