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modified Korteweg?de Vries equation

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1: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ ( a 2 q cosh ( 2 z ) ) w = 0 ,
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
§28.20(vi) Wronskians
2: 30.2 Differential Equations
§30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
§30.2(iii) Special Cases
3: 31.2 Differential Equations
§31.2 Differential Equations
§31.2(i) Heun’s Equation
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
4: 29.2 Differential Equations
§29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
5: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
15.10.1 z ( 1 z ) d 2 w d z 2 + ( c ( a + b + 1 ) z ) d w d z a b w = 0 .
This is the hypergeometric differential equation. …
6: 32.2 Differential Equations
§32.2 Differential Equations
§32.2(i) Introduction
The six Painlevé equations P I P VI  are as follows: …
§32.2(ii) Renormalizations
7: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
8: 32.13 Reductions of Partial Differential Equations
§32.13 Reductions of Partial Differential Equations
§32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations
The modified Korteweg–de Vries (mKdV) equationThe Korteweg–de Vries (KdV) equation
§32.13(iii) Boussinesq Equation
9: 10.29 Recurrence Relations and Derivatives
§10.29(i) Recurrence Relations
With 𝒵 ν ( z ) defined as in §10.25(ii), … For results on modified quotients of the form z 𝒵 ν ± 1 ( z ) / 𝒵 ν ( z ) see Onoe (1955) and Onoe (1956).
§10.29(ii) Derivatives
10.29.5 𝒵 ν ( k ) ( z ) = 1 2 k ( 𝒵 ν k ( z ) + ( k 1 ) 𝒵 ν k + 2 ( z ) + ( k 2 ) 𝒵 ν k + 4 ( z ) + + 𝒵 ν + k ( z ) ) .
10: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I ν ( z ) } = I ν ( z ) I ν 1 ( z ) I ν + 1 ( z ) I ν ( z ) = 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .