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Nonhomogeneous Boundary Value Problems for the Kortewegde Vries and the Kortewegde VriesBurgers Equations in a Quarter Plane
, 2007
"... Attention is given to the initialboundaryvalue problems (IBVPs) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t) for the Kortewegde Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0, ..."
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Attention is given to the initialboundaryvalue problems (IBVPs) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t) for the Kortewegde Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0,
UNIFORM STABILIZATION IN WEIGHTED SOBOLEV SPACES FOR THE KDV EQUATION POSED ON THE HALFLINE
"... Abstract. Studied here is the largetime behavior of solutions of the Kortewegde Vries equation posed on the right halfline under the effect of a localized damping. Assuming as in [19] that the damping is active on a set (a0, +∞) with a0> 0, we establish the exponential decay of the solutions in ..."
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Abstract. Studied here is the largetime behavior of solutions of the Kortewegde Vries equation posed on the right halfline under the effect of a localized damping. Assuming as in [19] that the damping is active on a set (a0, +∞) with a0> 0, we establish the exponential decay of the solutions in the weighted spaces L 2 ((x + 1) m dx) for m ∈ N ∗ and L 2 (e 2bx dx) for b> 0 by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived. 1. Introduction. The Kortewegde Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel [8, 14, 15]. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a socalled nonlinear Fourier transform,
Kawahara equation in a quarterplane and in a finite domain
 Bol. Soc. Paran. Mat
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APPROXIMATING INITIALVALUE PROBLEMS WITH TWOPOINT BOUNDARYVALUE PROBLEMS: BBMEQUATION
, 2010
"... The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude, long waves in dispersive media. This evolution equation has been used in both laboratory and field studies of water waves. The principal new result is an exact theory of convergence of the ..."
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The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude, long waves in dispersive media. This evolution equation has been used in both laboratory and field studies of water waves. The principal new result is an exact theory of convergence of the twopoint boundaryvalue problem to the initialvalue problem posed on an infinite stretch of the medium of propagation. In addition to their intrinsic interest, our results provide justification for the use of the twopoint boundaryvalue problem in numerical studies of the initialvalue problem posed on the entire line.
Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the halfline
, 2010
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Wellposedness of a Class of Nonhomogeneous Boundary Value Problems of the Kortewegde Vries Equation on a Finite Domain
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