The Gross-Pitaevskii equation and Bogoliubov spectrum
Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 (1) Laboratoire PhLAM, UMR CNRS 8523, Universit\303\251 Lille 1, F-59655, France(2) Maplesoft, CanadaDeparting from the equation for a quantum system of identical boson particles, i.e.the Gross-Pitaevskii equation (GPE), the dispersion relation for plane-wave solutions are derived, as well as the Bogoliubov equations and dispersion relations for small perturbations LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn around the GPE stationary solutions.
<Text-field style="Heading 1" layout="Heading 1">Stationary and plane-wave solutions to the Gross-Pitaevskii equation</Text-field>
Problem: Given the Gross-Pitaevskii equation, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==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) Derive a relationship between the chemical potential LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y1 entering the phase of stationary, uniform solutions, the atom-atom interaction constant G and the particle density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZlcXVhbHM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNigtRiw2JVEmJnBzaTtGJy9GMEY9RjlGOS9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGaG5GWS1JI21uR0YkNiRRIjJGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRj0vJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjk= in the absence of an external field (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRIlZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEpJmVxdWFscztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGVS1JI21uR0YkNiRRIjBGJ0YvRi9GK0Yv).b) Derive the dispersion relation for plane-wave solutions as a function of G and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==.Background: The Gross-Pitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5]. The temperature of these cold atomic gases is typically in the \342\210\274100 nano-Kelvin range. The atom-atom interaction are repulsive for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= and attractive for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= , where G is the interaction constant. The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers.
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>a) Derive a relationship between the chemical potential LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y1 entering the phase of stationary, uniform solutions, the atom-atom interaction constant G and the particle density 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 in the absence of an external field (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRIlZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEpJmVxdWFscztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGVS1JI21uR0YkNiRRIjBGJ0YvRi9GK0Yv).
The version of Physics used is from December/6 (or later), available at the Maplesoft Physics Research & Development 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Use a macro 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 to avoid redundant typing and use declare to have a compact displayQyQtSSZtYWNyb0clKnByb3RlY3RlZEc2Iy9JJFBzaUc2Ii1JJHBzaUdGKTYmSSJ4R0YlSSJ5R0YlSSJ6R0YlSSJ0R0YpISIiLV9JKVBERXRvb2xzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSShkZWNsYXJlR0YoNiMtNiRJJHBzaUdGKEkiVkdGKDYmSSJ4R0YmSSJ5R0YmSSJ6R0YmSSJ0R0YoThe Gross-Pitaevskii equation, derived in the previous post in Mapleprimes, is given byQyQvLUkiKkc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJ0koX3N5c2xpYkdGKi9JMWhpZGVtb2R1bGVwcmVmaXhHRipJJXRydWVHRic2JC1GJTYkXiMiIiJJJyZoYmFyO0dGKi1JJWRpZmZHRic2JEkkUHNpR0YsSSJ0R0YqLCYtRiU2JCwmLUYlNiRJIkdHRioqJC1JJGFic0dGJzYjRjoiIiNGNS1JIlZHRio2JkkieEdGJ0kieUdGJ0kiekdGJ0Y7RjVGOkY1LUYlNiQtRiU2JCokRjZGRy1JKyVMYXBsYWNpYW5HRipGRiokLUYlNiRGR0kibUdGKiEiIkZZRjU=where G is the atom-atom interaction constant, 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 is an external field, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= is the particle-wave function, m is the mass of a single particle and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NDYzO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the Planck constant divided by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJ0YvRi8=. Set the real objects of this problem including the chemical potential LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==LUkmU2V0dXBHNiI2Iy9JLHJlYWxvYmplY3RzR0YkPChJJyZoYmFyO0dGJEkiR0dGJEkibUdGJEkjbXVHRiRJInRHRiQtSSJWR0YkNiZJInhHJSpwcm90ZWN0ZWRHSSJ5R0YySSJ6R0YyRi0=In the absence of an external potential, we have 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. To relate the chemical potential to the interaction constant G and the particle density n we look for a real stationary and uniform solution for of the form 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the chemical potential and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYnUSkmdmFycGhpO0YnLyUnaXRhbGljR0Y0LyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGL0Yv is constant.Ly1JJHBzaUc2IjYmSSJ4RyUqcHJvdGVjdGVkR0kieUdGKEkiekdGKEkidEdGJS1JIipHNiVGKC9JK21vZHVsZW5hbWVHRiVJKFBoeXNpY3NHNiRGKEkoX3N5c2xpYkdGJS9JMWhpZGVtb2R1bGVwcmVmaXhHRiVJJXRydWVHRig2JEkpJnZhcnBoaTtHRiUtSSRleHBHRiU2IywkLUYtNiQtRi02JC1GLTYkXiMiIiJJI211R0YlRisqJEknJmhiYXI7R0YlISIiRkg=Substituting into the GPE equation ,LUklZXZhbEclKnByb3RlY3RlZEc2JC8qKF4jIiIiRilJJyZoYmFyO0c2IkYpLUklZGlmZkdGJDYkLUkkcHNpR0YrNiZJInhHRiRJInlHRiRJInpHRiRJInRHRitGNUYpLCYqJiwmKiZJIkdHRitGKS1JJGFic0dGJDYjRi8iIiNGKS1JIlZHRitGMUYpRilGL0YpRikqKEYqRj4tSSslTGFwbGFjaWFuR0YrRj1GKUkibUdGKyEiIiNGRUY+PCQvRj8iIiEvRi8qJkkpJnZhcnBoaTtHRitGKS1JJGV4cEc2JEYkSShfc3lzbGliR0YrNiMqKl4jRkVGKUkjbXVHRitGKUY1RilGKkZFRik=The relationship between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, G and n is then obtained evaluating the Laplacian in the right-hand side, isolating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and introducing the particle density 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LUkoaXNvbGF0ZUc2IjYkLUkmdmFsdWVHRiQ2Iy8qKEkpJnZhcnBoaTtHRiQiIiJJI211R0YkRiwtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyoqXiMhIiJGLEYtRixJInRHRiRGLEknJmhiYXI7R0YkRjZGLCwmKipJIkdHRiRGLC1JJGFic0dGMTYjRisiIiNGK0YsRi5GLEYsKihGOEY/LUkrJUxhcGxhY2lhbkdGJDYjKiZGK0YsRi5GLEYsSSJtR0YkRjYjRjZGP0YtFrom ,LUkkbWFwRyUqcHJvdGVjdGVkRzYkZio2I0kidUc2IkYpNiRJKW9wZXJhdG9yR0YpSSZhcnJvd0dGKUYpKiQtSSRhYnNHRiQ2IzkkIiIjRilGKUYpLy1JJHBzaUdGKTYmSSJ4R0YkSSJ5R0YkSSJ6R0YkSSJ0R0YpKiZJKSZ2YXJwaGk7R0YpIiIiLUkkZXhwRzYkRiRJKF9zeXNsaWJHRik2IyoqXiMhIiJGPUkjbXVHRilGPUY6Rj1JJyZoYmFyO0dGKUZFRj0=LUklc3Vic0clKnByb3RlY3RlZEc2JC8qJC1JJGFic0dGJDYjSSRQc2lHNiRGJEkoX3N5c2xpYkc2IiIiI0kibkdGLi8qJC1GKTYjLUkkcHNpR0YuNiZJInhHRiRJInlHRiRJInpHRiRJInRHRi5GLyokLUYpNiNJKSZ2YXJwaGk7R0YuRi8=From whereLUklc3Vic0clKnByb3RlY3RlZEc2JC0vSSRyaHNHRiRJJGxoc0dGJDYjL0kibkc2IiokLUkkYWJzR0YkNiNJKSZ2YXJwaGk7R0YtIiIjL0kjbXVHRi0qJkkiR0dGLSIiIkYvRjM=_____________________________________________________________________________b) Derive the dispersion relation for plane-wave solutions as a function of G and n.We now look for plane-wave solutions 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the wave number and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy the frequency. To simplify, the GPE is restricted to a 2D spacetime case, still in the absence of external potential.QyYtSSZtYWNyb0clKnByb3RlY3RlZEc2Iy9JJVBzaTJHNiItSSRwc2lHRik2JEkieEdGJUkidEdGKSEiIi1fSSlQREV0b29sc0c2JEYlSShfc3lzbGliR0YpSShkZWNsYXJlR0YpNiNGKCIiIg==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 the 2D GPE at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= is obtained replacing 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 in LUklc3Vic0clKnByb3RlY3RlZEc2JS8tSSJWRzYiNiZJInhHRiRJInlHRiRJInpHRiRJInRHRikiIiEvLUkkcHNpR0YpRiotRjI2JEYrRi4vKiheIyIiIkY4SScmaGJhcjtHRilGOC1JJWRpZmZHRiQ2JEYxRi5GOCwmKiYsJiomSSJHR0YpRjgtSSRhYnNHRiQ2I0YxIiIjRjhGJ0Y4RjhGMUY4RjgqKEY5RkUtSSslTGFwbGFjaWFuR0YpRkRGOEkibUdGKSEiIiNGSkZFIntroducing now an explicit form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= in