simulations/01_nonlinear_schrodinger_1d_stationary.ipynb

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   "source": [
    "# 1D nonlinear stationary Schrodinger solitons"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "This is probably the simplest soliton you can possibly have.\n",
    "Consider a Hamiltonian of the form\n",
    "$$\n",
    "\\hat{H} = -\\frac{1}{2} \\frac{d^2}{dx^2} + V.\n",
    "$$\n",
    "The potential $V$ is a combination of a low-density attractive term, and a high-density repulsive term\n",
    "$$ V(\\psi) = \\alpha (|\\psi|^2 - \\bar{\\psi}^2)^2 = -\\alpha \\bar{\\psi}^2 |\\psi|^2 + \\alpha |\\psi|^4 $$ \n",
    "Naively, this should result in a vacuum expectation value of $\\langle \\psi \\rangle = \\bar{\\psi}$. However, if the Hamiltonian is invariant under global phase rotations $\\psi \\rightarrow e^{i\\theta} \\psi$ (as this one is), then particle number is conserved, and the vacuum cannot reach the minimum of the potential except in localized areas (aka solitons).\n",
    "The time-independent equation of motion is\n",
    "$$\n",
    "-\\frac{1}{2m} \\frac{d^2\\psi}{dx^2} + \\alpha(|\\psi|^2 - \\bar{\\psi}^2)^2\\psi = E \\psi\n",
    "$$\n",
    "which we can solve in order to find the form of these solitons.\n",
    "Unfortunately this equation is too complicated to solve for analytically, so we will have to resort to approximate solutions.\n",
    "I will show an approximate analytical solution using the \"variational method\", and also a numerical algorithm that falls into a class of \"relaxation methods\"."
   ],
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   "source": [
    "## Approximate variational solution"
   ],
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   "source": [
    "Before jumping straight to a numerical solution, I will introduce the variational method, a simple way to get approximate analytic solutions for a wide array of complex solitonic configurations. The basic recipe is:\n",
    "* pick a simple \"trial\" wave function $\\psi$, parametrized by some variables $\\theta_i$\n",
    "* compute the expectation value of the energy due to this trial wave function via $E = \\langle \\psi | \\hat{H} | \\psi \\rangle / \\langle \\psi | \\psi \\rangle$\n",
    "* minimize the energy by solving $\\partial E/\\partial \\theta_i = 0$\n",
    "\n",
    "Because the energy spectrum is typically bounded from below (unless you choose a non-physical potential), the minimization will typically give you a trial wave function that approximates the ground state.\n",
    "You can also derive excited states by chosing a trial wavefunction that matches the symmetry or number of nodes of the excited states\n",
    "* e.g. a typical trial wave function for the ground state is a simple Gaussian $\\psi(x) \\sim \\exp(-(x/R)^2)$\n",
    "* an excited state might have an antisymmetric wave function like $\\psi(x) \\sim x\\exp(-(x/R)^2)$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "To get an approximate ground state for our soliton, we will also assume a Gaussian ansatz."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
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   "cell_type": "code",
   "execution_count": 3,
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     "data": {
      "text/plain": "Eq(\\psi(x), \\psi_0*exp(-x**2/R**2))",
      "text/latex": "$\\displaystyle \\psi{\\left(x \\right)} = \\psi_0 e^{- \\frac{x^{2}}{R^{2}}}$"
     },
     "metadata": {},
     "execution_count": 3
