Numerically solving NLS equation

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In summary, the speaker is using the Split Step Fourier Method to study the interaction of two solitons in the context of NLS in water waves. They have successfully created an animation of the collision of two solitons for the exact solution, but when they numerically solve it, the profiles during interaction are not the same. They have tried using a finer step size but it did not help. The speaker is wondering if the numerical solution will accurately reflect the exact solution during interaction and is seeking references for animations of NLS soliton collisions using numerical solutions.
  • #1
hanson
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Hi all.
I am using the Split Step Fourier Method to solve NLS to study the interaction of two solitons.
I have done the animation of the collision of two solitons for exact solution.

But when I numerically solve it and watch the animation, the profiles during interaction is not quite the same. I don't really know what's wrong. I have used a finer step size to try again. But it doesn't help.

Will the numerical solution resemble the exact solution very well during interaction?

Can you kindly refer me to some animations of NLS soliton collision using numerical solutions? I am using the NLS in water wave context in the most simple form, namely iut+uxx+u^2u=0.
 
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  • #2
hanson said:
iut+uxx+u^2u=0.
It going to be
iut + uxx + |u|^2u = 0
 
  • #3


Hi there,

Thank you for sharing your experience with numerically solving the NLS equation. It is a complex equation and it is not uncommon to encounter discrepancies between the exact solution and the numerical solution.

There are a few factors that could contribute to the differences in the profiles during interaction. One possible explanation could be the choice of time step and spatial resolution. If the step size is too large, it can lead to errors in the numerical solution. It is important to carefully choose the step size to ensure accuracy.

Another factor could be the boundary conditions used in the numerical solution. The NLS equation is a nonlinear partial differential equation, and the choice of boundary conditions can significantly affect the accuracy of the solution.

It is also worth considering the method used for numerical solution. The Split Step Fourier Method is a popular choice, but there are other methods that could potentially provide better accuracy for certain problems.

To answer your question, the numerical solution may not always resemble the exact solution perfectly during interaction, but with careful consideration of the factors mentioned above, it can get very close.

As for animations of NLS soliton collision using numerical solutions, there are many available online. I would suggest searching for "NLS soliton collision simulation" or "numerical solution of NLS equation animation" to find some examples.

I hope this helps and best of luck with your research.
 

1. What is the NLS equation and why is it important?

The NLS (Nonlinear Schrödinger) equation is a mathematical model used to describe the behavior of waves in a variety of physical systems, such as optics and plasma physics. It is important because it allows us to understand and predict the behavior of these systems, and has many practical applications in fields such as fiber optics and quantum mechanics.

2. What is numerical solving and how is it used for the NLS equation?

Numerical solving is a method of solving mathematical equations using numerical approximation techniques, rather than analytical methods. For the NLS equation, this involves using computer algorithms to approximate the solution to the equation, which can be difficult to solve analytically. This allows us to simulate and study the behavior of the system in a more efficient and accurate manner.

3. How is the NLS equation solved numerically?

There are several numerical methods that can be used to solve the NLS equation, such as the split-step Fourier method and the Runge-Kutta method. These methods involve breaking down the equation into smaller, more manageable parts and using iterative calculations to approximate the solution. The choice of method depends on the specific problem being solved and the desired level of accuracy.

4. What challenges are associated with numerically solving the NLS equation?

One of the main challenges is ensuring that the numerical approximation is accurate and stable. This can be affected by factors such as the initial conditions, the choice of numerical method, and the time and space resolutions used. Another challenge is dealing with the nonlinearity of the equation, which can lead to numerical instabilities and difficulties in finding a solution.

5. What are the applications of numerically solving the NLS equation?

Numerical solutions of the NLS equation have a wide range of applications, including in optics, plasma physics, and quantum mechanics. They can be used to study the behavior of light in optical fibers, the formation of rogue waves in the ocean, and the dynamics of Bose-Einstein condensates. Additionally, numerical solutions can also aid in the design and optimization of systems and devices in these fields.

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