periodic solutions

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1: 28.30 Expansions in Series of Eigenfunctions

… ►Let ${\widehat{\lambda }}_m$, $m=0,1,2,\mathrm{\dots }$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_m(x)$, $m=0,1,2,\mathrm{\dots }$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus …

2: 28.29 Definitions and Basic Properties

… ►The solutions of period $\pi$ or $2\pi$ are exceptional in the following sense. If (28.29.1) has a periodic solution with minimum period $n\pi$, $n=3,4,\mathrm{\dots }$, then all solutions are periodic with period $n\pi$. … ►The $\pi$-periodic or $\pi$-antiperiodic solutions are multiples of $w_{\text{I}}(z,\lambda ),w_{\text{II}}(z,\lambda )$, respectively. … ►In consequence, (28.29.1) has a solution of period $\pi$ iff $\lambda ={\lambda }_n$, and a solution of period $2\pi$ iff $\lambda ={\mu }_n$. …

3: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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§28.5(i) Definitions

… ►If a nontrivial solution of Mathieu’s equation with $q\ne 0$ has period $\pi$ or $2\pi$, then any linearly independent solution cannot have either period. … ►As a consequence of the factor $z$ on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as $z\to \pm \mathrm{\infty }$ on $ℝ$. …

4: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►The separated solutions $V_n(\xi ,\eta )$ must be $2\pi$-periodic in $\eta$, and have the form … ►For points $(q,a)$ that are at intersections of $ℒ$ with the characteristic curves $a=a_n\left(q\right)$ or $a=b_n\left(q\right)$, a periodic solution is possible. …

5: 28.3 Graphics

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Even $\pi$-Periodic Solutions

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Odd $\pi$-Periodic Solutions

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6: 21.9 Integrable Equations

… ►The KP equation has a class of quasi-periodic solutions described by Riemann theta functions, given by …

7: 28.31 Equations of Whittaker–Hill and Ince

… ►Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: … ►All other periodic solutions behave as multiples of $\exp \left(\frac{1}{4}\xi \cos \left(2z\right)\right){(\cos z)}^{-p-2}$. … ►All other periodic solutions behave as multiples of $\exp \left(-\frac{1}{4}\xi \cos \left(2z\right)\right){\left(\cos z\right)}^p$.

8: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …

9: Bibliography K

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An algebraic-geometrical construction of the Zakharov-Shabat equations and their periodic solutions. Sov. Math. Doklady 17, pp. 394–397.

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Referenced by:

Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2.

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10: 28.32 Mathematical Applications

… ►The first is the $2\pi$-periodicity of the solutions; the second can be their asymptotic form. …