F(x, y, u, ux, uy) = 0

where u is a real-valued function of the independent variables x and y. This framework is quite broad and encompasses linear, semilinear, quasilinear, and nonlinear forms, depending on how the function F incorporates each term. The study of such equations began historically with the work of Euler, Lagrange, and D’Alembert, and later evolved significantly with the contributions of Lie, who connected these equations to transformation groups and the foundation of Lie algebras.

When learning about PDEs, students are usually introduced to the notion of a classical solution—a continuously differentiable function that satisfies the equation at every point in a domain D ⊂ R². However, the practical goal is to not only find such solutions but to understand the behavior and structure of their solutions under various initial conditions. The more traditional path through integral surfaces and Monge cones is mathematically beautiful but not always necessary for students aiming to apply these concepts in physical or computational contexts.

Solving PDEs Using Characteristic Curves

The method of characteristic curves allows us to reduce a PDE to an ordinary differential equation (ODE) along specific paths in the (x,y)-plane. Along these curves, the PDE simplifies drastically, and the function u can be treated as varying in just one dimension. This transforms what seems like a two-dimensional problem into a much easier one-dimensional problem, which is far easier to solve and interpret.

To illustrate this with a basic example, consider the equation:

ux + c * uy = 0

where c is a real constant. One can introduce a new variable η = x − c * y, representing a straight line in the plane. Along these lines, it turns out that the function u is constant. This leads to the general solution:

u(x, y) = f(x − c * y)

where f is determined based on the initial condition. This elegant method of solution, using characteristic curves, is not only faster but also aligns more closely with physical intuition—imagine information being transported along these curves, unchanged as it moves.

Understanding the Cauchy Problem

For a PDE to be solved completely, we typically need initial or boundary data. This is known as the Cauchy problem. The Cauchy problem for a first order PDE involves prescribing the value of u along a certain curve in the plane, and then extending this information into a surrounding domain using the PDE. The key idea is that the initial curve should intersect the characteristic curves in a non-tangential way. If the initial data is placed along a characteristic, the solution may not be unique or might not exist at all in the classical sense.

The simplest example of a Cauchy problem occurs when we define the value of u along the x-axis, and then find the solution in the surrounding region. By identifying which characteristic curves pass through each point, and tracing them back to where they intersect the initial data, we can determine the value of u throughout the domain. This method reinforces the idea that information in the solution propagates along characteristics. The approach becomes especially useful when dealing with semilinear and linear equations, where the structure of the PDE remains manageable.

Semilinear and Quasilinear Equations

The distinction between semilinear and quasilinear equations becomes clear when we observe how the solution and its derivatives interact with the coefficients in the PDE. A semilinear PDE is one where the highest-order derivatives appear linearly, and the nonlinearity comes only through the function u. For example, the equation

a(x,y) * ux + b(x,y) * uy = c(x,y,u)

is semilinear if a, b, and c are known functions, and the derivatives ux and uy are treated linearly. This type of equation still permits the use of characteristic curves because the curves depend only on x and y, not on u. This leads to a relatively simple system of ODEs, one for each variable, which can be solved using standard techniques.

Quasilinear equations, on the other hand, include dependence on u in the coefficients of the derivatives. This increases the complexity because the characteristic curves themselves now depend on the solution u, and the entire system becomes coupled and nonlinear. A classic example is the inviscid Burgers’ equation:

u * ux = 0

which models phenomena like traffic flow and shock waves in fluids. In such cases, even with smooth initial conditions, the solution can develop singularities where different characteristics intersect. This results in a breakdown of the classical solution, requiring either a piecewise definition or the use of weak solutions.

The construction of a solution in the quasilinear case involves solving a system of ODEs for x, y, and u simultaneously, with initial values provided along a curve. The characteristic curves now change depending on the solution itself, so they must be recalculated dynamically. What emerges is a powerful tool for modeling nonlinear transport phenomena in various fields.

Nonlinear Equations and Charpit’s Method

When dealing with fully nonlinear PDEs—where the function F is nonlinear in all of its arguments including the derivatives—the standard characteristic method is no longer enough. Here, we turn to Charpit’s method, a generalized approach that constructs a system of five ODEs for the variables x, y, u, p = ux, q = uy. Solving this system allows us to construct what is known as a Monge strip, from which a solution surface can be pieced together.

Charpit’s method is more involved but necessary for handling equations like:

F(x, y, u, ux, uy) = 0

where F may represent a variety of physical constraints. The method requires initial values not only for u but also for its derivatives p and q, along a curve. These values must be consistent with the original PDE and with the condition that the solution remains smooth across the domain of interest. While technically more demanding, Charpit’s method retains the fundamental principle of using transport equations along characteristic directions, and provides a powerful means of constructing solutions.

Behavior of Solutions and Singularities

A remarkable feature of first order PDEs—especially quasilinear and nonlinear ones—is that they can develop singularities even from well-behaved initial data. This phenomenon occurs when different characteristic curves, each carrying different values of the solution, intersect. The intersection point marks a discontinuity or sharp change in the derivative of the solution, known as a shock or singularity.

For linear equations, the solution remains smooth as long as the coefficients are continuous. But in quasilinear cases, even small differences in initial values can amplify as they propagate, eventually leading to intersecting characteristics and abrupt changes in the solution. These singularities are often seen in fluid dynamics, where they model real physical effects like shock waves or breaking waves in shallow water equations.

Mathematically, the location and nature of singularities can be studied using the structure of the characteristic curves. The strength of a discontinuity can be described by a quantity ω, and the jump in a derivative such as ux is proportional to the directional derivative of a function ϕ(x,y) defining the discontinuity curve. This leads to practical tools for tracking the movement of discontinuities across the domain.

Conclusion

The study of first order partial differential equations is foundational not only to theoretical mathematics but also to countless real-world applications. By focusing on the simpler and more intuitive method of characteristic curves, students can gain deep insights into the nature of these equations without becoming overwhelmed by complex geometry or abstract formalisms. The use of characteristics transforms PDEs into ODEs along specific paths, giving us a direct and powerful way to understand how information propagates through a domain and helps students solve their math assignment more efficiently and with deeper conceptual clarity.

Semilinear and quasilinear equations demonstrate how changes in initial data affect the evolution of the solution, while nonlinear equations introduce complexities that require advanced techniques like Charpit’s method. Despite the increasing complexity, the underlying idea remains elegant: solutions travel along curves, and knowing how to describe and manipulate these curves is the key to unlocking the solution. In the end, mastering first order PDEs is not about memorizing equations or theorems—it’s about understanding how the structure of an equation influences its behavior. Once this core idea is understood, even the most complicated PDE begins to feel approachable, and students are better equipped to explore more advanced topics in mathematics and applied sciences and confidently solve their math assignment with a solid grasp of the fundamentals.