# Solving PDE involving Gaussian fields.

• vivek.iitd
In summary, the conversation discusses the solving of three coupled differential equations representing field amplitudes in the case of plane waves and Gaussian distributions. The equations are solved numerically, with the possibility of incorporating diffraction and other effects using the split-step method. The suggestion is made to assume a constant q parameter for the Gaussian beams, simplifying the problem, and to discretize along the radial direction to apply the equations step by step. This approach may not be physically perfect, but it is considered a practical solution.

#### vivek.iitd

I need to solve three coupled differential equations. The equations are as follows:

dE1/dz = f(E2,E3)
dE2/dz = f(E1,E3)
dE3/dz = f(E1,E2)

Where E1,E2,E3 represents field amplitudes. In case of plane waves these amplitudes will be constant in the transverse direction therefore i can directly solve these equation given their initial values. Although, i would like to, how should i solve these equations if i have a Gaussian distribution in the transverse direction? Do i need to solve it by creating a mesh and solving these equations for each pairs of points?

I thank you for your time.

sorry, equations are as follows

dE1/dz = f(E2,E3)
dE2/dz = g(E1,E3)
dE3/dz = h(E1,E2).

I am solving it numerically.

Do they represent laser beams?

yes they represent laser beams.

Actually it is a nonlinear phenomenon (sum frequency generation). To give a physical aspect to this problem it is like this, two beams (with amplitudes E1 and E2) are incident on a nonlinear crystal and they mix up with each other to give third beam (E3) this whole phenomenon is represented by these coupled equations. The direction of propagation is 'z'. In case of plane waves the amplitude will be constant in the transverse direction (plane perpendicular to 'z'), therefore i can directly solve these equations using numerical method. Although, I want to get more realistic, so i am taking Gaussian distribution. Therefore in the transverse direction, amplitude distribution is Gaussian. I am confused as how to solve these equations in that case.

Many times, the sum-difference problem is analyzed assuming plane waves.
If you want to consider Gaussian beams you must propagate them but the algorithm is more complex.
As an intermediate approach, I'd suggest assuming the gaussian beams have constant q paremeters. This isn't strictly true but simplifies the job. This way, at every radial coordinate r you have a gaussian amplitude and you apply the original differential equations at a given radial coordinate.

Thank you for your reply. You are right that usually these problems are solved assuming plane waves because they are simplest to solve.
As you suggested I will assume q parameter to be constant as of now i.e. neglecting diffraction and other effects. After that i may incorporate that also using split-step method.
Also to solve for Gaussian profile, i guess i will have to discretize along the radial direction. And after that i can apply these ODEs by taking one step at a time along radial direction.

I think this is the correct way to solve these problem for Gaussian profile right?

Even though it isn´t a "physically blameless" approach, I'd say it's a fair solution

## 1. What is a PDE involving Gaussian fields?

A PDE (partial differential equation) involving Gaussian fields is a type of mathematical equation that describes the behavior of a random field that follows a Gaussian distribution. This type of PDE is commonly used in fields such as physics, engineering, and finance to model and solve complex systems.

## 2. How do you solve a PDE involving Gaussian fields?

The solution to a PDE involving Gaussian fields typically involves using analytical or numerical methods. Analytical methods involve finding an exact solution using mathematical techniques, while numerical methods involve approximating the solution using computer algorithms.

## 3. What is the importance of solving PDEs involving Gaussian fields?

Solving PDEs involving Gaussian fields allows us to understand and predict the behavior of complex systems, such as the diffusion of particles in a fluid or the spread of heat in a material. It also has practical applications in fields such as image processing, signal processing, and data analysis.

## 4. What are some common techniques for solving PDEs involving Gaussian fields?

Some common techniques for solving PDEs involving Gaussian fields include separation of variables, Fourier transforms, and finite difference or finite element methods. These techniques can be used to find both analytical and numerical solutions to the PDE.

## 5. Are there any challenges in solving PDEs involving Gaussian fields?

Yes, there are several challenges in solving PDEs involving Gaussian fields. These include the complexity of the equations, the need for advanced mathematical and computational skills, and the difficulty in obtaining accurate initial and boundary conditions for the system being modeled. Additionally, some PDEs may have no closed-form solutions and require the use of numerical methods to approximate the solution.