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Learning about Differential Equations from Their Symmetries

Application of MathSym to Analyzing a Partial Differential Equation

As a final example, I will discuss the computation of the symmetries of the cubic nonlinear Schrödinger equation, a complex valued partial differential equation. Using the symmetries of the equation it is possible to generate exact solutions by a couple of different methods. First since symmetries of an equation transform its solutions to other solutions, I will demonstrate a family of solutions that are the transforms of a spatially invariant solution. Also, I will show that the symmetries can be used to reduce the partial differential equation to an ordinary differential equation in a couple of different ways.

For ordinary differential equations, it is theoretically possible to reduce the order of the equation by one for every symmetry of the equation. So, a second-order equation can be reduced to two integrals if two symmetries are known. Of course, as we saw above, in practice it can be difficult to perform the necessary computations. For a partial differential equation, it is generally not possible to get the full solution set from just the knowledge of several symmetries. However, by looking for fixed points of a given symmetry, we can find reductions of the equation and often special solutions.

The cubic nonlinear Schrödinger equation is derived in descriptions of nonlinear optics, water waves, and plasma physics [11]. Usually, the equation is written in terms of a complex valued function . The equation is

When computing the symmetries, we set and use the system of equations

which we denote as NLS.

Again, MathSym returns a system of linear coupled partial differential equations for the generators of the symmetries. MathSym uses the convention that the independent variables and are stored and printed as and . The dependent variables and are represented by and . The generators of the symmetries are , , , and .

MathSym has been successful in generating and reducing these equations. The solution of the determining equations is

where the are arbitrary constants. For clarity we have returned to the original variables , , , and . From these generators we can compute the symmetries for the nonlinear Schrödinger equation and with the symmetries we can derive solutions to NLS using at least two different strategies.

Recall that symmetries map solutions to solutions so knowledge of a solution and a symmetry allows us to generate a family of new solutions. As an example, let us use the Galilean boost, which is represented above by the constant . We can compute a transformation by setting and all of the other constants in the list of generators for the symmetries to zero. The transformation corresponding to this choice of constants is given by the change to the new variables , , , and where

The point in this exercise was to generate transformations of the variables appearing in the nonlinear Schrödinger equation that sent solutions of the equation to other solutions. This means that any solution to NLS can be used to generate a family of solutions.

It is straightforward to check that a solution of NLS is

where is an arbitrary constant. This solution is a prototypical nonlinear oscillator where the frequency is a function of the amplitude.

The transformation in equations (13) to (16) gives new solutions to NLS when applied to equations (17) and (18). If we carry out these substitutions, we find

which can be shown to satisfy NLS for any choices of and . Images of the real parts for these two solutions are generated in the following cells.

Figure 2. The real part of a spatially invariant solution of the nonlinear Schrödinger equation.

Figure 3. The real part of a transformed solution of the nonlinear Schrödinger equation.

Finally, we will use the symmetries to derive ordinary differential equations. In order to compute the reductions, we pose the question: "What are the solutions of NLS that are invariant under a given symmetry?" Notice that this question is similar to the one that we asked to define the symmetries. A symmetry is a mapping of the solution set of an equation to itself. Reductions arise from looking at the invariant subset of that transformation.

To find these invariant solutions we look for the intersection of the solution sets of the original equation, here the NLS equation, and a pair of first-order, quasi-linear partial differential equations, which are

These equations are known as the invariant surface condition equations. We will solve this new pair of equations using the method of characteristics and then substitute the solutions into the NLS equation.

We start with , , and . When we solve the invariant surface equations, we find that along characteristic curves, , and must be constant. Thus, and for arbitrary functions and . Here . The functions and must be determined by the NLS equation. You perhaps recognize and as being the functions that one must find in order to compute the traveling wave solutions of the NLS equation.

Substituting and into the NLS equation gives us two equations that we must solve in order to find and . These two new equations are now ordinary differential equations rather than the original partial differential equations. They are

We can approximate solutions to this pair of ordinary differential equations by changing to polar coordinates and and integrating twice. The new equations are

with and constants of the integration. The form of the equation for was chosen on purpose to remind us of the equation from classical mechanics,

If we let , , , and , we find that, along characteristic curves , invariant solutions of the NLS equation multiplied by must be constant. That is, and . Note that these are different , , and than those that we just used. Substituting into the NLS equation gives a pair of coupled ordinary differential equations that and must solve. This time these equations are

Our last reduction will come from letting , , , and . We will switch to a polar representation immediately here. The resulting pair of ordinary differential equations that we derive are

We can solve these two equations to get

In terms of the original variables this is

Here it is nice to have Mathematica check our computations. First we will derive the equations. Notice that we are using a little instead of the capital that Mathematica usually uses. This is done so that we can separate using a CoefficientList.

Before we substitute for and , we need to "undo" the differentiation that occurred. It would also be nice to strip the arguments and from and on the cubic term. We will do this using two functions UnDt and UnArg. Here is their syntax.

We apply these two functions.

We can now substitute our solution in and run the result through Simplify.

So it works. Let us see what some pictures look like. The constant is only a phase shift so we will set it to zero. affects the amplitude and frequency and we will let it be one. Here is the command to generate a density plot of the real part of the solution that is given by .

Figure 4. The real part of a boost invariant solution of the nonlinear Schrödinger equation.