# Non-linear Schrodinger Equation (Differential Equation)

• syedmohsin
In summary, the person is working on a pulse propagation problem in optical fiber and needs help from someone. They have written a program to find traveling wave solutions, but they need help understanding the difference between a time-dependent and a time-independent solution. They also need help with adaptive mesh.
syedmohsin
Hello,

i am working on pulse propagation in optical fiber. i have to simulate the nonlinear Schrodinger equation using the FDTD (Finite Difference Time Domain) method. The Schrodinger equation has the form dA/dz = i/2 β2 d2A/dt2 –α/2 +iγ |A2|A

where β2 is dispersion, α is attenuation and γ is fiber non-linearity.

I need to do this in MATLAB. Please help me in simulation or send me a MATLAB code related to it.

You will need an adaptive mesh type code. The NLS equation is weakly nonlinear but the equations are somewhat tricky to solve, I have written programs which find traveling wave solutions.

hunt_mat said:
You will need an adaptive mesh type code. The NLS equation is weakly nonlinear but the equations are somewhat tricky to solve, I have written programs which find traveling wave solutions.

Sir can you email main that code? my email is syedmohsinshah@gmail.com
If you give me some time i will discuss my work with you by email.

I am looking for traveling wave solutions, and I think that is a vast difference to what you want. You want time dependent one right?

i don't understand the difference but yes i want time dependent solution

I look for a solution of the form $A(t,x)=A(x-\omega t)$

For my work i have a simple numerical solution. i need to generate a Gaussian pulse and then to propagate along the fiber length. i can use simple difference derivative or Crank Nicholson difference. The solution has the form:
Ai+1,n = Ai,n +j delta x ( ½ β2(Ai,n+1 – 2Ai,n + Ai,n-1)/delta t2 ) j delta x γ |A2|A

Where β2 is fiber dispersion and γ is fiber nonlinearity
i is space step and n is time step

I don't think that my code will help you then but adaptive mesh will be the best way forward.

Thank you so much

## 1. What is the Non-linear Schrodinger Equation?

The Non-linear Schrodinger Equation is a partial differential equation that describes the evolution of a quantum system with non-linear interactions. It is used to model a variety of physical phenomena, including superfluidity, Bose-Einstein condensates, and optical solitons.

## 2. How is the Non-linear Schrodinger Equation different from the Linear Schrodinger Equation?

The Linear Schrodinger Equation describes the behavior of a quantum system with linear interactions, while the Non-linear Schrodinger Equation takes into account non-linear interactions. This means that the Non-linear Schrodinger Equation is more complex and can exhibit more interesting and unexpected behavior.

## 3. What are some applications of the Non-linear Schrodinger Equation?

The Non-linear Schrodinger Equation has a wide range of applications, including in quantum optics, Bose-Einstein condensates, and superfluidity. It is also used in the study of non-linear dynamics and chaos theory.

## 4. What are some numerical methods for solving the Non-linear Schrodinger Equation?

Some common numerical methods for solving the Non-linear Schrodinger Equation include the split-step Fourier method, the Crank-Nicolson method, and the Runge-Kutta method. These methods involve breaking the equation into smaller, more manageable parts and using iterative techniques to find a solution.

## 5. What are the limitations of the Non-linear Schrodinger Equation?

While the Non-linear Schrodinger Equation is a powerful tool for modeling non-linear quantum systems, it does have some limitations. It assumes that the system is in a stationary state and does not take into account the effects of dissipation or external forces. Additionally, it may not accurately describe systems with strong non-linearities or interactions with other particles.

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