By Mark J. Ablowitz
The sphere of nonlinear dispersive waves has constructed greatly because the paintings of Stokes, Boussinesq and Korteweg-de Vries (KdV) within the 19th century. within the Nineteen Sixties, researchers constructed potent asymptotic equipment for deriving nonlinear wave equations, reminiscent of the KdV equation, governing a huge category of actual phenomena that admit particular strategies together with these generally known as solitons. This e-book describes the underlying approximation options and techniques for locating recommendations to those and different equations. The techniques and techniques coated contain wave dispersion, asymptotic research, perturbation concept, the tactic of a number of scales, deep and shallow water waves, nonlinear optics together with fiber optic communications, mode-locked lasers and dispersion-managed wave phenomena. such a lot chapters function workout units, making the publication appropriate for complex classes or for self-directed studying. Graduate scholars and researchers will locate this an exceptional access to a thriving zone on the intersection of utilized arithmetic, engineering and actual technological know-how.
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Additional info for Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons
Example text
When the string is plucked we can approximately describe its vertical displacement from equilibrium at each point x as a function y(x, t). 11). We will assume no external forces and that horizontal acceleration is negligible. It follows from Newton’s second law that • difference in vertical tensions: (T sin θ)| x+Δx − (T sin θ)| x = Δm ytt , • difference in horizontal tensions: (T cos θ)| x+Δx − (T cos θ)| x = 0. 10 Vibrations of a long string. 11 Tension forces acting on a small segment of the string.
9 Introduction Hint: Set ξ = x − ct, integrate twice with respect to ξ, and use uξ = q(u) to solve the resulting equation. Consider the sine–Gordon equation u xx − utt = sin u. (a) Using the transformation χ = γ(x − vt), τ = γ(t − vx) write the equation in terms of the new coordinates χ, τ; find γ in terms of v, −1 < v < 1 so that the equation is invariant under the transformation. (b) Consider the transformation ξ = (x + t)/2, η = (x − t)/2. Find the equation in terms of the new coordinates ξ, η.
8) 48 Asymptotic analysis of wave equations √ where A(x/t)/ t is the decaying in time, slowly varying, complex amplitude. √ This decay rate, O(1/ t), is slow, especially when compared to those cases when the solution decays exponentially or as O(1/tn ) for n large. Note that dimensionally, ω corresponds to a speed. Stationary phase has shown that the leading-order contribution comes from a region moving with speed ω , which is termed the group velocity. This is the velocity of a slowly varying packet of waves.