Unit 3: Classification and Canonical Reduction of Second-Order Linear PDEs

1. General Form of Second-Order Equations

The most general second-order linear (or semilinear) partial differential equation in two independent variables $(x, y)$ is:

where $A, B, C, D, E, F, G$ are real-valued continuous functions on a domain $\Omega \subseteq \mathbb{R}^2$, with $A^2 + B^2 + C^2 \ne 0$. (Note: The convention $2B$ is standard in academic mathematics so that the discriminant takes the clean form $\Delta = B^2 - AC$).

The principal part containing the highest-order derivatives is the differential operator:

Associated with $\mathcal{L}_0$ is the characteristic polynomial (quadratic form) in variables $(\xi_1, \xi_2)$:

2. Classification Criteria

Definition 3.1 (Discriminant and PDE Classification): The discriminant (or indicator) of the second-order equation is defined at each point $(x, y) \in \Omega$ by:

The PDE is classified at $(x, y)$ as:

1. Hyperbolic if $\Delta(x, y) > 0$ (analogous to the hyperbola $B^2 - AC > 0$).
2. Parabolic if $\Delta(x, y) = 0$ (analogous to the parabola $B^2 - AC = 0$).
3. Elliptic if $\Delta(x, y) < 0$ (analogous to the ellipse $B^2 - AC < 0$).

If an equation retains the same type at all points of $\Omega$, it is of pure type; if $\Delta(x, y)$ changes sign across $\Omega$, it is of mixed type.

3. Coordinate Invariance of the Discriminant

Let $(\xi, \eta) = (\phi(x, y), \psi(x, y))$ be a non-singular $C^2$ change of variables with non-vanishing Jacobian:

By the multivariable chain rule:

Differentiating again:

Substituting into the PDE yields the transformed equation in coordinates $(\xi, \eta)$:

where the new coefficients of the principal part are:

Theorem 3.1 (Discriminant Invariance Theorem): Under any smooth non-singular coordinate transformation, the transformed discriminant satisfies:

Since $J \ne 0$, $J^2 > 0$; therefore, the sign of the discriminant is an absolute geometric invariant of the PDE.

Proof: Express the transformation in matrix form:

where $M = \begin{pmatrix} \xi_x & \xi_y \\ \eta_x & \eta_y \end{pmatrix}$ and $K = \begin{pmatrix} A & B \\ B & C \end{pmatrix}$. Taking the determinant of both sides:

Evaluating the determinants:

Multiplying by $-1$:

Since $J^2 > 0$, $\operatorname{sgn}(\Delta^*) = \operatorname{sgn}(\Delta)$. $\blacksquare$

1. Characteristic Equations for Hyperbolic PDEs

For a hyperbolic PDE, $\Delta = B^2 - AC > 0$. We seek coordinates $(\xi, \eta)$ such that the diagonal second-derivative terms vanish:

Recall that:

Dividing by $\xi_y^2$ (assuming $\xi_y \ne 0$):

Along any level curve $\xi(x, y) = \text{constant}$, implicit differentiation gives $\frac{dy}{dx} = -\frac{\xi_x}{\xi_y}$. Thus, the curve $y = y(x)$ satisfies the characteristic ODE:

Solving for the characteristic slopes $\frac{dy}{dx}$:

Since $B^2 - AC > 0$, there exist two distinct families of real characteristic curves:

2. The Two Canonical Forms of Hyperbolic PDEs

Canonical Form 1 (D'Alembert Cross-Derivative Form):

Setting $\xi = \phi(x, y)$ and $\eta = \psi(x, y)$ forces $A^ = 0$ and $C^ = 0$. Dividing through by $2B^* \ne 0$, the PDE reduces to:

Canonical Form 2 (Wave Operator Form):

Introducing the rotated coordinates:

We have:

The cross-derivative transforms into:

Thus, the equation becomes the standard 1D wave operator:

1. Characteristic Degeneracy in Parabolic PDEs

For a parabolic PDE, the discriminant vanishes identically:

The characteristic equation:

has equal roots:

Therefore, there exists only one real family of characteristic curves:

This single family leaves $A^* = 0$. Furthermore, since $\Delta^ = (B^)^2 - A^ C^ = J^2 \Delta = 0$, having $A^* = 0$ immediately forces:

Both $A^$ and $B^$ vanish automatically!

2. Reduction to Canonical Form

To complete the transformation, we choose the second coordinate $\eta = \psi(x, y)$ as any arbitrary smooth function that is functionally independent of $\xi$, ensuring $J \ne 0$. (A convenient choice is often $\eta = x$ or $\eta = y$). Since $A^ = 0$ and $B^ = 0$, while $C^ \ne 0$ (otherwise $J$ would vanish), dividing by $C^$ yields:

Theorem 3.2 (Parabolic Canonical Form): Every parabolic equation can be reduced to the canonical form:

If $\Phi$ contains $u_\xi$ with non-zero coefficient, dividing produces the standard heat/diffusion form:

1. Complex Characteristics for Elliptic PDEs

For an elliptic PDE, $\Delta = B^2 - AC < 0$. The characteristic equation:

has no real roots. Instead, it yields two complex conjugate characteristic slopes:

Integrating these gives complex conjugate first integrals:

2. Real Canonical Reduction to Laplace Form

If we were to use the complex coordinates $\tilde{\xi} = \phi + i\psi$ and $\tilde{\eta} = \phi - i\psi$, the equation would reduce to $u_{\tilde{\xi}\tilde{\eta}} = \dots$, but the coordinates would be complex. To obtain a real canonical form, we define the real transformation:

By the chain rule relating $(\tilde{\xi}, \tilde{\eta})$ to $(\xi, \eta)$:

Thus, in the real coordinates $(\xi, \eta)$, the equation reduces to:

Theorem 3.3 (Elliptic Canonical Form): Every elliptic second-order linear PDE can be transformed into the canonical form:

which is the classical Laplace / Poisson operator $\nabla^2 u = \Phi$.

1. Equations of Mixed Type

When the coefficients $A(x, y), B(x, y), C(x, y)$ depend on position, the sign of the discriminant $\Delta(x, y) = B^2 - AC$ may vary across the domain, producing equations of mixed type. The boundary curve separating different types is the parabolic transition curve where $\Delta(x, y) = 0$.

2. The Tricomi Equation

The most celebrated mixed-type PDE in mathematical physics is the Tricomi equation, introduced by Francesco Tricomi in 1923 to model transonic fluid flow (aerodynamics near Mach 1):

Here $A = y, B = 0, C = 1$. The discriminant is:

Classification of Tricomi's Equation:

• In the upper half-plane $y > 0$: $\Delta = -y < 0 \implies$ Elliptic (subsonic flow, $M < 1$).
• Along the line $y = 0$: $\Delta = 0 \implies$ Parabolic (sonic transition line, $M = 1$).
• In the lower half-plane $y < 0$: $\Delta = -y > 0 \implies$ Hyperbolic (supersonic flow, $M > 1$).

Canonical Reduction in the Hyperbolic Half-Plane ($y < 0$):

The characteristic ODE is:

Taking the square root:

Integrating:

Thus, the two real characteristic coordinates are:

In these coordinates, Tricomi's equation reduces to the Euler-Poisson-Darboux form:

Notice the characteristic curves are semi-cubical parabolas with cusps touching the sonic line $y = 0$!