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x = sp.symbols(\"x\", real=True)\n",
    "R, ps0 = sp.symbols(r\"R \\psi_0\", positive=True)\n",
    "psi = sp.Function(r\"\\psi\", real=True)\n",
    "\n",
    "# this is the gaussian wave function profile\n",
    "def psi_ansatz(x):\n",
    "    return ps0 * sp.exp(-(x/R)**2)\n",
    "sp.Eq(psi(x), psi_ansatz(x))"
   ]
  },
  {
   "source": [
    "However, because of the nonlinearities in the potential, the normalization of our wave function actually matters since the self-interaction depends on it.\n",
    "We will normalize the wave function so that it represents a condensate of $N$ particles, rather than just a single particle, which means $\\langle \\psi | \\psi \\rangle = N$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Eq(N, sqrt(2)*sqrt(pi)*R*\\psi_0**2/2)",
      "text/latex": "$\\displaystyle N = \\frac{\\sqrt{2} \\sqrt{\\pi} R \\psi_0^{2}}{2}$"
     },
     "metadata": {},
     "execution_count": 4
    }
   ],
   "source": [
    "N = sp.symbols(\"N\", positive=True)\n",
    "# we normalize the wave function to <psi|psi> = N\n",
    "eq = sp.Equality(\n",
    "    N, sp.integrate(abs(psi_ansatz(x))**2, (x, -sp.oo, sp.oo))\n",
    ")\n",
    "eq"
   ]
  },
  {
   "source": [
    "We can now solve for the normalization factor $\\psi_0$ in terms of the parameters $N$ and $R$."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Eq(\\psi_0, 2**(1/4)*sqrt(N)/(pi**(1/4)*sqrt(R)))",
      "text/latex": "$\\displaystyle \\psi_0 = \\frac{\\sqrt[4]{2} \\sqrt{N}}{\\sqrt[4]{\\pi} \\sqrt{R}}$"
     },
     "metadata": {},
     "execution_count": 5
    }
   ],
   "source": [
    "# solve for \\psi_0 in terms of N, R\n",
    "ps0_sol = sp.solve(eq, ps0)[0]\n",
    "sp.Eq(ps0, ps0_sol)"
   ]
  },
  {
   "source": [
    "The variational energy is simply the normalized expectation value as explained above. I will do a little trick and subtract off the vacuum energy $\\alpha \\bar{\\psi}^4$ so that $V(0) = 0$ (this doesn't affect the physics at all since only differences in energy are physical, not absolute values)."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Eq(\\widetilde{E}, 2*sqrt(3)*N**2*\\alpha/(3*pi*R**2) - 2*N*\\alpha*\\bar{\\psi}**2/(sqrt(pi)*R) + 1/(2*R**2*m))",
      "text/latex": "$\\displaystyle \\widetilde{E} = \\frac{2 \\sqrt{3} N^{2} \\alpha}{3 \\pi R^{2}} - \\frac{2 N \\alpha \\bar{\\psi}^{2}}{\\sqrt{\\pi} R} + \\frac{1}{2 R^{2} m}$"
     },
     "metadata": {},
     "execution_count": 12
    }
   ],
   "source": [
    "m, psb, a = sp.symbols(r\"m \\bar{\\psi} \\alpha\", positive=True)\n",
    "# calculate (one-dimensional) the energy density\n",
    "f = psi_ansatz\n",
    "energy_density = (\n",
    "    -(1/(2*m))*sp.diff(f(x), x, x)*f(x) # kinetic term\n",
    "    + a*((f(x)**2 - psb**2)**2 - psb**4)*(f(x)**2) # potential term\n",
    ").subs(ps0, ps0_sol)/N\n",
    "# variational energy\n",
    "E_var = sp.integrate(energy_density, (x, -sp.oo, sp.oo))\n",
    "sp.Eq(sp.symbols(r\"\\widetilde{E}\"), E_var)"
   ]
  },
  {
   "source": [
    "Now we minimize $\\widetilde{E}$ with respect to $R$, holding $N$ constant, by solving $\\partial \\widetilde{E}/\\partial R = 0$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Eq(R, (4*sqrt(3)*N**2*\\alpha*m + 3*pi)/(6*sqrt(pi)*N*\\alpha*\\bar{\\psi}**2*m))",
      "text/latex": "$\\displaystyle R = \\frac{4 \\sqrt{3} N^{2} \\alpha m + 3 \\pi}{6 \\sqrt{\\pi} N \\alpha \\bar{\\psi}^{2} m}$"
     },
     "metadata": {},
     "execution_count": 13
    }
   ],
