The method of characteristics

Theorem

The method of characteristics

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Statement

Let \(D \subseteq \mathbb{R}^2\) be open and let \(a,b,c : D \times \mathbb{R} \to \mathbb{R}\) be \(C^1\) functions, giving the quasilinear first-order PDE \[ a(x,y,u)\,u_x + b(x,y,u)\,u_y = c(x,y,u) \] for an unknown \(u = u(x,y)\). Let \(\Gamma : s \mapsto (x_0(s), y_0(s), z_0(s))\), \(s \in I\) (an open interval), be a \(C^1\) curve in \(D \times \mathbb{R}\) prescribing Cauchy data \(u(x_0(s), y_0(s)) = z_0(s)\), and suppose \(\Gamma\) is non-characteristic: at every \(s \in I\), \[ J(s) := x_0'(s)\, b(x_0(s), y_0(s), z_0(s)) - y_0'(s)\, a(x_0(s), y_0(s), z_0(s)) \neq 0. \] Then for each \(s_0 \in I\) there exist \(\varepsilon \gt 0\) and a neighbourhood \(U\) of \((x_0(s_0), y_0(s_0))\) such that the initial value problem has a unique \(C^1\) solution \(u\) on \(U\) agreeing with \(z_0\) on \(\Gamma \cap U\), obtained by solving the characteristic system \[ \frac{dx}{dt} = a(x,y,u), \qquad \frac{dy}{dt} = b(x,y,u), \qquad \frac{du}{dt} = c(x,y,u) \] with initial conditions \(x(s,0) = x_0(s)\), \(y(s,0) = y_0(s)\), \(u(s,0) = z_0(s)\), \(|t| \lt \varepsilon\), and inverting \((s,t) \mapsto (x,y)\) near \(t=0\).

Why it matters

A first-order PDE looks, on the surface, like it should be "easier" than a second-order one such as the heat or wave equation, yet its solutions can develop discontinuities in finite time even from smooth data — the theorem explains exactly how and why. It converts a partial differential equation, an object living on a two-dimensional domain, into a family of ordinary differential equations along curves called characteristics, each of which the equation forces \(u\) to be transported along like a particle carried by a flow. This is the single idea underlying nonlinear conservation laws, geometric optics (eikonal equations), traffic flow models, and the Hamilton–Jacobi formulation of classical mechanics.

Its true importance in MU-305 is structural: it is the prototype for how "propagation of information" works for hyperbolic PDE in general, and it is the mechanism by which smooth Cauchy data can produce a genuine singularity (a shock, or a loss of well-posedness) — a phenomenon second-order elliptic and parabolic theory simply does not exhibit.

Hypotheses

\(a,b,c\) are \(C^1\) in a neighbourhood of the relevant points. Without local Lipschitz continuity in \(u\) (which \(C^1\) guarantees via the mean value theorem), the characteristic ODE system \(\dot x = a, \dot y = b, \dot u = c\) can fail to have unique solutions — e.g. \(u_t = \sqrt{|u|}\,u_x\) type coefficients allow two distinct characteristics to emanate from the same point, so "the" characteristic through a data point is not well defined and the construction breaks down before it starts. \(\Gamma\) is \(C^1\). If \(\Gamma\) merely has a corner, the parametrisation \((x_0(s), y_0(s), z_0(s))\) has no well-defined tangent at that point, so the non-characteristic condition \(J(s) \neq 0\) cannot even be evaluated there, and the local solution built on either side need not join into a single \(C^1\) function across the corner. \(\Gamma\) is non-characteristic: \(J(s) \neq 0\) on \(I\). Take \(u_x = 0\) (so \(a=1, b=0, c=0\)) with data prescribed on \(\Gamma = \{(s,0,z_0(s))\}\), which is itself a characteristic (\(x_0'=1, y_0'=0\), so \(J \equiv 0\)). The general solution is \(u(x,y)=f(y)\) for arbitrary \(f\); consistency with the data forces \(z_0(s) = f(0)\) constant, so a solution exists only for the special data \(z_0 \equiv\) const, and if it does exist it is wildly non-unique (any \(f\) with \(f(0)=z_0\) works). This is the "characteristic initial value problem" and it is the paradigm case of ill-posedness under this theorem. The map \((s,t) \mapsto (x(s,t), y(s,t))\) is a local diffeomorphism near \(t=0\). This is not a separate hypothesis but the content of \(J(s) \neq 0\) together with the inverse function theorem (see Step 4 below); it is listed to flag that the *entire* proof rests on being able to invert this map, and the solution is only guaranteed on the (possibly small) open set where the inversion is valid — this is exactly why "the" solution is local, not global, in general.

Proof

Set up the characteristic ODE system: for each fixed \(s \in I\), solve \[ \dot x(t) = a(x,y,u), \quad \dot y(t) = b(x,y,u), \quad \dot u(t) = c(x,y,u), \quad (x,y,u)(0) = (x_0(s), y_0(s), z_0(s)). \]

Since \(a,b,c\) are \(C^1\), hence locally Lipschitz, on a neighbourhood of \((x_0(s_0), y_0(s_0), z_0(s_0))\), the Picard–Lindelöf existence and uniqueness theorem for ODEs applies to give a unique \(C^1\) solution \(t \mapsto (x(s,t), y(s,t), u(s,t))\) for \(|t| \lt \varepsilon(s)\); by continuous dependence on parameters, \(\varepsilon\) can be chosen uniformly for \(s\) near \(s_0\), and the joint map \((s,t) \mapsto (x,y,u)(s,t)\) is \(C^1\). A

If \(u\) solves the PDE and \(u(x_0(s),y_0(s)) = z_0(s)\), then along any curve \((x(t),y(t))\) solving \(\dot x = a(x,y,u), \dot y = b(x,y,u)\) with \(u = u(x(t),y(t))\), the value \(w(t) := u(x(t),y(t))\) automatically satisfies \(\dot w = c(x,y,w)\).

Chain rule: \(\dot w = u_x \dot x + u_y \dot y = u_x\, a(x,y,u) + u_y\, b(x,y,u) = c(x,y,u)\) by the PDE itself, evaluated along the curve. This shows the PDE is exactly the statement that \(u\) is constant along solutions of \(\dot u = c\) coupled to \(\dot x=a,\dot y=b\) — motivating why solving the ODE system in Step 1 is the right candidate construction. B

Define \(F(s,t) := (x(s,t), y(s,t))\) using the solution from Step 1. Then \[ \det DF(s,0) = \det \begin{pmatrix} \partial_s x(s,0) & \partial_t x(s,0) \\ \partial_s y(s,0) & \partial_t y(s,0) \end{pmatrix} = \det \begin{pmatrix} x_0'(s) & a(x_0(s),y_0(s),z_0(s)) \\ y_0'(s) & b(x_0(s),y_0(s),z_0(s)) \end{pmatrix} = J(s). \]

\(\partial_s x(s,0) = x_0'(s)\) and \(\partial_s y(s,0) = y_0'(s)\) directly from the initial condition \(x(s,0)=x_0(s)\), \(y(s,0)=y_0(s)\); \(\partial_t x(s,0) = \dot x(0) = a(\cdots)\) and \(\partial_t y(s,0) = \dot y(0) = b(\cdots)\) from the ODE itself evaluated at \(t=0\). Expanding the \(2\times2\) determinant reproduces \(J(s)\) from the Statement. A

Since \(J(s_0) \neq 0\) by hypothesis, \(F\) is a local \(C^1\)-diffeomorphism from a neighbourhood \(V\) of \((s_0,0)\) onto a neighbourhood \(U\) of \((x_0(s_0),y_0(s_0))\); let \((s(x,y), t(x,y))\) denote its \(C^1\) inverse on \(U\). Define \(u(x,y) := w(s(x,y), t(x,y)) = u\big(s(x,y),t(x,y)\big)\) where \(w(s,t)\) is the third component of the Step-1 solution.

This is exactly the Inverse Function Theorem, applicable because \(F\) is \(C^1\) (Step 1) and \(\det DF(s_0,0) = J(s_0) \neq 0\) (Step 3). Composition of \(C^1\) functions \((x,y) \mapsto (s,t) \mapsto w\) is \(C^1\), so \(u\) is a well-defined \(C^1\) function on \(U\). C

The function \(u\) so defined satisfies the PDE on \(U\) and the initial condition on \(\Gamma \cap U\).

