How Do Jacobi Elliptic Functions Solve Nonlinear Differential Equations?

[MalachiK]

I have ##\frac{d^2x(t)}{dt^2} + B(x(t))^3 = 0## for a system where I know the initial conditions and where B is a constant that's constructed from the properties of the system. I would like to find ##x(t)##.

I've modeled the system in Python and produced some graphs. I know that ##x(t)## is some periodic function.

Could somebody name the method that I should study so that I can get a solution?

 

[MalachiK]

After some further reading I've got this...
Let ##s = x'## so ##s' = x''##

##-Bx^3 = x'' = s' \frac{ds}{dt} = \frac{ds}{dx}\frac{dx}{dt} = \frac{ds}{dx} s##

##-\int{Bx^3 dx} = \int{s ds} ##

##\frac{-Bx^4}{4} + C = \frac{s^2}{2}##

2C is just a constant that I'll rename C

##\sqrt{C-\frac{Bx^4}{2}} = s = \frac{dx}{dt}##

Since ##x(0) = x_0## and ##x'(0) = 0## we have ##C-\frac{Bx_0^4}{2} = 0## so ##C = \frac{Bx_0^4}{2}##

##\frac{dx}{dt} = \sqrt{\frac{B}{2}}\sqrt{x_0^4-x^4}##

##\sqrt{\frac{B}{2}}\int{\frac{1}{\sqrt{x_0^4-x^4}}dx} = \int{dt}##

I don't know if this is correct, but it seems plausible. Now to evaluate the integral.

 

[member 428835]

I know the initial conditions

Could somebody name the method that I should study so that I can get a solution?

Could you state the initial conditions? I ask because you may be able to perturb the equation, take a naive expansion, and approximating your solution analytically. It would not be exact but the method can be incredibly close (like a truncated Taylor series).

 

[lurflurf]

looks like

https://en.wikipedia.org/wiki/Jacobi_elliptic_functions