   "source": [
    "# minimize with respect to R\n",
    "sol = sp.solveset(sp.Eq(sp.diff(E_var, R), 0), R)\n",
    "sp.Eq(R, sol.args[0].simplify())"
   ]
  },
  {
   "source": [
    "This looks a little messy, so I will rewrite it in a little neater form:\n",
    "$$\n",
    "R = R_0 \\frac{\\widetilde{N}^2 + 1}{3^{1/4} \\widetilde{N}}, \\\\ \\text{where} \\quad\n",
    "R_0 = (\\sqrt{\\alpha m} \\bar{\\psi}^2)^{-1}, \\quad\n",
    "\\widetilde{N} = \\frac{2\\sqrt{\\alpha m}}{3^{1/4} \\sqrt{\\pi}} N\n",
    "$$\n"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "True"
     },
     "metadata": {},
     "execution_count": 17
    }
   ],
   "source": [
    "# confirm rewritten stuff is equivalent\n",
    "R0 = 1/(sp.sqrt(a*m)*(psb**2))\n",
    "Nt = (2*sp.sqrt(a*m))/(sp.Pow(3,sp.Rational(1,4))*sp.sqrt(sp.pi))*N\n",
    "(R0*(Nt**2 + 1)/(sp.Pow(3, sp.Rational(1, 4))*Nt) - sol.args[0]).simplify() == 0"
   ]
  },
  {
   "source": [
    "We can see that for $\\widetilde{N} \\ll 1$ the radius actually decreases with increasing quanta like $O(1/N)$, but for $\\widetilde{N} \\gg 1$ the radius increses linearly.\n",
    "It is likely that in the low-number regime, the soliton hasn't reached sufficient density to bind effectively."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 21,
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# plot behavior of R with N\n",
    "N = np.linspace(0.1, 5, 100)\n",
    "R = (N**2 + 1)/N/(3**0.25)\n",
    "opt = {\"fontsize\": 16}\n",
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(N, R)\n",
    "plt.xlabel(r\"$\\widetilde{N}$\", **opt)\n",
    "plt.ylabel(r\"$R/R_0$\", **opt)\n",
    "plt.ylim(top=5)\n",
    "plt.title(\"Radius of soliton by number of condensate quanta\", **opt)\n",
    "plt.gca().set_facecolor(\"w\")\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "We can also plug our solution back into the variational energy expression to get the approximate binding energy of the soliton."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "-sqrt(3)*sqrt(\\alpha)*\\bar{\\psi}**2*\\widetilde{N}/(R_0*sqrt(m)*(\\widetilde{N}**2 + 1)) + sqrt(3)*\\widetilde{N}**2/(R_0**2*m*(\\widetilde{N}**2 + 1)**2)",
      "text/latex": "$\\displaystyle - \\frac{\\sqrt{3} \\sqrt{\\alpha} \\bar{\\psi}^{2} \\widetilde{N}}{R_{0} \\sqrt{m} \\left(\\widetilde{N}^{2} + 1\\right)} + \\frac{\\sqrt{3} \\widetilde{N}^{2}}{R_{0}^{2} m \\left(\\widetilde{N}^{2} + 1\\right)^{2}}$"
     },
     "metadata": {},
     "execution_count": 56
    }
   ],
   "source": [
    "R, R0, Nt, N = sp.symbols(\"R R_0 \\widetilde{N} N\", positive=True)\n",
    "(\n",
    "    E_var\n",
    "    .subs(R, R0*(Nt**2 + 1)/(sp.Pow(3, sp.Rational(1, 4))*Nt))\n",
    "    .subs(N, 1/((2*sp.sqrt(a*m))/(sp.Pow(3,sp.Rational(1,4))*sp.sqrt(sp.pi))))\n",
    ")"
   ]
  },
  {
   "source": [
    "This can be cleaned up a little bit too:\n",
    "$$\n",
    "E = -\\sqrt{3} \\alpha \\bar{\\psi}^4 \\frac{\\widetilde{N}}{\\widetilde{N}^2+1} \\left( 1 - \\frac{\\widetilde{N}}{\\widetilde{N}^2+1} \\right)\n",
    "$$\n",
    "Interestingly, the soliton is most strongly bound when $\\widetilde{N} \\sim 1$. For $\\widetilde{N} \\ll 1$, it quickly loses its stickiness and dissipates. In the $\\widetilde{N} \\gg 1$ regime, it continues to stay bound, but since the binding energy asymptotically approches zero instead of growing linearly, it actually becomes more energetically favorable for a large soliton to fracture into multiple smaller solitons, and is likely unstable to perturbations."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "Nt = np.linspace(0, 10, 250)\n",
    "E = -(3**0.5)*(Nt/(Nt**2+1))*(1 - Nt/(Nt**2+1))\n",
    "opt = {\"fontsize\": 16}\n",
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(Nt, E)\n",
    "plt.xlabel(r\"$\\widetilde{N}$\", **opt)\n",
    "plt.ylabel(r\"$E/\\alpha\\bar{\\psi}^4$\", **opt)\n",
    "plt.title(\"Soliton binding energy\", **opt)\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "However, in the next section on numerical solutions, we will see that the Gaussian ansatz is perhaps not the best trial wavefunction in the $\\tilde{N} \\gg 1$ regime, and so we cannot trust this conclusion in that parameter space."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "## Numerical simulation"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "Before attempting numerical calculation of $\\psi(x)$, it will be convenient to make a couple transformations to simplify computations.\n",
    "First of all, from our approximate analysis in the previous section, we expect the width of the soliton to be approximately $O(R)$.\n",
    "Therefore, it would be prudent to transform our coordinate system like $x \\rightarrow R \\xi$, so that our new spatial coordinate $\\xi$ is now a dimensionless quantity representing lengths in units of $R$.\n",
    "This means that $\\xi \\sim O(1)$, simplifying our choice of grid points for numerical simulation.\n",
    "Second, it will simplify the differential equation itself to transform $\\psi \\rightarrow \\bar{\\psi} \\chi$.\n",
    "Third (and as mentioned before), we can see that the the potential is nonzero when $\\psi = 0$, which means that there is an infinite vacuum energy.\n",
    "We can simply subtract off this constant term to get a finite value for the energy.\n",
    "This does not alter the physics at all (unless considering general relativity), but makes things more numerically stable and easier to understand.\n",
    "After these transformations, the differential equation looks like\n",
    "\n",
    "$$\n",
    "-\\frac{1}{2} \\frac{d^2\\chi}{d\\xi^2} + \\beta^2\\left[\\left(|\\chi|^2 - 1\\right)^2 - 1 \\right]\\chi = \\mathcal{E} \\chi \\\\ \\text{where} \\quad\n",
    "\\beta = \\sqrt{\\alpha m} \\bar{\\psi}^2 R\n",
    "= R/R_0\n",
    "= \\frac{\\widetilde{N}^2 + 1}{3^{1/4}\\widetilde{N}}, \\quad\n",
    "\\mathcal{E} = m R^2 E\n",
    "$$\n",
    "This only has a single dimensionless free parameter $\\widetilde{N}$, and a dimensionless energy eigenvalue $\\mathcal{E}$.\n",
    "Keep in mind that the wave function normalization condition is now\n",
    "$$\n",
    "\\langle \\chi | \\chi \\rangle = \\int d\\xi\\, |\\chi|^2\n",
    "= \\int \\frac{dx}{R}\\, \\left|\\psi/\\bar{\\psi}\\right|^2\n",
    "= \\frac{N}{R \\bar{\\psi}^2}\n",
    "= \\frac{3^{1/4} \\sqrt{\\pi}}{2\\beta} \\widetilde{N}\n",
    "= \\frac{\\sqrt{3\\pi}}{2} \\frac{\\widetilde{N}^2}{\\widetilde{N}^2 + 1}.\n",
    "$$\n",
    "$\\widetilde{N}$ is now the only free parameter in the entire scenario, which means that $\\mathcal{E}$ is a function of $\\widetilde{N}$ alone.\n",
    "We can exploit this to solve for the energy spectrum quite easily."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "Normally, 1-dimensional Schrodinger equations are solved using a shooting technique that involves integrating the differential equation while performing a bisection search on the energy eigenvalue, with the termination condition being that the boundary conditions are (approximately) satisfied.\n",