Initial condition: at \(t=0\), \(s(x_0(s),y_0(s))=s\), so \(u(x_0(s),y_0(s)) = w(s,0) = z_0(s)\) by construction. PDE: write \(u_x, u_y\) via the chain rule through \((s,t)\) and use that \((x(s,t),y(s,t),w(s,t))\) solves the characteristic ODEs for every fixed \(s\); a direct computation (below) shows \(a u_x + b u_y = c\) at every point of \(U\), using that \(\partial(x,y)/\partial(s,t)\) is invertible there (Step 4) to solve for \(u_x, u_y\) in terms of \(\partial_t w = c\), \(\partial_s w\), and the Jacobian entries. B

Explicitly: writing \(x_t=a\), \(y_t=b\), \(w_t=c\) (from the ODEs) and differentiating \(w(s,t)=u(x(s,t),y(s,t))\) in \(t\): \(w_t = u_x x_t + u_y y_t = a u_x + b u_y\). Since \(w_t = c\) by construction, \(a u_x + b u_y = c\) identically on \(U\), i.e. \(u\) solves the PDE.

This is just Step 2 run in reverse: having built \(u\) from the ODE data, differentiate the defining identity \(w(s,t) = u(x(s,t),y(s,t))\) with respect to \(t\) using the multivariable chain rule, and substitute \(w_t = c(x,y,w)\) from the third characteristic equation. No further hypothesis is needed beyond \(C^1\)-ness of everything involved, already secured in Steps 1 and 4. C

Uniqueness: any \(C^1\) solution \(\tilde u\) of the PDE with \(\tilde u = z_0\) on \(\Gamma\) agrees with \(u\) on \(U\) (shrinking \(U,\varepsilon\) if necessary).

By Step 2, for each fixed \(s\), \(t \mapsto \tilde u(x(s,t),y(s,t))\) satisfies the same scalar ODE \(\dot w = c(x,y,w)\), \(w(0) = z_0(s)\), as \(w(s,\cdot)\) does; by the uniqueness clause of Picard–Lindelöf (same Lipschitz hypothesis as Step 1) these two solutions of the same IVP coincide for all \(|t\)| in the common interval of existence. Since every point of \(U\) is \((x(s,t),y(s,t))\) for a unique \((s,t)\) (Step 4), \(\tilde u = u\) on \(U\). B

Result

a u_x + b u_y = c,\ \ u|_\Gamma = z_0 \ \Longrightarrow\ u \text{ is obtained by solving } \dot x=a,\ \dot y=b,\ \dot u=c \text{ along characteristics through } \Gamma

Reading. To solve a first-order PDE, do not attack the PDE directly: instead treat each point of the initial curve as launching a particle whose position \((x,y)\) and "height" \(u\) evolve by a coupled system of ODEs. Sweeping the parameter \(s\) along the initial curve and the ODE-time \(t\) away from it sweeps out the whole solution surface, provided the initial curve is not itself tangent to a characteristic direction.

Scope. Purely local and purely first-order: it applies to a single quasilinear (or linear, or fully nonlinear via the extended Charpit system) first-order PDE in finitely many independent variables, with \(C^1\) coefficients, near a non-characteristic point of \(C^1\) Cauchy data. It gives no global existence in general (characteristics from different \(s\) may cross, destroying the diffeomorphism property of Step 4 and hence the single-valuedness of \(u\)), and it does not directly extend to systems of first-order PDEs or to second-order PDEs (though it underlies the classification of second-order PDEs via their own characteristic curves/surfaces).

