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Learning about Differential Equations from Their Symmetries

Application of MathSym to Analyzing an Ordinary Differential Equation

In the previous section we used a scaling symmetry to help understand the solutions of a pair of differential equations. In each case, the scaling symmetry was found by inspection. Here I present the computation of the complete set of point symmetries for two additional differential equations. Our third example is a nonlinear ordinary differential that we analyze using its two symmetries. The final example is the partial differential equation known as the cubic nonlinear Schrödinger equation [11].

Example three is the ordinary differential equation

that arises in the study of nonlinear water wave equations. I also show that we can use its two symmetries to begin to learn something about the structure of its solutions.

MathSym returns a system of equations, the determining equations, whose solutions generate the symmetries of equation (8). Internally, the MathSym package denotes all independent variables in an equation as and dependent variables as . This way it can be run on systems of equations with arbitrary numbers of independent and dependent variables without needing to know how to treat different variable names. Furthermore, constants are represented as internally and printed as . With this notation, constants are treated correctly by Mathematica's differentiation routine Dt. MathSym's output is the following list of determining equations.

With the output from MathSym we can continue our analysis of equation (8). First, we solve the determining equations:

The functions and determine two symmetries that can be used to convert equation (8) into two integrals. The reader is directed to similar computations for the Blasius boundary layer equation which appear on pages 118-120 of [4].

We begin by considering the symmetry that occurs because of the term. Setting and produces a transformation and . We next look for two quantities that do not change under this transformation. Obvious choices are and . If we assume that is a function of and write the differential equation for that arises by insisting that satisfy equation (8), we find

This is a standard reduction of order for autonomous equations that may be found in a sophomore differential equations text such as [12].

This equation in and has a symmetry that is generated by the constant appearing in equations (9) and (10). From this symmetry we can derive new variables and and consider as a function of . In terms of and , equation (11) becomes

We have now converted the problem of solving the original equation into two integrations. First we find as a function of giving us a solution of equation (12) and hence of equation (11). Then we return to the original variables and have implicitly as a function of . Integrating again gives a relationship between and .

We can make Mathematica carry out some of these computations. First we will ask that it determine a solution to equation (12) by integrating both sides of the equation.

In the equation for we can return to the original variables and .

What results is an implicit relationship between and and while MathSym has been successful in generating the symmetries of equation (8), it still is a challenge to solve this equation.