Soliton Solutions of a Variation of the Nonlinear Schrödinger Equation

Abstract

The nonlinear Schrödinger (NLS) equation is a classical field equation that describes weakly nonlinear wave-packets in one-dimensional physical systems. It is in a class of nonlinear partial differential equations (PDEs) that pertain to several physical and biological systems. In this project we apply a pseudo-spectral solution-estimation method to a modified version of the NLS equation as a means of searching for solutions that are solitons, where a soliton is a self-reinforcing solitary wave that maintains its shape over time. We use the pseudo-spectral method to determine whether cardiac action potential states, which are perturbed solutions to the Fitzhugh–Nagumo nonlinear PDE, create soliton-like solutions. We then use symmetry group properties of the NLS equation to explore these solutions and find new ones.

6.1 Appendix

The following is our code utilizing the pseudo-spectral method in order to solve our perturbed NLS equation. The code was adapted from a Scipy Cookbook KdV example [13].

 import numpy as np

 from scipy.integrate import odeint

 from scipy.fftpack import diff as psdiff

 # from mpl_toolkits. mplot3d import Axes3D

 from matplotlib.collections import PolyCollection

 from matplotlib.colors import colorConverter

 # from mpl_toolkits. mplot3d import axes3d

import matplotlib.pyplot as plt

def shr_exact(x, c):

   ””” Profile of the exact solution to the KdV for a

   single soliton on the real line. ”””

   # u =1.2*1/( np. cosh (1.2*( x +20)))

      +np.exp(8j*(x))*0.8*1/(np.cosh(.8*x))

   eps = 2.0

   delta = 0.8

   beta = eps

   gamma = 1/eps

   u =(np.exp(−delta*x))/(1+np.exp(−x))

      +(np.exp(−delta*(x+20)))/(1+np.exp(−(x+20)))

      *np.exp(3j*(x))

  # u = np. exp (− delta *( beta *( x )))* np. exp (0 j *( x ))/

      (1+np.exp(−(beta*(x))))

  # u = np. exp (− delta *( beta * x ))/(1+ np. exp (−( beta * x )))

   u = gamma*u

   u = np.array(u, dtype=np.complex64)

   u = np.array([u.real,u.imag])

   u = u.flatten()

   return u

def shr(u, t, L):

   ””” Differential equations for the KdV equation,

   discretized in x. ”””

   # Compute the x derivatives using the

   pseudo−spectral method.

  # ux = psdiff ( u, period = L )

   eps = 2.0

   gamma = 1/eps

   n = len(u)

   uxxRe = psdiff(u[0:(n/2)], period=L, order=2)

   uxxIm = psdiff(u[(n/2):n], period=L, order=2)

   uxx = np.array([uxxRe,uxxIm])

   uxx = uxx.flatten()

   absu =np.sqrt(u[0:n/2]**2+u[n/2:n]**2)

   absu = np.array([absu,absu])

   absu = absu.flatten()

   absu2 = u[0:n/2]**2+u[n/2:n]**2

   absu2 = np.array([absu2,absu2])

   absu2 = absu2.flatten()

   # Compute du / dt = − i *( − uxx − 2 abs ( u ) u )

   = i * (uxx + 2abs(u)u)

   dudt = (−1*2*absu2)*u + uxx + eps*(3*absu)*u

   idudt= np.array([−1*dudt[(n/2):n], dudt[0:(n/2)]])

   return idudt.flatten()

   # return ( idudt. real, idudt. imag )

# Set the size of the domain, and create the

discretized grid.

eps = 2.0

beta = eps

L =160.0/beta

N = 256

dx = L/N

x = np.linspace(−L/2, L/2, N)

x1 = np.linspace(−L/beta, L/beta, N)

# Set the initial conditions.

# Not exact for two solitons on a periodic domain, but

close enough...

u0 = shr_exact(x, 0.75) # + kdv_exact ( x −0.65* L, 0.4)

# Set the time sample grid.

# ps = .01

# alpha = eps **2

Tm = 7

t = np.linspace(0, Tm, 1000)

# t = alpha * t

print ”Computing_the_solution.”

from mpl_toolkits.mplot3d import Axes3D

from matplotlib.collections import PolyCollection

from matplotlib.colors import colorConverter

sol = odeint(shr, u0, t, args=(L,), mxstep=500)

sol = sol[:, 0:N] + 1j *sol[:,N:(2*N) ]

print ”IMshow.”

plt.figure(figsize=(6,5))

plt.imshow(np.abs(sol[::−1, :]), extent=[−L/2,L/2,0,Tm])

plt.colorbar()

plt.xlabel(’x’)

plt.ylabel(’t’)

plt.axis(’normal’)

plt.title(’The_Nonlinear_Schrodinger_on_a_Periodic

Domain’)

# plt. show ()

# print ” Wireframe.”

# fig = plt. figure ()

# ax = fig. add_subplot (111, projection =’3 d ’)

# tind = range (0, len ( t ),10)

# xind = range (0, len ( x ),5)

# tt = t [ tind ]

# xx = x [ xind ]

# ux = abs ( sol )[:, xind ]

# uu = ux [ tind,:]

# X, T = np. meshgrid ( xx, tt )

# ax. plot_wireframe ( X, T, uu )

# plt. show ()

print(”WaterFall.”)

# # Redo the sampling

tind = range(0,len(t),30)

xind = range(0,len(x),1)

tt = t[tind]

xx = x[xind]

ux = abs(sol)[:,xind]

# The figure

fig = plt.figure()

ax = fig.gca(projection=’3d’)

cc = lambda arg: colorConverter.to_rgba(arg, alpha=0.6)

verts = []

for i in tind:

   verts.append( zip(xx,ux[i,:]) )

poly = PolyCollection(verts, facecolors = [cc(’b’)])

poly.set_alpha(0.3)

ax.add_collection3d(poly, zs=tt, zdir=’y’)

ax.set_xlabel(’X’)

ax.set_xlim3d(−L/2,L/2)

ax.set_ylabel(’t’)

ax.set_ylim3d(0,Tm)

ax.set_zlabel(’Z’)

ax.set_zlim3d(0,1.1*np.max(abs(sol)))

plt.title(’The_Nonlinear_Schrodinger_on_a_Periodic

Domain’)

plt.show()

plt.figure()

plt.plot(xx, abs(sol[0]))

plt.xlabel(’X’)

plt.ylabel(’Z’)

plt.title(’Solution_at_T_=_0’)

plt.show()

plt.figure()

plt.plot(xx, abs(sol[999]))

plt.xlabel(’X’)

plt.ylabel(’Z’)

plt.title(’Solution_at_Max_Time’)

plt.show()

Diff = np.max(abs(sol[0]))−np.max(abs(sol[999]))

Diff = abs(Diff)

print Diff