Corollaries & converses

Linear/semilinear case. If \(a,b\) do not depend on \(u\) (linear or semilinear PDE), the \((x,y)\)-characteristics \(\dot x=a(x,y),\dot y=b(x,y)\) can be solved independently of \(u\), simplifying the construction to: find the characteristic curves first, then integrate \(\dot u = c\) along each.
Constancy along characteristics. For the homogeneous linear case \(c\equiv 0\), the theorem gives immediately that \(u\) is constant along each characteristic curve, so the general solution is \(u = f(\phi(x,y))\) for any \(C^1\) function \(f\), where \(\phi = \) const parametrises the characteristics — a fact used constantly in transport-equation and conservation-law analysis.
Envelope/breakdown time. The maximal time of existence of the classical solution is exactly the first \(t\) at which two distinct characteristics (from different \(s\)) collide, i.e. the first \(t^*\) where \(\det DF(s,t)\) vanishes for some \(s\); this quantity is computed explicitly in conservation-law theory to locate shock formation.
Converse fails as a global statement. The theorem does not converse to "every classical solution near \(\Gamma\) arises this way and exists on all of \(D\)"; existence is only local, and a classical (everywhere-\(C^1\)) global solution frequently fails to exist even when \(D\), \(a,b,c\) are entirely smooth (e.g. Burgers' equation with decreasing initial data).
Non-characteristic hypothesis is sharp, not conversely sufficient for global uniqueness: non-characteristic data guarantees local uniqueness (Step 7), but distinct non-characteristic initial curves through overlapping regions can still be inconsistent with each other if they are not compatible with a single global \(u\); the theorem only ever asserts a local, not a global, classification.

Fails without

Drop \(J(s) \neq 0\) (characteristic initial curve). For \(u_x = 0\) with data on \(\Gamma = \{y=0\}\) (which is characteristic since \(a=1,b=0\) gives \(J\equiv 0\)): the general solution is \(u=f(y)\), so matching \(u(x,0)=z_0(x)\) forces \(f(0)=z_0(x)\) for all \(x\) — impossible unless \(z_0\) is constant, and even then \(f\) is undetermined off \(y=0\). Existence and uniqueness both fail.
Drop \(C^1\) regularity of \(a,b,c\) in \(u\) (only continuity). Take \(u_t + u\,u_x = 0\) but with the flux modified to use \(c(u) = \sqrt{|u|}\,\mathrm{sgn}(u)\) style non-Lipschitz nonlinearity in place of \(u\) itself in the transport speed: e.g. \(\dot x = \sqrt{|u|}\), \(\dot u = 0\) with \(u(x,0)=0\) for \(x\le 0\). Both \(u\equiv0\) and infinitely many other characteristic fields through the same point solve the characteristic ODE non-uniquely (a Peano-type non-uniqueness, exactly as for \(\dot y=\sqrt{|y|}\), \(y(0)=0\)), so Step 1 (Picard–Lindelöf) is unavailable and the construction produces multiple candidate solutions.
Ignore crossing of characteristics beyond the local neighbourhood. For inviscid Burgers \(u_t + u u_x = 0\) with strictly decreasing smooth data \(u(x,0)=-x\), characteristics \(x = x_0 + t\,u(x_0,0) = x_0(1-t)\) all meet at \((x,t)=(0,1)\); for \(t\ge 1\) the map \(F\) from Step 4 is no longer injective, \(\det DF\) vanishes, and no single-valued \(C^1\) solution exists past \(t=1\) even though the data and coefficients are entire and smooth.

Common errors

Forgetting to check \(J(s) \neq 0\) before writing down "the" characteristics — silently assuming the initial curve is admissible when it may be characteristic (or characteristic at isolated points, which is enough to break the inversion locally there).
Solving \(\dot x = a, \dot y = b\) first and only afterwards trying to integrate \(\dot u = c\) as a function of \((x,y)\) alone, when \(c\) genuinely depends on \(u\) (quasilinear case) — the three ODEs are coupled and must be solved as one system, not sequentially, unless \(a,b\) are \(u\)-independent.
Treating the parametrised solution \((x(s,t),y(s,t),u(s,t))\) as the final answer without inverting \((s,t)\mapsto(x,y)\) — a parametric surface is not yet "\(u\) as a function of \(x,y\)" until the inversion of Step 4 is actually carried out.
Assuming the local solution automatically extends for all \(t\), and being surprised when characteristics cross and the "solution" becomes multivalued; students often mistake this for an arithmetic mistake rather than the genuine finite-time breakdown the theorem predicts.
Confusing the non-characteristic condition \(J(s)\neq0\) (a statement about the initial curve vs. the vector field \((a,b)\)) with linear independence of the vector \((a,b,c)\) itself, which is not what is required.

Discussion

The method of characteristics is, historically, one of the oldest general techniques in PDE theory, tracing to Lagrange and Monge's work on first-order equations in the eighteenth century, well before functional-analytic existence theory existed; it is the reason first-order PDE were understood essentially completely long before second-order theory matured. Monge's geometric picture — a solution surface in \((x,y,u)\)-space built by threading together curves each tangent to the "Monge direction" \((a,b,c)\) — is exactly the geometric content of Step 2 above and remains the cleanest way to think about the theorem.

The construction generalises in two directions covered later in MU-305: to fully nonlinear first-order PDE \(F(x,y,u,p,q)=0\) (where \(p=u_x,q=u_y\)) via the Charpit–Lagrange characteristic equations, which augment the system with ODEs for \(p,q\) themselves; and to \(n\) independent variables, where characteristics remain curves (not hypersurfaces) but the non-characteristic condition on the initial hypersurface becomes a transversality condition on an \((n-1)\)-dimensional \(\Gamma\).

There is a deep structural link to Hamiltonian mechanics: for the eikonal/Hamilton–Jacobi equation \(u_t + H(x,\nabla u) = 0\), the characteristic equations are exactly Hamilton's equations \(\dot x = H_p, \dot p = -H_x\), and \(u\) evolves along them by the Legendre-transform relation \(\dot u = p\cdot \dot x - H\); this is not an analogy but a literal special case of the Charpit system, which is why "characteristics" in optics, mechanics, and control theory (Pontryagin's maximum principle) are the same object viewed from different fields.

Common misconception. Students often believe the theorem guarantees a solution on all of \(D\) because \(a,b,c\) are smooth there; the theorem is emphatically local (only near \((x_0(s_0),y_0(s_0))\)), and the true maximal domain of the classical solution is determined by tracking exactly where the map \(F\) of Step 3 stops being injective — a computation the theorem sets up but does not itself answer globally.

Worked examples

Solve \(u_x + u_y = u\) with \(u(x,0) = x^2\) (a linear PDE; here \(a=1,b=1,c=u\)).

Identify \(\Gamma\): \((x_0(s),y_0(s),z_0(s)) = (s,0,s^2)\). Check non-characteristic: \(J(s) = x_0' b - y_0' a = 1\cdot1 - 0\cdot1 = 1 \neq 0\) for all \(s\), so the theorem applies everywhere on \(\Gamma\). A

Characteristic ODEs: \(\dot x = 1, \dot y = 1, \dot u = u\), with \(x(s,0)=s, y(s,0)=0, u(s,0)=s^2\). Solving: \(x = s+t\), \(y=t\), \(u = s^2 e^t\).

\(\dot x=1,\dot y=1\) integrate trivially; \(\dot u = u\) is separable, \(u(s,t) = u(s,0) e^{t} = s^2 e^t\), using Step 1's Picard–Lindelöf solution explicitly since \(c(u)=u\) is smooth (linear, hence \(C^1\)) in \(u\). A

Invert: from \(y=t\) and \(x=s+t\), get \(t=y\), \(s=x-y\). Substitute: \(u(x,y) = (x-y)^2 e^{y}\).

This is Step 4 of the proof carried out explicitly; the map \((s,t)\mapsto(x,y)=(s+t,t)\) has Jacobian \(\det\begin{pmatrix}1&1\\0&1\end{pmatrix}=1\neq0\) everywhere, matching \(J(s)\equiv1\), so the inversion is global here (a special feature of this linear, constant-coefficient example, not guaranteed in general). B

u(x,y) = (x-y)^2\, e^{y}

Reading. Direct check: \(u_x = 2(x-y)e^y\), \(u_y = -2(x-y)e^y + (x-y)^2 e^y\), so \(u_x+u_y = (x-y)^2 e^y = u\). At \(y=0\): \(u(x,0)=x^2\). Both conditions confirmed.

Scope. Valid on all of \(\mathbb{R}^2\) here because the coefficients are linear and the characteristics never cross — this is the easy end of the spectrum for the theorem.

