Picard–Fuchs equations

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21: 31.12 Confluent Forms of Heun’s Equation

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Confluent Heun Equation

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Doubly-Confluent Heun Equation

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Biconfluent Heun Equation

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Triconfluent Heun Equation

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22: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by …In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

23: 9.16 Physical Applications

… ►A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … ►In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation). …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). ►Airy functions play a prominent role in problems defined by nonlinear wave equations. These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …

24: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). …

25: 32.12 Asymptotic Approximations for Complex Variables

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§32.12(i) First Painlevé Equation

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§32.12(ii) Second Painlevé Equation

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§32.12(iii) Third Painlevé Equation

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26: 28.17 Stability as $x\to \pm \mathrm{\infty }$

§28.17 Stability as $x\to \pm \mathrm{\infty }$

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Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify

27: 21.9 Integrable Equations

§21.9 Integrable Equations

… ►Typical examples of such equations are the Korteweg–de Vries equation …and the nonlinear Schrödinger equations … ► … ►

28: Mark J. Ablowitz

… ►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations. Certain nonlinear equations are special; e. …ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems. …

29: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M. …

30: 13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …