Simulation lab

Move the sliders to see how the fractional order and the noise intensity reshape a soliton. Everything is computed live in your browser from the solution forms used in my papers.

View

Drag the surface to rotate it.

Soliton type

Fractional derivative

Solution u(x, t): 3D surface

Height is |u|. Drag to rotate.

2D profile at time t

Dashed: noise-free profiles for three fractional orders. Gold: your settings, with noise.

Noise in time (2D)

One Brownian path against the average over all paths. Stronger noise drains the expected amplitude faster.

Fractional order

1 is the classical derivative. Lower values add stronger memory.

Noise intensity

0 removes the randomness completely.

Time

Sets the instant shown in the 2D profile.

Run

Mean factor ; this path .

The model

The papers study equations with a multiplicative noise term driven by Brownian motion \(B(t)\), in the Stratonovich sense. Their solutions take the form

\[ u(x,t) = \phi(\xi)\,\exp\!\big(i\theta(x,t) + \sigma B(t) - \sigma^{2} t\big), \]

where \(\phi\) is the soliton profile and \(\sigma\) the noise intensity. The fractional order enters through the wave variable. For the conformable derivative

\[ \xi = \frac{k}{\alpha}\,x^{\alpha} - \frac{c}{\alpha}\,t^{\alpha}, \qquad 0 < \alpha \le 1, \]

and for the β-derivative

\[ \xi = \frac{k}{\beta}\Big(x + \tfrac{1}{\Gamma(\beta)}\Big)^{\beta} - \frac{c}{\beta}\Big(t + \tfrac{1}{\Gamma(\beta)}\Big)^{\beta}. \]

Two effects follow. Lowering the order bends the coordinates, so the pulse narrows or widens and travels non-uniformly. Since \(\mathbb{E}\,e^{\sigma B(t)} = e^{\sigma^{2}t/2}\), the expected amplitude factor is \(e^{-\sigma^{2}t/2}\): on average, noise damps the soliton, while any single path fluctuates around that mean.

This lab uses \(k = 1.2\), \(c = 1.5\) and the profiles \(\operatorname{sech}\), \(|\tanh|\), \(\tanh\) and \(\cos\) for illustration. The exact solution families and parameter conditions for each equation are in the papers.