terms of the particle density 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 and a plane wave 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, and taking into account that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5,L0klUHNpMkc2Ii1JIipHNiUlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSShQaHlzaWNzRzYkRihJKF9zeXNsaWJHRiQvSTFoaWRlbW9kdWxlcHJlZml4R0YkSSV0cnVlR0YoNiQtSSVzcXJ0R0YkNiNJIm5HRiQtSSRleHBHRiw2Iy1GJjYkXiMiIiIsJi1GJjYkSSJrR0YkSSJ4R0YoRjwtRiY2JEkmb21lZ2FHRiRJInRHRiQhIiI=LUkpYXNzdW1pbmdHNiI2JDcjLUklZXZhbEclKnByb3RlY3RlZEc2JC8qKF4jIiIiRi5JJyZoYmFyO0dGJEYuLUklZGlmZkdGKTYkLUkkcHNpR0YkNiRJInhHRilJInRHRiRGN0YuLCYqKEkiR0dGJEYuRjNGLi1JJGFic0dGKTYjRjMiIiNGLiooRi9GPi1JKyVMYXBsYWNpYW5HRiRGPUYuSSJtR0YkISIiI0ZDRj4vRjMqJkkibkdGJCNGLkY+LUkkZXhwRzYkRilJKF9zeXNsaWJHRiQ2IyomRi1GLiwmKiZJImtHRiRGLkY2Ri5GLiomSSZvbWVnYUdGJEYuRjdGLkZDRi5GLjcjLUkjPj1HRik2JEZHIiIhIsolating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and evaluating the Laplacian we get the dispersion relationLUkoaXNvbGF0ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLyoqSScmaGJhcjtHRiciIiJJIm5HRicjRiwiIiNJJm9tZWdhR0YnRiwtSSRleHBHRiQ2IyomXiNGLEYsLCYqJkkia0dGJ0YsSSJ4R0YlRixGLComRjBGLEkidEdGJ0YsISIiRixGLCwmKihJIkdHRidGLEYtIyIiJEYvRjFGLEYsKihGK0YvLUkrJUxhcGxhY2lhbkdGJzYjKiZGLUYuRjFGLEYsSSJtR0YnRjwjRjxGL0YwLUknZXhwYW5kRzYkL0kocGFja2FnZUclKnByb3RlY3RlZEdJKFBoeXNpY3NHNiRGJ0koX3N5c2xpYkc2IkYlNiMtSSZ2YWx1ZUdGKTYjSSIlR0YrThis GPE dispersion relation is a function of both the non-linearity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5and the field density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (which would not be the case for a linear PDE).LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=
<Text-field style="Heading 1" layout="Heading 1">The Bogoliubov equations and dispersion relations</Text-field>Problem: Given the Gross-Pitaevskii equation,a) Derive the Bogoliubov equations, that is, equations for elementary excitations LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn and 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around a GPE stationary solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5ODE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRj1GP0ZBLUYsNiVRInpGJ0Y9Rj9GMkYyRjI=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGLw==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
b) Show that the dispersion relations of these equations, known as the Bogoliubov spectrum, are given by
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,
where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the wave number of the considered elementary excitation, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLzYlUSJrRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIUYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWRjU= its energy or, equivalently, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHRjRGLy1GSjYlUSJrRicvRk5RJXRydWVGJy9GMFEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Yv its frequency.
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>a) To derive Bogoliubov's equations, start from a non-uniform stationary GPE solution, so LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYoLUkjbWlHRiQ2JVEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUvSSttc2VtYW50aWNzR0YkUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRj1GOEY1 depending only on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2KC1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEoJmNvbW1hO0YnRi9GMi9GNkZORjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUZJNiVRInlGJ0ZMRk9GUS1GSTYlUSJ6RidGTEZPRi9GK0Yv, that