Solving Nonlinear Gross Pitaevskii Eq. with Mathematica?

In summary, the conversation is about solving the Gross-Pitaevskii equation, specifically in the context of Bose Einstein Condensates. The question is whether Mathematica can be used to solve this nonlinear equation and if so, what steps should be taken. The equation is provided and there is also a discussion about using NDSolve with a different function name and the possibility of solving multidimensional differential equations.
  • #1
chasingwind
15
0
I am thinking of solving a nonlinear equation, namely, Gross Pitaevskii equation that applied in Bose Einstein Condensates. Can I solve the equation with Mathematica? Could someone give me some clues?
 
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  • #2
i think you can get at least a numerical calculation, but here is what we can do, post the equation, ill run it and see what the software can actually do :wink:
 
  • #3
I am also interested in solving the Gross-Pitaevskii-Equation. I spend my weekend to get a solution with mathematica without success. There are a couple of Phys. Rev. A.s which treating the problem of numerical solution of the GPE or other non lineary time-(in)dependent Schrödinger-Equations, however it would be much nicer if Mathematica can do such things and I don't have to learn how to write an effective Runge-Kutta.

btw. the time-independent GPE is:

-\hbar^2/(2m) \nabla^2 \psi(x,y,z) + V + g|\Psi(x,y,z)|^2 \Psi(x,y,z) = \mu \Psi(x,y,z)

wherby V = m/2(\omega_x^2 x^2 + \omega_y^2 y^2 + \omega_z^2 z^2)

An other question comming up when I tried getting a solution is: Is it possible to use NDSolve with an other function name than y[x], for example :Psi:[x]? Is it theoretically possible to solve more dimensional (i.e. 3, f[x,y,z]) diff. eq. with NDSolve, which is needed to solve the GPE seriously?
 
  • #4
hi i f u get any way for writting this equation on mathmatica 5 please help me i will appreciate please help me i have to do this for my thesies please help me if u get any sucsses to solve thank u vase moeini from iran
booth said:
I am also interested in solving the Gross-Pitaevskii-Equation. I spend my weekend to get a solution with mathematica without success. There are a couple of Phys. Rev. A.s which treating the problem of numerical solution of the GPE or other non lineary time-(in)dependent Schrödinger-Equations, however it would be much nicer if Mathematica can do such things and I don't have to learn how to write an effective Runge-Kutta.

btw. the time-independent GPE is:

-\hbar^2/(2m) \nabla^2 \psi(x,y,z) + V + g|\Psi(x,y,z)|^2 \Psi(x,y,z) = \mu \Psi(x,y,z)

wherby V = m/2(\omega_x^2 x^2 + \omega_y^2 y^2 + \omega_z^2 z^2)

An other question comming up when I tried getting a solution is: Is it possible to use NDSolve with an other function name than y[x], for example :Psi:[x]? Is it theoretically possible to solve more dimensional (i.e. 3, f[x,y,z]) diff. eq. with NDSolve, which is needed to solve the GPE seriously?
 
  • #5


As a scientist familiar with both Mathematica and the Gross Pitaevskii equation, I can confirm that it is possible to solve the nonlinear Gross Pitaevskii equation using Mathematica. In fact, Mathematica has specific built-in functions and packages designed for solving differential equations, including nonlinear ones like the Gross Pitaevskii equation.

To solve the equation, you will need to input the relevant parameters and initial conditions into Mathematica, and then use the appropriate functions and methods to solve for the solution. Depending on the complexity of the equation and the specific parameters, you may need to use numerical methods or analytical solutions.

I would recommend consulting the Mathematica documentation and online resources for specific guidance on solving the Gross Pitaevskii equation. Additionally, there are many research papers and articles available that discuss the use of Mathematica in solving this equation, which could also provide helpful insights and clues for your specific application.

In summary, yes, it is possible to solve the nonlinear Gross Pitaevskii equation with Mathematica. With the right parameters, initial conditions, and knowledge of Mathematica's functions and methods, you should be able to obtain a solution for your specific problem. Good luck with your research!
 

1. What is the Gross Pitaevskii equation?

The Gross Pitaevskii equation is a nonlinear partial differential equation that describes the behavior of a quantum-mechanical system of bosons at low temperatures. It is commonly used to model the dynamics of a Bose-Einstein condensate, a state of matter where a large number of bosons occupy the same quantum state.

2. Why is it important to solve the Gross Pitaevskii equation?

The Gross Pitaevskii equation allows us to understand the behavior of a Bose-Einstein condensate and make predictions about its properties and dynamics. It has applications in fields such as atomic physics, condensed matter physics, and quantum information processing.

3. What is the role of Mathematica in solving the Gross Pitaevskii equation?

Mathematica is a powerful software tool that is commonly used by scientists to solve complex mathematical problems, including the Gross Pitaevskii equation. It provides a user-friendly interface and powerful numerical methods that make it easier to solve the equation and obtain accurate results.

4. What are the challenges in solving the Gross Pitaevskii equation with Mathematica?

One of the main challenges in solving the Gross Pitaevskii equation is dealing with the high computational cost and memory requirements. As the equation is nonlinear, it requires iterative methods that can be computationally expensive. Additionally, the solutions to the equation may exhibit numerical instabilities, which need to be carefully managed.

5. Are there any alternative methods for solving the Gross Pitaevskii equation?

Yes, there are alternative methods for solving the Gross Pitaevskii equation, such as using other software tools like MATLAB or Python. Additionally, some researchers have developed analytical approximations and simplified versions of the equation that can be solved more efficiently. However, Mathematica remains a popular choice due to its comprehensive capabilities and user-friendly interface.

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