How does one "design" a PDE from a physical phenomenon?

[SemM]

Hi, I have read some on the PDEs for fluids, and particularly for rogue waves, where for instance the extended Dysthe equation and the NLSE look rather intimidating:

Take for instance the Non-linear Schrödinger eqn:

\begin{equation}
\frac{\partial^2 u}{dx^2}-i\frac{\partial d u}{dt}+\kappa|u|^2 u=0
\end{equation}except for that it was "designed" by Zakharov in 1968, and used in that form since then, I am wondering what makes it critical to include the second derivative of one dimension (I take its the wave-direction), then the imaginary component of the time dimension ( I take it has to do with velocity) and finally the nonlinear term.

What is the rationale behind the selection of each of the given terms?

For the KdW eqn and Hirota it can get even worse, where 4-6th order derivatives occurs.

Thanks!

 

[boneh3ad]

Generally it comes from the underlying physical laws underpinning a given phenomenon. Once you look at the most basic laws describing a situation, it often results in a PDE or system of PDEs (e.g. the Navier-Stokes equations in fluids). From there, it's an exercising in making (usually) well-supported assumptions and approximations to make the PDEs more tractable for a given phenomenon.

 

[SemM]

Generally it comes from the underlying physical laws underpinning a given phenomenon. Once you look at the most basic laws describing a situation, it often results in a PDE or system of PDEs (e.g. the Navier-Stokes equations in fluids). From there, it's an exercising in making (usually) well-supported assumptions and approximations to make the PDEs more tractable for a given phenomenon.

None of the papers describe how the various terms are related to a physical property, but I assume:

$\frac{\partial^2 u}{dx^2}$ is related to the change of the wavefunction over the distance, so it must have to do with kinetic energy or the movement of the wave.

$i \frac{\partial d u}{dt}$ must have to do with the change of the wavefunction over time, so whether it increases or decreases with time? The imaginary part is elusive, but taking into account that in quantum mechanics the imaginary number is related to the solution being square-integrable, it must have to do with some form of probability of the change of the wave function over time.

$|u^2|u$ is really $u^3$ , and is nonlinear term, which is related to some amplitude in the solution which is not "part of the wavepattern", this can be attributed to the soliton wave. This may be the rogue wave itself.

All this is of course speculation, but if anyone can correct or comment, that would be great.

 

[Andy Resnick]

Hi, I have read some on the PDEs for fluids, and particularly for rogue waves, where for instance the extended Dysthe equation and the NLSE look rather intimidating:

Take for instance the Non-linear Schrödinger eqn:

What is the rationale behind the selection of each of the given terms?

For the KdW eqn and Hirota it can get even worse, where 4-6th order derivatives occurs.

Thanks!

Using the NLSE for water waves (and also for nonlinear optical pulses) results from transforming the 'original' nonlinear PDE into coordinates that result in a PDE of the form of the NLSE.

It's not clear what your comfort level with the mathematics is, have you read through any of the derivations:

https://en.wikipedia.org/wiki/Nonlinear_Schrödinger_equation
https://en.wikipedia.org/wiki/Soliton_(optics)

I'm not sure about "the KdW eqn", but the Korteweg-deVries (KdV) equation originates from the Boussinesq approximation:

https://en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves)

AFAIK, most of these PDEs originate by writing down an expression for the energy or some other scalar potential, doing a Taylor expansion, then making approximations to truncate the series.