I'm trying to understand how to create form-invariant solutions to PDEs.
Let $u(x,t): \mathbb{R}^2 \to \mathbb{C}$, which solves the differential equation $\hat{L}u(x,t)=0$.
$u(x,t)$ is form-invariant, iff $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$.
During evolution, the abolute-square of $u$ does not change, except of dilatation $f(t)$ and translation $g(t)$.
Example: An example of such solutions are (higher-order) gaussian solutions for the paraxial wave equation.
I tried
- $u(x,t)=A(x,t)\cdot \exp(i\cdot h(x,t))$, with $A,h: (x,t) \to \mathbb{R}$ with the simple 1-dimensional cases of the paraxial wave equation
- $u(x,t)=u(f(t)\cdot x,0)$, $u: (x,t) \to \mathbb{R}$ with the 1-dimensional heat-equation
but arrived at PDEs which I was unable to solve.
Questions
- Is there a general method/idea for creating form-invariant solutions?
- Is it known whether form-invariant solutions exist for many PDEs?
- Can you provide literature to this topic (google is not helping this time)?
- Can you make suggestions such that my question is better understandable and receive more attention?