Abstract
The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.
Highlights
Considered here are small amplitude long waves on the surface of an ideal fluid of finite depth over a featureless, horizontal bottom under the force of gravity. When such wave motion is long crested, it may propagate essentially in, say, the x-direction and without significant variation in the y-direction of a standard xyzCartesian frame in which gravity acts in the negative z-direction
Let u be the solution of the quarter plane problem (2.1) for the BBM equation (2.1) where g is supposed to lie in ∈ C(I) with the compatibility condition g(0) = 0
For any fixed point (x, t) ∈ R+ × I, lim vL(x, t) u(x, t) where u is the solution of the quarter-plane problem (2.1)
Summary
Considered here are small amplitude long waves on the surface of an ideal fluid of finite depth over a featureless, horizontal bottom under the force of gravity. (For such motions, systems of equations are useful; see, for example, Bona, Chen and Saut 2002, 2004.) This point leads one to pose the problem for all x ≥ 0, placing the issue of a boundary condition at the right-hand end-point at ∞. There is available analytical theory for the two-point boundary value problem wherein (1.2) or (1.3) is posed on a finite spatial interval with an initial condition and suitable boundary conditions. Notice that the difference u(x, t) − g(t)e−x, which according to (2.7) is equal to a double integral, is one order smoother than either u or g in the temporal variable t and is C∞ in the spatial variable x It has zero initial value and vanishes at x = 0 and in the limit as x → ∞, for any t > 0.
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