Solve the quasilinear (inviscid Burgers) equation \(u_t + u\,u_x = 0\) with \(u(x,0) = -x\) (here independent variables are \((x,t)\); \(a=u, b=1, c=0\)).

\(\Gamma: (x_0(s),t_0(s),z_0(s)) = (s,0,-s)\). Non-characteristic check: \(J(s) = x_0' \cdot b - t_0' \cdot a = 1\cdot 1 - 0 \cdot(-s) = 1 \neq 0\), so the theorem's local existence applies near every point of \(\Gamma\). A

Characteristic ODEs: \(\dot x = u, \dot t = 1, \dot u = 0\), initial data \(x(s,0)=s, t(s,0)=0, u(s,0)=-s\). Since \(\dot u=0\), \(u\equiv -s\) along the whole characteristic; then \(\dot x = -s\) gives \(x(s,\tau) = s - s\tau\) and \(t(s,\tau)=\tau\).

Integrate \(\dot u=0\) first (trivial, gives \(u\) constant along each characteristic — the hallmark of Burgers' equation); substitute the now-known constant \(u=-s\) into \(\dot x = u\) and integrate in the ODE-time parameter \(\tau\), using Step 1's existence theorem, valid since \(a(x,t,u)=u\) is smooth in all its arguments. B

Invert using \(t=\tau\): \(x = s - st = s(1-t)\), so \(s = \dfrac{x}{1-t}\) for \(t \neq 1\); hence \(u(x,t) = -s = \dfrac{-x}{1-t} = \dfrac{x}{t-1}\).

Jacobian of \((s,\tau)\mapsto(x,t)\): \(\det\begin{pmatrix}\partial_s x & \partial_\tau x\\ \partial_s t&\partial_\tau t\end{pmatrix} = \det\begin{pmatrix}1-t & -s\\0&1\end{pmatrix}=1-t\), matching \(J(s)=1\) at \(t=0\) and vanishing exactly at \(t=1\) — this recovers the claim of Corollary 3 (breakdown time) and the third bullet of "Fails without": the inversion of Step 4 is only valid for \(t \lt 1\). C

u(x,t) = \dfrac{x}{t-1}, \qquad 0 \le t \lt 1

Reading. All characteristics converge at the single point \((x,t)=(0,1)\) (since \(x=s(1-t)=0\) at \(t=1\) for every \(s\)), so the classical solution exists exactly up to, but not including, \(t=1\): a genuine finite-time breakdown from perfectly smooth (linear) initial data.

Scope. Valid only for \(t\lt1\); at \(t=1\) the solution becomes infinite/discontinuous (a shock forms), illustrating precisely the local-only character of the Result box above.

