Multiple-scale analysis: Difference between revisions

Revision as of 07:55, 31 January 2024

Revision as of 07:56, 31 January 2024

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<math display="block">\left[ -3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} \right]\, e^{+it} - A^3\, e^{+3it} + c.c.</math>

where ''c.c.'' denotes the [[complex conjugate]] of the preceding terms. The occurrence of ''secular terms'' can be prevented by imposing on the – yet unknown – amplitude ''A''(''t''₁) the ''solvability condition''

<math display="block">-3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} = 0.</math>,

i.e the amplitude equation:

<math display="block"> i \frac{dA}{dt_1} = - \frac{3}{2} |A|^2 A.</math>

The solution to the solvability condition, also satisfying the initial conditions {{math|1=''y''(0) = 1}} and {{math|1=''dy''/''dt''(0) = 0}}, is: