ODE
Simulates a particle of mass \( m \) using Newton's second law
\[
m\dfrac{d^2 x}{dt^2} = F, \quad x(0) = x_0, \:\, v(0) = v_0
\]
in either an external potential
\[
V(x) = \frac{1}{2} \Omega^2 x^2 + A \, \text{sech}^2(bx)
\]
or a "patched" variational potential:
\[
U_{\text{patched}}(x;A,b,\Omega,\epsilon)=
\begin{cases}
U_{\text{eff}}(x;A,b,\Omega) & \text{if }|x|\ge\epsilon,\\[4pt]
U_{\text{Taylor}}(x;A,b,\Omega) & \text{if }|x|<\epsilon,
\end{cases}
\]
with
\[
\begin{aligned}
U_{\text{eff}} &= {}
-\frac{256A b^{2} x}{\!\bigl(e^{2bx}-1\bigr)^{5}}
-\frac{256A b(2bx-1)}{\!\bigl(e^{2bx}-1\bigr)^{3}}
-\frac{128A b(5bx-1)}{\!\bigl(e^{2bx}-1\bigr)^{4}} \\[-2pt]
&\hphantom{={}}
-\frac{32A b(12bx-13)}{3\bigl(e^{2bx}-1\bigr)^{2}}
+\frac{32A b}{3\bigl(e^{2bx}-1\bigr)}
+\frac{2\Omega^{2} b x^{2}}{3},\\[6pt]
U_{\text{Taylor}} &= {}
-\frac{64A b^{3}x^{2}}{105}
+\frac{16A b}{15}
+\frac{2\Omega^{2} b x^{2}}{3}.
\end{aligned}
\]
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.