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  "body": "\n## Definition\n\nLet $d \\ge 1$ be an integer, let $p \u003e 1$ be a real number, and let $\\mu \\in \\set{1, -1}$.\nThe *nonlinear Schrödinger equation* with power nonlinearity is the [[partial differential equation|GC5BDZ6QBC]]\n$$i \\partial_t u + \\Delta u = \\mu \\abs{u}^{p-1} u$$\nfor an unknown function $u \\colon \\R \\times \\R^d \\to \\C$, $(t, x) \\mapsto u(t, x)$.\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] in the variable $x$.\nThe equation is *defocusing* if $\\mu = 1$ and *focusing* if $\\mu = -1$.\nThe case $p = 3$ is the *cubic* nonlinear Schrödinger equation.\n\n## Remarks\n\nExample: for $A \\in \\C$ and $k \\in \\R^d$, the plane wave $u(t,x) = A e^{i(k \\cdot x - \\omega t)}$ is a solution when $\\omega = \\abs{k}^2 + \\mu \\abs{A}^{p-1}$.\nIndeed $i \\partial_t u = \\omega u$, $\\Delta u = -\\abs{k}^2 u$ and $\\abs{u}^{p-1} u = \\abs{A}^{p-1} u$.\n\nNon-example: the linear [[Schrödinger equation|EPZR6H2WJN]] $i \\partial_t u + \\Delta u = 0$ corresponds to $\\mu = 0$, which the definition excludes.\n\nSmooth solutions that decay at infinity conserve the mass $\\int \\abs{u}^2 \\, dx$ and the energy\n$$E(u) = \\int_{\\R^d} \\left( \\tfrac{1}{2} \\abs{\\nabla u}^2 + \\tfrac{\\mu}{p+1} \\abs{u}^{p+1} \\right) dx.$$\nSome authors write the equation as $i \\partial_t u = -\\Delta u + \\mu \\abs{u}^{p-1} u$, which is the same equation.\n",
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