is, of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2KS1JI21pR0YkNiVRJyYjOTY4O0YnLyUnaXRhbGljR0Y0Ri8tRiw2LVEpJmVxdWFscztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGUi1GSTYlUScmIzk4MTtGJ0ZMRi8tSShtZmVuY2VkR0YkNiQtRiM2KC1GSTYlUSJ4RicvRk1RJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiLEYnRi9GMi9GNkZqbkY3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GSTYlUSJ5RidGaW5GW29GXW8tRkk2JVEiekYnRmluRltvRi9GLy1GLDYtUTEmSW52aXNpYmxlVGltZXM7RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JJW1zdXBHRiQ2JS1GLDYtUS8mRXhwb25lbnRpYWxFO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBL0ZFUSwwLjExMTExMTFlbUYnLUYjNiUtRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZqcC1JJm1mcmFjR0YkNigtRiM2J0YrLUYjNiYtRkk2JVEiaUYnRmluRltvRmlvLUZJNiVRJSZtdTtGJ0ZMRi9GL0Zpby1GSTYlUSJ0RidGaW5GW29GLy1GSTYlUSgmIzg0NjM7RidGTEYvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zkci8lKWJldmVsbGVkR0Y0Ri8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRi9GK0Yv, where written in this form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the chemical potential, and introduce small perturbations 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around 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. Bogoliubov's equations describe the evolution in time and space of these small perturbations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEtJiM5NDg7JiM5ODE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjNRJ25vcm1hbEYn and are obtained, basically, by inserting the perturbed solution into the GPE and discarding terms of higher degree in LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn.LV9JKVBERXRvb2xzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSShkZWNsYXJlR0YoNiUtSSRwc2lHRig2JkkieEdGJkkieUdGJkkiekdGJkkidEdGKC1JKSZ2YXJwaGk7R0YoNiVGLkYvRjAtSTAmZGVsdGE7JnZhcnBoaTtHRihGLQ==The departing point is then the form of the GPE solution 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 that includes this small perturbation 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 depending on space and timeL0kkUHNpRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkiKkc2JUYlL0krbW9kdWxlbmFtZUdGJ0koUGh5c2ljc0dGJC9JMWhpZGVtb2R1bGVwcmVmaXhHRidJJXRydWVHRiU2JCwmLUkpJnZhcnBoaTtHRic2JUkieEdGJUkieUdGJUkiekdGJSIiIi1JMCZkZWx0YTsmdmFycGhpO0dGJzYmRjZGN0Y4SSJ0R0YnRjktSSRleHBHRiQ2IywkLUYpNiQtRik2JC1GKTYkXiNGOUkjbXVHRidGPSokSScmaGJhcjtHRichIiJGTA==Recalling the form of the GPE LyooXiMiIiJGJUknJmhiYXI7RzYiRiUtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkkcHNpR0YnNiZJInhHRipJInlHRipJInpHRipJInRHRidGMkYlLCYqJiwmKiZJIkdHRidGJS1JJGFic0dGKjYjRiwiIiNGJS1JIlZHRidGLkYlRiVGLEYlRiUqKEYmRjstSSslTGFwbGFjaWFuR0YnRjpGJUkibUdGJyEiIiNGQkY7Substitute into it the perturbed 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 unperturbed GPE solution 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 is also substituted into :Ly1JJHBzaUc2IjYmSSJ4RyUqcHJvdGVjdGVkR0kieUdGKEkiekdGKEkidEdGJS1JIipHNiVGKC9JK21vZHVsZW5hbWVHRiVJKFBoeXNpY3NHNiRGKEkoX3N5c2xpYkdGJS9JMWhpZGVtb2R1bGVwcmVmaXhHRiVJJXRydWVHRig2JC1JKSZ2YXJwaGk7R0YlNiVGJ0YpRiotSSRleHBHRjI2IywkLUYtNiQtRi02JC1GLTYkXiMiIiJJI211R0YlRisqJEknJmhiYXI7R0YlISIiRko=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 obtain the equation for the perturbation LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn, subtract the non-perturbed solution from the perturbed solution :LUkiK0clKnByb3RlY3RlZEc2JC8sKCooXiMiIiJGKkknJmhiYXI7RzYiRiotSSVkaWZmR0YkNiQtSTAmZGVsdGE7JnZhcnBoaTtHRiw2JkkieEdGJEkieUdGJEkiekdGJEkidEdGLEY2RipGKiomSSNtdUdGLEYqLUkpJnZhcnBoaTtHRiw2JUYzRjRGNUYqRioqJkY4RipGMEYqRiosLiooSSJHR0YsRiotSSRhYnNHRiQ2IywmRjlGKkYwRioiIiNGOUYqRioqKEY/RipGQEZERjBGKkYqKiYtSSJWR0YsRjJGKkY5RipGKiomRkdGKkYwRipGKiooRitGREkibUdGLCEiIi1JKyVMYXBsYWNpYW5HRiw2I0Y5RiojRkxGRCooRitGREZLRkwtRk42I0YwRipGUCwkL0Y3LCgqKEY5RipGP0YqLUZBRk9GREYqRkZGKkZKRlBGTA==In order to discard higher degree terms in the perturbation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEtJiM5NDg7JiM5ODE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRjEvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2Ji1GIzYlLUkjbWlHRiQ2JVEtJiM5NDg7JiM5ODE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjRGNy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRRk0vSSttc2VtYW50aWNzR0YkUSpjb25qdWdhdGVGJ0Y+ , rewrite the 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 