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  "body": "\n## Summary\n\nThe nonlinear Schrödinger equation is the [[Schrödinger equation|EPZR6H2WJN]] for a complex wave $u$, with an added term that depends on the size $\\abs{u}$.\nIts most studied case is the cubic equation $i \\partial_t u + \\Delta u = \\pm \\abs{u}^2 u$.\nIt describes slowly varying wave packets in optics, water waves and Bose–Einstein condensates.\nIt is a model problem for dispersive equations, and in one space dimension the cubic case is completely integrable.\n\n## Explanations\n\n### L1 human\n\nA wave packet spreads out as it moves, because waves of different frequencies travel at different speeds.\nIn some media the speed also depends on how strong the wave is, and this adds a nonlinear term.\nIf the nonlinearity pulls the packet together while dispersion spreads it out, the two effects can balance.\nThen a solitary wave forms that keeps its shape, such as $u(x,t) = \\sqrt{2}\\,\\mathrm{sech}(x)\\, e^{it}$ for the focusing cubic equation on the line.\nLight pulses in optical fibres behave like this.\n",
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