Problems

Solve \(x u_x + y u_y = u\) with Cauchy data \(u(s,1) = s\) for \(s \in \mathbb{R}\). Verify the non-characteristic condition first.
Solution
Here \(a=x,b=y,c=u\), \(\Gamma:(s,1,s)\). \(J(s) = x_0' b - y_0' a = 1\cdot 1 - 0 \cdot s = 1 \neq 0\): non-characteristic everywhere, theorem applies. Characteristic ODEs: \(\dot x=x,\dot y=y,\dot u=u\) with \(x(s,0)=s,y(s,0)=1,u(s,0)=s\), giving \(x=se^t, y=e^t, u=se^t\). Invert: \(t=\ln y\), \(s=x/y\) (for \(y\gt0\)), so \(u = (x/y)\cdot y = x\). Check: \(xu_x+yu_y = x\cdot1+y\cdot0=x=u\). At \(y=1\): \(u(x,1)=x=s\) as required. Answer: \(u(x,y)=x\) for \(y\gt0\).
Explain, without solving, why Cauchy data prescribed on the curve \(\Gamma=\{(s,s,z_0(s))\}\) for the PDE \(u_x - u_y = 0\) makes the theorem inapplicable, and describe the two possible outcomes.
Solution
Here \(a=1,b=-1\). \(J(s)=x_0'b - y_0'a = 1\cdot(-1) - 1\cdot1 = -2 \neq 0\) — wait, check directly: the PDE's characteristics solve \(\dot x=1,\dot y=-1\), i.e. lines \(x+y=\)const. The curve \(\Gamma=\{x=y\}\) is not of this form (it has \(x-y=0\), not \(x+y=\)const, generically), so in fact \(\Gamma\) is *not* characteristic here and the theorem *does* apply — this is a deliberate trap: always compute \(J(s)\) explicitly rather than guessing from the shape of \(\Gamma\). Direct computation: \(x_0=s,y_0=s\), \(J(s)=x_0'b-y_0'a = 1\cdot(-1)-1\cdot1=-2\neq0\). So the theorem applies and gives a unique local solution for any \(C^1\) data \(z_0\); the "trap" is to assume any diagonal-looking curve is automatically characteristic without checking \(J\).
For \(u u_x + u_y = 0\) with \(u(x,0)=x\), find the time (value of \(y\)) at which the classical solution first breaks down.
Solution
\(a=u,b=1,c=0\). Characteristics: \(\dot u=0\Rightarrow u=s\) (constant, from data \(u(s,0)=s\)); \(\dot x = u = s \Rightarrow x=s+s y\); \(y=t\)(=itself). So \(x=s(1+y)\). Jacobian \(\partial x/\partial s = 1+y\) vanishes at \(y=-1\). For \(y\gt -1\) the inversion \(s=x/(1+y)\) is valid and \(u(x,y)=x/(1+y)\); breakdown occurs at \(y=-1\) (characteristics with \(s\gt0\) diverge, those with \(s\lt0\) converge, so it is at \(y=-1\), reached from \(y=0\) by decreasing \(y\), consistent with data \(u(x,0)=x\) being increasing, hence non-crossing for \(y\gt0\), but crossing for \(y \lt -1\)). Breakdown at \(y=-1\).
Show that if \(a,b\) are independent of \(u\) (semilinear case: \(a(x,y)u_x+b(x,y)u_y=c(x,y,u)\)), the \((x,y)\)-characteristic curves can be found by solving a single first-order ODE \(dy/dx = b(x,y)/a(x,y)\) (where \(a\neq0\)), without reference to \(u\) at all, and explain why this simplification fails in the fully quasilinear case.
Solution
Since \(\dot x=a(x,y),\dot y=b(x,y)\) do not involve \(u\), this \(2\times2\) subsystem is autonomous and can be solved on its own; eliminating the parameter \(t\) via \(dy/dx = \dot y/\dot x = b(x,y)/a(x,y)\) (valid where \(a\neq0\), by the chain rule for parametrised curves) gives the characteristic curves directly as level sets of some \(\phi(x,y)\), and only afterwards does one integrate \(\dot u=c(x,y,u)\) along each found curve. In the quasilinear case \(a=a(x,y,u),b=b(x,y,u)\), the characteristic curves themselves depend on the unknown \(u\), so \(x,y,u\) form a genuinely coupled \(3\times3\) system (as in Step 1 of the proof) and cannot be separated into "find the curve" then "integrate along it" in general.
(Non-uniqueness from a characteristic initial curve.) For \(u_x=0\) prescribe \(u(x,0)=5\) for all \(x\) (constant data on the characteristic line \(y=0\)). Show the IVP has infinitely many \(C^1\) solutions, and explain how this is consistent with the theorem rather than a counterexample to it.
Solution
General solution of \(u_x=0\) is \(u(x,y)=f(y)\) for any \(C^1\) function \(f\). The data \(u(x,0)=5\) forces only \(f(0)=5\); any \(C^1\) \(f\) with \(f(0)=5\) (e.g. \(f(y)=5+y\), \(f(y)=5+y^3\), \(f(y)=5\cos y\), etc.) gives a solution — infinitely many. This is consistent with, not a counterexample to, the theorem: \(\Gamma=\{y=0\}\) is characteristic for \(u_x=0\) (since \(a=1,b=0\) and \(y_0'\equiv0\) gives \(J\equiv x_0'\cdot0-0\cdot1=0\)), so the hypotheses of the theorem are not met, and it simply makes no assertion — as illustrated already in "Fails without" above, this is precisely the case the non-characteristic hypothesis is designed to exclude.