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 .
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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.

  • Every member of the two-parameter family gives a particular solution to the PDE.
  • The envelope of any one-parameter family is a solution called a general integral of the PDE.
  • The envelope of the entire two-parameter family is a solution called the singular integral of the PDE.
    • The complete integral is not unique, but any other complete integral for the PDE can be obtained from it by the process of envelope formation.

    These remarkable properties account for the usefulness of the complete integral in geometrical optics, dynamics, and other areas of application. Following are various examples of nonlinear PDEs that show different kinds of complete integrals.

    Here is the complete integral for the eikonal equation.
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    This complete integral is a two-parameter family of planes. This type of solution arises whenever the PDE depends explicitly only on and , but not on , , or . For a fixed value of , it is a line in the plane at a distance of C[1] units from the origin that makes an angle of ArcCos[C[2]] with the axis. This is the familiar picture of wave-front propagation from geometrical optics.

    This verifies the solution for the eikonal equation.
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    This is an example of a Clairaut equation ().
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    Once again, the complete integral is a family of planes.
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    This verifies the solution.
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    In the following equation, the variables can be separated; that is, the PDE can be written in the form . Hence, the equation can be integrated easily.
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    This verifies the solution.
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    In this example (equation 6.49, page 202 of [K74]), the independent variables and are not explicitly present.
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    This verifies the solution.
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    Often a coordinate transformation can be used to cast a given PDE into one of the previous types. The expression for the complete integral will then have the same form as for the standard types. Here are some examples of nonlinear PDEs for which DSolve applies a coordinate transformation to find the complete integral.

    This PDE (equation 6.47, page 201 of [K74]) can be reduced to the form using the transformation and .
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    This PDE (equation 6.93, page 213 of [K74]) can be solved easily in a polar coordinate system, in which the variables are separable.
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    This equation (equation 6.36, page 196 of [K74]) can be transformed into a linear PDE using a Legendre transformation.
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    This verifies the solution.
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    It should be noted that there is no general practical algorithm for finding complete integrals, and that the answers are often available only in implicit form.

    The solution to this example (problem 2, page 66 of [S57]) is in implicit form.
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    The solution can be verified as follows.
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