    "However, we can make use of the variational wave function derived above as a starting point.\n",
    "In terms of $\\chi$ and $\\xi$, this is now\n",
    "\n",
    "$$\n",
    "\\chi_0(\\xi) = \\left[\\sqrt{\\frac{3}{2}} \\frac{\\widetilde{N}^2}{\\widetilde{N}^2 + 1}\\right]^{1/2} e^{-\\xi^2}\n",
    "$$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numba import njit # use numba for SPEEEEEEED\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "@njit\n",
    "def init_chi(Nt, xi):\n",
    "    return np.sqrt(np.sqrt(3/2)*(Nt**2)/(Nt**2+1)) * np.exp(-xi**2)\n",
    "\n",
    "# declare parameters\n",
    "xi_min, xi_max = -5, 5\n",
    "n_points = 101\n",
    "delta = (xi_max - xi_min)/(n_points - 1)\n",
    "Nt = 1\n",
    "\n",
    "# set things up\n",
    "xi = np.linspace(xi_min, xi_max, n_points)\n",
    "chi0 = init_chi(Nt, xi)\n",
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(xi, chi0)\n",
    "plt.xlabel(r\"$\\xi$\")\n",
    "plt.ylabel(r\"$\\chi$\")\n",
    "plt.title(\"variational soliton wave function\")\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "Following along with with [Ref. 1](https://arxiv.org/abs/1701.08934), we can perform iterative updates to a discretized version of $\\chi_0$ in order to relax the solution to the ground state or other low-lying excited states.\n",
    "\n",
    "The potential is given by $V(\\chi) = \\beta^2((|\\chi|^2-1)^2-1)$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit\n",
    "def potential(chi, Nt):\n",
    "    beta = (Nt**2 + 1)/((3**0.25)*Nt)\n",
    "    return (beta**2)*(((chi**2) - 1)**2 - 1)"
   ]
  },
  {
   "source": [
    "The expectation values are given by\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\langle \\chi | \\chi \\rangle &= \\delta \\sum_i \\chi_i^2 \\\\\n",
    "\\bar{V} = \\langle \\chi | V | \\chi \\rangle &= \\delta \\sum_i V_i \\chi_i^2 \\\\\n",
    "\\bar{K} = \\langle \\chi | K | \\chi \\rangle &= -\\delta^{-1} \\sum_i (\\chi_i \\bar{\\chi}_{nn} - \\chi_i^2)\n",
    "\\end{align*}\n",
    "$$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "(0.7674950309598665,\n -1.1512425464397997,\n 0.38278974364252677,\n -1.0012479192682313)"
     },
     "metadata": {},
     "execution_count": 74
    }
   ],
   "source": [
    "@njit\n",
    "def chi_nn(chi):\n",
    "    # average each point over nearest neighbors\n",
    "    # assume values outside range are 0\n",
    "    _chi = np.zeros_like(chi)\n",
    "    _chi[1:len(chi)-1] = (chi[0:len(chi)-2] + chi[2:len(chi)])/2\n",
    "    _chi[0] = chi[1]/2\n",
    "    _chi[-1] = chi[-1]/2\n",
    "    return _chi\n",
    "\n",
    "norm = delta * np.sum(chi0**2)\n",
    "Vbar = delta * np.dot(potential(chi0, Nt), (chi0**2))\n",
    "Kbar = -np.sum(chi0*chi_nn(chi0) - chi0**2)/delta\n",
    "(norm, Vbar, Kbar, (Kbar + Vbar)/norm)"
   ]
  },
  {
   "source": [
    "Here is a simple relaxation algorithm to find the true ground state of $\\chi$:"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit\n",
    "def update(chi, Nt, delta):\n",
    "    E_arr = np.zeros(1000)\n",
    "    for i in range(1000):\n",
    "        # common quantities that will be reused\n",
    "        _chi_nn = chi_nn(chi)\n",
    "        chi2 = chi**2\n",
    "        V0 = potential(chi, Nt)\n",
    "        # update expectation values\n",
    "        norm = delta * np.sum(chi2)\n",
    "        Vbar = delta * np.dot(V0, chi2)\n",
    "        Kbar = -np.sum(chi*_chi_nn - chi2)/delta\n",
    "        E = (Kbar + Vbar)/norm\n",
    "        # update field values\n",
    "        chi = _chi_nn / (1 - (E - V0)*(delta**2))\n",
    "        E_arr[i] = E\n",
    "    return E_arr, chi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "E_arr, chif = update(chi0, Nt, delta)"
   ]
  },
  {
   "source": [
    "We can tell that the algorithm has converged by looking at the evolution of the variational energy over multiple iterations"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(E_arr)\n",
    "plt.xlabel(\"iteration\", fontsize=16)\n",
    "plt.ylabel(r\"$\\langle\\mathcal{E}\\rangle$\", fontsize=16)\n",
    "plt.title(r\"Variational energy convergence, $\\widetilde{N}$ = \" + str(Nt))\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "We can see that the exact numerical solution is a bit broader than expected by the variational approximation, but that they are roughly the same size and density"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 720x432 with 1 Axes>",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(xi, chif, label=\"exact\")\n",
    "plt.plot(xi, chi0, color=\"C0\", alpha=0.5, linestyle=\"--\", label=\"var. approx.\")\n",
    "plt.xlabel(r\"$\\xi$\", fontsize=16)\n",
    "plt.ylabel(r\"$\\chi$\", fontsize=16)\n",
    "plt.title(r\"Soliton profile; $\\widetilde{N}$\"f\" = {Nt}\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "If we crank up the density and self-interaction strength of the soliton, we can see that it develops a nearly uniform density core, and is not as broad/diffuse as before.\n",
    "This is what is known as a \"thin-walled\" soliton."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit(parallel=True)\n",
    "def chi_solve(Nt, n_points=101, n_iter=1000, xi_min=-5, xi_max=+5, eps=1e-9):\n",
    "    delta = (xi_max - xi_min)/(n_points - 1)\n",
    "    xi = np.linspace(xi_min, xi_max, n_points)\n",
    "    chi0 = init_chi(Nt, xi)\n",
    "    chi = chi0\n",
    "    E_arr = np.zeros(n_iter)\n",
    "    for i in range(n_iter):\n",
    "        # common quantities that will be reused\n",
    "        _chi_nn = chi_nn(chi)\n",
    "        chi2 = chi**2\n",
    "        V0 = potential(chi, Nt)\n",
    "        # update expectation values\n",
    "        norm = delta * np.sum(chi2)\n",
    "        Vbar = delta * np.dot(V0, chi2)\n",
    "        Kbar = -np.sum(chi*_chi_nn - chi2)/delta\n",
    "        E = (Kbar + Vbar)/norm\n",
    "        # update field values\n",
    "        chi = _chi_nn / (1 - (E - V0)*(delta**2)+eps)\n",
    "        E_arr[i] = E\n",
    "    return E_arr, xi, chi0, chi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "E_arr, xi, chi0, chi = chi_solve(Nt=10, n_iter=int(8.0e3), n_points=2000)\n",
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(E_arr)\n",
    "plt.xlabel(\"iterations\", fontsize=16)\n",
    "plt.ylabel(r\"$\\langle \\mathcal{E} \\rangle$\", fontsize=16)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 6))\n",
    "plt.plot(xi, chi, label=\"exact\")\n",
    "plt.plot(xi, chi0, color=\"C0\", alpha=0.5, linestyle=\"--\", label=\"var. approx.\")\n",
    "plt.xlabel(r\"$\\xi$\", fontsize=16)\n",
    "plt.ylabel(r\"$\\chi$\", fontsize=16)\n",
    "plt.title(r\"Soliton profile; $\\widetilde{N}$\"f\" = {10}\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "Recall that $\\chi = 1$ is the minimum of the potential, so if the field value gets much higher than that (as the soliton becomes more dense), then an internal pressure is produced which keeps the soliton from collapsing further."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ]
}