# solutions

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##### 1: 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). Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).##### 2: 31.13 Asymptotic Approximations

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►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999).
►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

##### 3: 31.18 Methods of Computation

###### §31.18 Methods of Computation

►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …##### 4: 16.25 Methods of Computation

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►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions.
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##### 5: 29.17 Other Solutions

###### §29.17 Other Solutions

►###### §29.17(i) Second Solution

►If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by …For properties of these solutions see Arscott (1964b, §9.7), Erdélyi et al. (1955, §15.5.1), Shail (1980), and Sleeman (1966b). … ►Algebraic Lamé functions are solutions of (29.2.1) when $\nu $ is half an odd integer. …##### 6: 32.13 Reductions of Partial Differential Equations

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►The

*Korteweg–de Vries*(KdV) equation … ►The*sine-Gordon*equation … ►The*Boussinesq*equation …has the traveling wave solution …where $c$ is an arbitrary constant and $v(z)$ satisfies …##### 7: 28.29 Definitions and Basic Properties

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►Then (28.29.1) has a nontrivial solution
$w(z)$ with the pseudoperiodic property
…A solution satisfying (28.29.7) is called a

*Floquet solution with respect to*$\nu $ (or*Floquet solution*). … … ► … ► …##### 8: 1.13 Differential Equations

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###### §1.13(i) Existence of Solutions

… ►###### Fundamental Pair

… ► … ►###### §1.13(v) Products of Solutions

… ►###### §1.13(vii) Closed-Form Solutions

…##### 9: 31.6 Path-Multiplicative Solutions

###### §31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by …This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the $z$-plane that encircles ${s}_{1}$ and ${s}_{2}$ once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$. These solutions are called*path-multiplicative*. …

##### 10: 9.15 Mathematical Applications

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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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