ODE & PDE Dynamics Simulation

ODE: Simulates a particle of mass \( m \) in the effective potential \( V(x) = \frac{1}{2} \Omega^2 x^2 + A \, \text{sech}^2(bx) \).

PDE: Evolves the focusing 1D nonlinear Schrödinger equation (NLS) \[ i u_t = -\frac{1}{2} u_{xx} + V(x) u - |u|^2 u, \] with periodic boundary conditions, using a finite-difference RK4 scheme.

Initial Condition PDE: \( u(x, 0) = A_{\text{sol}} \, \text{sech}(A_{\text{sol}} (x - x_0)) \cdot e^{i v_0 x} \), optionally refined via Newton's method or Levenberg–Marquardt.

Shared Parameters

Initial Position \( x_0 \)

Initial Velocity \( v_0 \)

Total Time \( T \)

Time Step \( \Delta_t \)

How often to update the plot (milli-seconds)

Spatial Step \( \Delta_x \)

\( x_{\text{min}} \)

\( x_{\text{max}} \)

ODE Parameters

Mass \( m \)

\( \Omega \)

Amplitude \( A \)

Width Factor \( b \)

PDE Parameters

Soliton Amplitude \( A_{\text{sol}} \)

Chemical Potential \( \omega \)

\( \Omega_{\text{pde}} \)

\( A_{\text{pde}} \)

\( b_{\text{pde}} \)

Steady-State Solver

Auto-loop

Apply steady-state solver to minimize residual \( \mathcal{F} \) for initial soliton \( u(x,0) = A_{\text{sol}} \, \text{sech}(A_{\text{sol}}(x - x_0)) \)