# Korteweg?de Vries equation

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##### 1: 30.2 Differential Equations
###### §30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
##### 2: 31.2 Differential Equations
###### §31.2(i) Heun’s Equation
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
##### 3: 29.2 Differential Equations
###### §29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
##### 4: 15.10 Hypergeometric Differential Equation
###### §15.10(i) Fundamental Solutions
15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
This is the hypergeometric differential equation. …
##### 5: 32.2 Differential Equations
###### §32.2(i) Introduction
The six Painlevé equations $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ are as follows: …
##### 6: 28.2 Definitions and Basic Properties
###### §28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
##### 7: 28.20 Definitions and Basic Properties
###### §28.20(i) Modified Mathieu’s Equation
When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:
28.20.8 $h=\sqrt{q}\;(>0).$
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to $\zeta^{\ifrac{1}{2}}e^{\pm 2\mathrm{i}h\zeta}$ as $\zeta\to\infty$ in the respective sectors $|\operatorname{ph}\left(\mp\mathrm{i}\zeta\right)|\leq\tfrac{3}{2}\pi-\delta$, $\delta$ being an arbitrary small positive constant. …
##### 8: 32.13 Reductions of Partial Differential Equations
###### §32.13(i) Korteweg–de Vries and Modified Korteweg–de VriesEquations
The modified Korteweg–de Vries (mKdV) equationThe Korteweg–de Vries (KdV) equation
##### 9: 29.19 Physical Applications
###### §29.19(ii) Lamé Polynomials
Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …Hargrave (1978) studies high frequency solutions of the delta wing equation. …Roper (1951) solves the linearized supersonic flow equations. Clarkson (1991) solves nonlinear evolution equations. …
##### 10: 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). …