Interact Exercise 3

Imports



In [2]:

    
%matplotlib inline
from matplotlib import pyplot as plt
import numpy as np



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from IPython.html.widgets import interact, interactive, fixed
from IPython.display import display









    



:0: FutureWarning: IPython widgets are experimental and may change in the future.

Using interact for animation with data

A soliton is a constant velocity wave that maintains its shape as it propagates. They arise from non-linear wave equations, such has the Korteweg–de Vries equation, which has the following analytical solution:

$$ \phi(x,t) = \frac{1}{2} c \mathrm{sech}^2 \left[ \frac{\sqrt{c}}{2} \left(x - ct - a \right) \right] $$

The constant c is the velocity and the constant a is the initial location of the soliton.

Define soliton(x, t, c, a) function that computes the value of the soliton wave for the given arguments. Your function should work when the postion x or t are NumPy arrays, in which case it should return a NumPy array itself.



In [4]:

    
def soliton(x, t, c, a):
    inside = ((np.sqrt(c)/2)*(x-c*t-a))
    if type(x) or type(y) == np.array:
        answer = np.array((0.5)*c*(1/(np.cosh(inside))))
    else:
        answer = (0.5)*c*(1/(np.cosh(inside)))
    return answer



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assert np.allclose(soliton(np.array([0]),0.0,1.0,0.0), np.array([0.5]))

To create an animation of a soliton propagating in time, we are going to precompute the soliton data and store it in a 2d array. To set this up, we create the following variables and arrays:



In [6]:

    
tmin = 0.0
tmax = 10.0
tpoints = 100
t = np.linspace(tmin, tmax, tpoints)

xmin = 0.0
xmax = 10.0
xpoints = 200
x = np.linspace(xmin, xmax, xpoints)

c = 1.0
a = 0.0

Compute a 2d NumPy array called phi:

It should have a dtype of float.
It should have a shape of (xpoints, tpoints).
phi[i,j] should contain the value $\phi(x[i],t[j])$.



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phi = np.ndarray(shape=(xpoints,tpoints),dtype=float)
for i in x:
    for j in t:
        phi[i,j] = soliton(x[i],t[j],c,a)



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assert phi.shape==(xpoints, tpoints)
assert phi.ndim==2
assert phi.dtype==np.dtype(float)
assert phi[0,0]==soliton(x[0],t[0],c,a)

Write a plot_soliton_data(i) function that plots the soliton wave $\phi(x, t[i])$. Customize your plot to make it effective and beautiful.



In [9]:

    
def plot_soliton_data(i=0):
    """Plot the soliton data at t[i] versus x."""
    plt.figure(figsize=(15,5))
    plt.plot(x,phi[:,i])
    plt.ylabel('$\phi$')
    plt.xlabel('x')
    plt.title('Soliton Wave Plot')
    plt.grid(True)
    plt.box(True)
    plt.xlim(0,1);
    plt.ylim(0,0.7)
    axis = plt.gca()
    axis.spines['top'].set_visible(False)
    axis.spines['right'].set_visible(False)
    axis.get_xaxis().tick_bottom()
    axis.get_yaxis().tick_left()



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plot_soliton_data(0)



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assert True # leave this for grading the plot_soliton_data function

Use interact to animate the plot_soliton_data function versus time.



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interact(plot_soliton_data , i = (0.0,8.0,0.1));



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assert True # leave this for grading the interact with plot_soliton_data cell