Some Properties of the Nonlinear Phase Diffusion Equation

2 Some Properties of the Nonlinear Phase Diffusion Equation 

As a preliminary to investigating two-dimensional wave patterns, we first make a brief inspection of some particular solutions of the nonlinear phase diffusion equation in one dimension. 

There exists a slightly more general class of solutions of (6.2.1) than (6.2.2), and they are obtained by smoothly joining a pair of periodic waves of different q. To find such solutions, we first note that (6.2.1) is equivalent to the Burgers equation (Burgers, 1974) 

It is also a well-known fact that the Burgers equation (6.2.3) has a family of shock solutions 

To see some physical implications of (6.2.6), it would be better to work with the original phase variable (j) = t+y/) rather than y/. Then (6.2.6) says that p behaves like 

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