Nonlinear PDEs

The general first-order nonlinear PDE for an unknown function is given by

Here is a function of , , and .

The term "nonlinear" refers to the fact that is a nonlinear function of and . For instance, the eikonal equation involves a quadratic expression in and .

The general solution to a first-order linear or quasi-linear PDE involves an arbitrary function. If the PDE is nonlinear, a very useful solution is given by the complete integral. This is a function of u(x,y,C[1],C[2]), where C[1] and C[2] are independent parameters and u satisfies the PDE for all values of (C[1],C[2]) in an open subset of the plane. The complete integral can be used to find a general solution for the PDE as well as to solve initial value problems for it.

Here is a simple nonlinear PDE.
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The complete integral depends on the parameters C[1] and C[2]. Since DSolve returns a general solution for linear and quasi-linear PDEs, a warning message appears before a complete integral is returned.
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This verifies the solution.
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If the values of C[1] and C[2] are fixed, the previous solution represents a plane in three dimensions. Thus, the complete integral for this PDE is a two-parameter family of planes, each of which is a solution surface for the equation.

Next, the envelope of a one-parameter family of surfaces is a surface that touches each member of the family. If the complete integral is restricted to a one-parameter family of planes, for example by setting C[2]=5C[1], the envelope of this family is also a solution to the PDE called a general integral.

This finds the envelope of the one-parameter family given by setting C[2]=5C[1] in the complete integral for the preceding PDE p*q==1.
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This verifies that the envelope surface is a solution to the PDE.
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Like nonlinear ODEs, some nonlinear PDEs also have a singular solution (or singular integral) that is obtained by constructing the envelope of the entire two-parameter family of surfaces represented by the complete integral.

Here is an example of such a construction, (equation 6.4.13, page 429 of [K00]).
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Thus, the singular integral for this PDE is a plane parallel to the - plane.

To summarize, the complete integral for a nonlinear PDE includes a rich variety of solutions.

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