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Chapter S4: Special Functions and Their Properties
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
An elliptic function is a function that is the inverse of an elliptic integral. An elliptic function is a doubly periodic meromorphic function of a complex variable. All its periods can be written in the form 2mω1 + 2nω2 with integer m and n, where ω1 and ω2 are a pair of (primitive) half‐periods. The ratio τ = ω2/ω1 is a complex quantity that may be considered to have a positive imaginary part, Imτ>0. $ {\text{Im }}\tau> 0. $
On the stability of periodic traveling waves for the modified Kawahara equation
Published in Applicable Analysis, 2021
Gisele Detomazi Almeida, Fabrício Cristófani, Fábio Natali
Now, we present some contributors concerning the orbital stability of explicit periodic/solitary waves related to the generalized Kawahara equation where is an integer. In fact, for the case p = 1, the authors in [2] established the orbital stability of explicit periodic traveling waves solution of the form where a, b and d are real parameters. Here represents the Jacobi elliptic function of dnoidal type, is the complete elliptic integral of the first kind, is the complete elliptic integral of the second kind and both of them depend on the elliptic modulus (see [3] for additional details). The method used in [2] to obtain the stability was an adaptation of the method in [4].
Are all KdV-type shallow water wave equations the same with uniform solutions?
Published in Coastal Engineering Journal, 2020
where is the modulus of Jacobian elliptic function, is the complete elliptic integral of the first kind and is the second kind. Since the water depth below wave trough changes with wave properties, it is more reasonable to use the still water depth for nondimensionalization process. Figure 2 shows the relationship between the dimensionless water depth below the wave trough and the Ursell number for three representative relative wave heights. The maximum relative wave height is taken the same as the limiting case in Wiegel (1960), . In Figure 2, increases with the increase of , which denotes the extent of wave nonlinearity. Specifically, when approaches infinite, all three lines with different values are asymptotic to unity, indicating a solitary wave condition. Nevertheless, if approaches zero, the following relation holds
On the approximate and analytical solutions to the fifth-order Duffing oscillator and its physical applications
Published in Waves in Random and Complex Media, 2021
Alvaro H. Salas, S. A. El-Tantawy, Castillo H. Jairo E
Let us assume that Equation (7) admits analytic solution in terms of the Jacobi elliptic functions sn and cn or through the famous Weierstrass elliptic function. It may be proved that the analytic solution has one of the following famous formulas: or where ω, m, λ and μ represent some constants. Note that for cn and sn for every t.