Abstract
This paper focuses on the mathematical study of the existence of solitary gravity waves (solitons) and their characteristics (amplitude, velocity, ldots ) generated by a piston wave maker lying upstream of a horizontal channel. The mathematical model requires both incompressibility condition, irrotational flow of no viscous fluid and Lagrange coordinates. By using both the inverse scattering method and a given initial potential f_{0}(r), we can transform the KdV equation into Sturm–Liouville spectral problem. The latter problem amounts to find negative discrete eigenvalues lambda and associated eigenfunctions psi , where each calculated eigenvalue lambda gives a soliton and the profile of the free surface. For solving this problem, we can use the Runge–Kutta method. For illustration, two examples of the wave maker movement are proposed. The numerical simulations show that the perturbation of wave maker with hyperbolic tangent displacement under physical conditions affect the number of solitons emitted.
Highlights
Interest in nonlinear wave propagation has grown rapidly during the last three decades and has gained considerable attention in engineering and applied mathematics
We transform the Eqs. (3)–(8) into KdV equation (1), for this we introduce distortion variables which express the assumption of shallow water and asymptotic profile of wave respectively
Remark 4 To check the validity of our results, it is insightful to compare the obtained soliton solutions with localized pulses propagating in other nonlinear media such as optical waveguides
Summary
Interest in nonlinear wave propagation has grown rapidly during the last three decades and has gained considerable attention in engineering and applied mathematics. (3)–(8) into KdV equation (1), for this we introduce distortion variables which express the assumption of shallow water and asymptotic profile of wave respectively. This distortion will be characterized by using a small parameter ε as follows: α = εa, β = b and τ = ε ght, where gh represents the critical celerity of the propagated long waves, h and g are the depth of fluid at rest and the gravity respectively.
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