in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEnJnBsdXM7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkktRiw2JVEwJmRlbHRhOyZ2YXJwaGk7RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMg== and its conjugateLUknZXhwYW5kRzYkL0kocGFja2FnZUclKnByb3RlY3RlZEdJKFBoeXNpY3NHNiRGJ0koX3N5c2xpYkc2IkYlNiMtSShjb252ZXJ0R0YnNiQvLCYqKF4jIiIiRjRJJyZoYmFyO0dGK0Y0LUklZGlmZkdGJzYkLUkwJmRlbHRhOyZ2YXJwaGk7R0YrNiZJInhHRidJInlHRidJInpHRidJInRHRitGP0Y0RjQqJkkjbXVHRitGNEY5RjRGNCwsKihJIkdHRitGNC1JJGFic0dGJzYjLCYtSSkmdmFycGhpO0dGKzYlRjxGPUY+RjRGOUY0IiIjRklGNEY0KihGREY0RkVGTEY5RjRGNComLUkiVkdGK0Y7RjRGOUY0RjQqKEY1RkxJIm1HRishIiItSSslTGFwbGFjaWFuR0YrNiNGOUY0I0ZTRkwqKEZJRjRGREY0LUZGNiNGSUZMRlNJKmNvbmp1Z2F0ZUdGJw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8sJiooXiMiIiJGLUknJmhiYXI7R0YnRi0tSSVkaWZmR0YlNiQtSTAmZGVsdGE7JnZhcnBoaTtHRic2JkkieEdGJUkieUdGJUkiekdGJUkidEdGJ0Y4Ri1GLSomSSNtdUdGJ0YtRjJGLUYtLDAqKC1JKSZ2YXJwaGk7R0YnNiVGNUY2RjciIiNJIkdHRidGLS1JKmNvbmp1Z2F0ZUdGJTYjRjJGLUYtKipGPUYtRkFGLUYyRi0tRkM2I0Y9Ri1GQCoqRj1GLUZBRi1GMkYtRkJGLUZAKihGQUYtRjJGQEZGRi1GLSooRkFGLUYyRkBGQkYtRi0qJi1JIlZHRidGNEYtRjJGLUYtKihGLkZASSJtR0YnISIiLUkrJUxhcGxhY2lhbkdGJ0ZERi0jRlBGQEklc2l6ZUdGJw==In we see that, discarding higher degree terms in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEtJiM5NDg7JiM5ODE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRjEvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2Ji1GIzYlLUkjbWlHRiQ2JVEtJiM5NDg7JiM5ODE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjRGNy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRRk0vSSttc2VtYW50aWNzR0YkUSpjb25qdWdhdGVGJ0Y+ , we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiktSSNtaUdGJDYlUScmIzk4MTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEnJnBsdXM7RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtRjQ2JVEtJiM5NDg7JiM5ODE7RicvRjhRJXRydWVGJy9GO1EnaXRhbGljRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGWC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjpGOi1JI21uR0YkNiRRIjJGJ0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GPjYtUTEmSW52aXNpYmxlVGltZXM7RidGOkZBRkNGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTRmhvLUkmbW92ZXJHRiQ2Ji1GIzYnRlRGZW5GaG5Gam5GOi1GPjYtUS0mY29uanVnYXRlMDtGJ0Y6RkFGQy9GRkZYRkdGSUZLRk1GZ29GaW9GTS9JK21zZW1hbnRpY3NHRiRRKmNvbmp1Z2F0ZUYnLUY+Ni1RKSZhcHByb3g7RidGOkZBRkNGRUZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRmpwLUYsNiVGM0Zdb0Zhb0Zkb0Zqb0ZlbkZobkZqbkY6 , and 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 These higher degree terms have degrees 2 and 3 and can be selected as 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So we can either subtract these terms from , as in - or, simpler, use remove instead of select, as 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Isolating now the term involving the time derivative and converting to abs only the terms involving LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5ODE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjI= we arrive at Bogoliubov's equation for 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The equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2Ji1GIzYlLUkjbWlHRiQ2JVEtJiM5NDg7JiM5ODE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjRGNy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRRk0vSSttc2VtYW50aWNzR0YkUSpjb25qdWdhdGVGJ0Y+ can now be obtained by taking the conjugate of the equation for LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn, expanding and collecting termsQyYsJC1JKmNvbmp1Z2F0ZUclKnByb3RlY3RlZEc2Iy8qKF4jIiIiRitJJyZoYmFyO0c2IkYrLUklZGlmZkdGJjYkLUkwJmRlbHRhOyZ2YXJwaGk7R0YtNiZJInhHRiZJInlHRiZJInpHRiZJInRHRi1GN0YrLCgqJiwoKiZJIkdHRi1GKy1JJGFic0dGJjYjLUkpJnZhcnBoaTtHRi02JUY0RjVGNiIiI0ZDLUkiVkdGLUYzRitJI211R0YtISIiRitGMUYrRisqKEZARkNGPEYrLUYlNiNGMUYrRisqKEYsRkNJIm1HRi1GRy1JKyVMYXBsYWNpYW5HRi1GSkYrI0ZHRkNGR0ZHLUkoY29sbGVjdEc2JEYmSShfc3lzbGliR0YtNiQtSSdleHBhbmRHNiQvSShwYWNrYWdlR0YmSShQaHlzaWNzR0ZSRlg2I0kiJUdGLUYlRis=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JS1JI21pR0YkNiVRKGNvbGxlY3RGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2KC1GSTYlUSdleHBhbmRGJ0ZMRk8tRlI2JC1GIzYkLUZJNiVRIiVGJ0ZMRk9GL0YvLUYsNi1RIixGJ0YvRjIvRjZGTkY3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GLDYtUSJ+RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GSTYlUSpjb25qdWdhdGVGJ0ZMRk9GL0YvRi9GKy8lK2V4ZWN1dGFibGVHRjRGLw==_____________________________________________________________________________b) Solving Bogoliubov's equations and is, in general, a difficult problem. The computation is here restricted to a perturbation LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn around the uniform solution derived in the previous section. So we take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2J1EiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPUYnRjJGNC9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiZRIjBGJ0YyRjRGPi1GOzYvUSIsRidGMkY0Rj5GQC9GREYxRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU1EsMC4zMzMzMzMzZW1GJy1GOzYvUSJ+RidGMkY0Rj5GQEZDRkVGR0ZJRktGTUZmbi9GU0ZnbkZqbi1GLDYnUScmIzk1NjtGJy9GMEZCRjJGNEY+RjotRiw2J1EiR0YnRi9GMkY0RjdGam4tRiw2J1EibkYnRi9GMkY0RjdGMkY0Rj4=, and 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a constant uniform particle density. The dispersion relations for and are then derived using two different approaches: first simplify the problem to one in 1+1 spacetime dimensions and use Fourier transforms, then consider the whole problem in 3+1 spacetime dimensions and find the dispersion relations treating LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR0YuLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= and 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 as independent perturbation functions. Introduce 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, and 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 into Bogoliubov's equationsPkkrY29uZGl0aW9uc0c2IjclLy1JIlZHRiQ2JkkieEclKnByb3RlY3RlZEdJInlHRitJInpHRitJInRHRiQiIiEvLUkpJnZhcnBoaTtHRiQ2JUYqRixGLS1JJXNxcnRHNiRGK0koX3N5c2xpYkdGJDYjSSJuR0YkL0kjbXVHRiQtSSIqRzYlRisvSSttb2R1bGVuYW1lR0YkSShQaHlzaWNzR0Y2L0kxaGlkZW1vZHVsZXByZWZpeEdGJEkldHJ1ZUdGKzYkSSJHR0YkRjk=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 this point we need to introduce an expression for the perturbation LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn. We start considering this perturbation as a complex function, that is 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, where u and v are real functions of x and t.LV9JKVBERXRvb2xzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSShkZWNsYXJlR0YoNiMtNiRJInVHRihJInZHRig2JEkieEdGJkkidEdGKA==LUkmU2V0dXBHNiI2Iy9JLHJlYWxvYmplY3RzR0YkPCMtNiRJInVHRiRJInZHRiQ2JEkieEclKnByb3RlY3RlZEdJInRHRiQ=So the simplified equation becomesLUklZXZhbEclKnByb3RlY3RlZEc2JC8qKF4jIiIiRilJJyZoYmFyO0c2IkYpLUklZGlmZkdGJDYkLUkwJmRlbHRhOyZ2YXJwaGk7R0YrNiZJInhHRiRJInlHRiRJInpHRiRJInRHRitGNUYpLCgqKEkiR0dGK0YpSSJuR0YrRilGL0YpRikqKEY5RilGOEYpLUkqY29uanVnYXRlR0YkNiNGL0YpRikqKEYqIiIjSSJtR0YrISIiLUkrJUxhcGxhY2lhbkdGK0Y9RikjRkFGPy9GLywmLUkidUdGKzYkRjJGNUYpLUkiKkc2JUYkL0krbW9kdWxlbmFtZUdGK0koUGh5c2ljc0c2JEYkSShfc3lzbGliR0YrL0kxaGlkZW1vZHVsZXByZWZpeEdGK0kldHJ1ZUdGJDYkRigtSSJ2R0YrRklGKQ==Compute now the Laplacian on the right-hand sideLUknZXhwYW5kRzYkL0kocGFja2FnZUclKnByb3RlY3RlZEdJKFBoeXNpY3NHNiRGJ0koX3N5c2xpYkc2IkYlNiMtSSZ2YWx1ZUdGKTYjLyooXiMiIiJGM0knJmhiYXI7R0YrRjMsJi1JJWRpZmZHRic2JC1JInVHRis2JEkieEdGJ0kidEdGK0Y9RjMqJkYyRjMtRjc2JC1JInZHRitGO0Y9RjNGM0YzLCgqKEkiR0dGK0YzSSJuR0YrRjMsJkY5RjMqJkYyRjNGQUYzRjNGM0YzKihGRkYzRkVGMywmRjlGMyomXiMhIiJGM0ZBRjNGM0YzRjMqKEY0IiIjSSJtR0YrRk0tSSslTGFwbGFjaWFuR0YrNiNGR0YzI0ZNRk8=All the objects in this equation are real, so represents two equations, respectively conformed by its real and imaginary partsLUknZXhwYW5kRzYkL0kocGFja2FnZUclKnByb3RlY3RlZEdJKFBoeXNpY3NHNiRGJ0koX3N5c2xpYkc2IkYlNiMtSSNSZUdGJzYjLywmKiheIyIiIkY0SScmaGJhcjtHRitGNC1JJWRpZmZHRic2JC1JInVHRis2JEkieEdGJ0kidEdGK0Y9RjRGNComRjVGNC1GNzYkLUkidkdGK0Y7Rj1GNCEiIiwoKihJIkdHRitGNEY5RjRJIm5HRitGNCIiIyooRjVGSEkibUdGK0ZDLUY3NiQtRjc2JEY5RjxGPEY0I0ZDRkgqKl4jRk9GNEY1RkhGSkZDLUY3NiQtRjc2JEZBRjxGPEY0RjQ=LUknZXhwYW5kRzYkL0kocGFja2FnZUclKnByb3RlY3RlZEdJKFBoeXNpY3NHNiRGJ0koX3N5c2xpYkc2IkYlNiMtSSNJbUdGJzYjLywmKiheIyIiIkY0SScmaGJhcjtHRitGNC1JJWRpZmZHRic2JC1JInVHRis2JEkieEdGJ0kidEdGK0Y9RjRGNComRjVGNC1GNzYkLUkidkdGK0Y7Rj1GNCEiIiwoKihJIkdHRitGNEY5RjRJIm5HRitGNCIiIyooRjVGSEkibUdGK0ZDLUY3NiQtRjc2JEY5RjxGPEY0I0ZDRkgqKl4jRk9GNEY1RkhGSkZDLUY3NiQtRjc2JEZBRjxGPEY0RjQ=To obtain the dispersion relations, take the Fourier transforms of these two equations towards eliminating 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 or 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. The Fourier transform with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= gives:QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKWludHRyYW5zR0YlISIiLUkoZm91cmllckc2IjYlLywkKiZJJyZoYmFyO0dGJCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2R0YkNiRJInhHRi1JInRHRiRGM0YqISIiLCYqKEkiR0dGJEYqLUkidUdGJEYxRipJIm5HRiRGKiIiIyooRilGO0kibUdGJEY0LUYsNiQtRiw2JEY4RjJGMkYqI0Y0RjtGMkkia0dGJA==LUkoZm91cmllckc2IjYlLyomSScmaGJhcjtHRiQiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkidUdGJDYkSSJ4R0YsSSJ0R0YkRjJGKSwkKihGKCIiI0kibUdGJCEiIi1GKzYkLUYrNiQtSSJ2R0YkRjBGMUYxRikjRjdGNUYxSSJrR0YkEquation can be simplified using to eliminate 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 in favor of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJkZzY3I7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiktRi82JVEieEYnRjJGNS1JI21vR0YkNi1RKCZjb21tYTtGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRi82JVEia0YnRjJGNS8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lKXJlYWRvbmx5R0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY+Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkFGQy9GR0ZFRkhGSkZMRk5GUEZSL0ZWRlQtSShtZmVuY2VkR0YkNiQtRi82JVEidkYnRjJGNUZBRmVuRmhuRmpuRkE= (see simplify with respect to side relations)LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JS8qJkknJmhiYXI7R0YnIiIiLUklZGlmZkdGJTYkLUkoZm91cmllckdGJzYlLUkidUdGJzYkSSJ4R0YlSSJ0R0YnRjZJImtHRidGN0YsLCQqKkYrIiIjSSJtR0YnISIiRjhGOy1GMTYlLUkidkdGJ0Y1RjZGOEYsI0YsRjs8Iy8sJComRitGLC1GLjYkRj5GN0YsRj0sJCooLCYqJkYrRjtGOEY7RiwqKEkiR0dGJ0YsRjxGLEkibkdGJ0YsIiIlRixGPEY9RjBGLEZCPCNGMA==The equation above now involves only 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. This dependency can also be eliminated by taking another fourier transform of both sides of , now with respect to t, introducing the frequency 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We see now that the double Fourier transform is a common factor of the left and right-hand sides. The desired dispersion relation for LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiw2JVEia0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= is then obtained just isolating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjI2JVEia0YnL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtbkdGJDYkUSIyRidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSJeRidGOA==LUkoaXNvbGF0ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLywkKixJJyZoYmFyO0dGJyIiIywmKiZGLEYtSSJrR0YnRi0iIiIqKEkiR0dGJ0YxSSJtR0YnRjFJIm5HRidGMSIiJSEiIkY0RjEmSSZvbWVnYUdGJzYjRjBGLS1JKGZvdXJpZXJHRic2JS1GPDYlLUkidkdGJzYkSSJ4R0YlSSJ0R0YnRkNGMEZERjhGMUYtLCQqKkYsRi1GNEY3RjBGLUY7RjEjRjFGLSokRjhGLQ==Alternatively, a 3+1 spacetime treatment is also possible treating LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR0YuLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= and 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 as independent perturbation functions, introducing them as a plane waves 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 To reuse u and v this time as non-real functions of t, unset 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 and 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 used as real objects in the previous approachLUkmU2V0dXBHNiI2Iy9JLHJlYWxvYmplY3RzR0YkPCVJJnVuc2V0R0YkLUkidUdGJDYkSSJ4RyUqcHJvdGVjdGVkR0kidEdGJC1JInZHRiRGLA==LV9JKVBERXRvb2xzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSShkZWNsYXJlR0YoNiMtNiRJInVHRihJInZHRig2I0kidEdGKA==We use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInVGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GSTYlUSJ0RidGTEZPRi9GL0YrRi8= for LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GSTYlUSJ0RidGTEZPRi9GL0YrRi8= for 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L0kjcl9HNiIsKC1JIipHNiUlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSShQaHlzaWNzRzYkRilJKF9zeXNsaWJHRiQvSTFoaWRlbW9kdWxlcHJlZml4R0YkSSV0cnVlR0YpNiRJInhHRilJI19pR0YpIiIiLUYnNiRJInlHRilJI19qR0YpRjUtRic2JEkiekdGKUkjX2tHRilGNQ==where 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 and 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 are realLUkmU2V0dXBHNiI2Iy9JLHJlYWxvYmplY3RzR0YkPChJI2tfR0YkSSJuR0YkSSNyX0dGJCZJImtHRiQ2I0kieEclKnByb3RlY3RlZEcmRi02I0kieUdGMCZGLTYjSSJ6R0YwIntroducing now the form for the perturbations and , expanding the scalar product and evaluating the Laplacian on the right-hand side, we have: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In the equation above, the 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 is now a common factor. Isolating the time derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInVGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GSTYlUSJ0RidGTEZPRi9GL0YrRi8=LUknbm9ybWFsRyUqcHJvdGVjdGVkRzYjLUkoaXNvbGF0ZUc2JEYkSShfc3lzbGliRzYiNiQvKipeIyIiIkYvSScmaGJhcjtHRipGLy1JJWRpZmZHRiQ2JC1JInVHRio2I0kidEdGKkY3Ri8tSSRleHBHRig2IyomRi5GLywoKiZJInhHRiRGLyZJImtHRio2I0Y+Ri9GLyomSSJ5R0YkRi8mRkA2I0ZDRi9GLyomSSJ6R0YkRi8mRkA2I0ZHRi9GL0YvRi8sKCoqSSJHR0YqRi9JIm5HRipGL0Y0Ri9GOEYvRi8qKkZNRi9GTEYvLUkidkdGKkY2Ri9GOEYvRi8qLEYwIiIjSSJtR0YqISIiRjRGL0Y4Ri8sKCokRj9GUkYvKiRGREZSRi8qJEZIRlJGL0YvI0YvRlJGMQ==Introducing 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, LUkkbWFwRyUqcHJvdGVjdGVkRzYkZio2I0kidUc2IkYpNiRJKW9wZXJhdG9yR0YpSSZhcnJvd0dGKUYpKiQtSSVOb3JtR0YpNiM5JCIiI0YpRilGKS9JI2tfR0YpLCgqJkkjX2lHRiQiIiImSSJrR0YpNiNJInhHRiRGOEY4KiZJI19qR0YkRjgmRjo2I0kieUdGJEY4RjgqJkkjX2tHRiRGOCZGOjYjSSJ6R0YkRjhGOA==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An equation for the time derivative of v(t) can be obtained, for instance, by taking the conjugate of and noting that, from and , 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 and 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 , or, for simplicity, just repeating the steps after , this time departing from the equation for 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Taken together, the 1st order linear differential equations for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInVGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GSTYlUSJ0RidGTEZPRi9GL0YrRi8= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GSTYlUSJ0RidGTEZPRi9GL0YrRi8=, equations and can be rewritten as a matrix equation can be rewritten: 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 whereQyQ+SSJNRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMvSSQlaWRHRiUiNW10KyNSeVN1WSU9ISIiand we are using 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. This DE system can be solved by passing it to dsolve, and from there conclude about the dispersion relations by inspection, or one can directly get the dispersion relations noting that, by definition, 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, where 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 is the energy, in turn equal to the eigenvalues of M, soLyZJKGVwc2lsb25HNiI2I0kia0dGJS1fSS5MaW5lYXJBbGdlYnJhRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlSSxFaWdlbnZhbHVlc0dGJTYjSSJNR0YlLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=Taking into account that 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, this result is the same one obtained using Fourier transforms .LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==ReferencesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==[1] Gross-Pitaevskii equation (wiki)[2] Continuity equation (wiki)
[3] Bose\342\200\223Einstein condensate (wiki)[4] Dispersion relations (wiki)[5] Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.