Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

By Jerry L. Bona, Min Chen, Jean-Claude Saut. Journal of nonlinear science, 2002

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Numerical Methods for Evolutionary Differential Equations (2008, unknown): “32a) ηttx ≈ −utxx , so another nearby Boussinesq system is [180, 29, 138] ηt + ux + (uη)x = 0, (7.”
Long Wave Approximations for Water Waves (2005, methodology): “1 6 )(1 ) (2.5) as in [ 5 ] in the two dimensional case.”
CONTROL AND STABILIZATION OF A FAMILY OF BOUSSINESQ SYSTEMS (2009, methodology): “In this paper, we will focus on the system (1.1) posed on a finite interval (0,L) with periodic boundary conditions imposed on and w. As in [ 5 ], we will restrict our attention to system (1.1) with its parameters a, b, c, d satisfying one of the assumptions below.”
Sharp well-posedness results for the BBM equation (2008, methodology): “The argument for local well-posedness in L 2 follows the analysis of Boussinesq systems by Bona, Chen and Saut [6] [7].”
Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation (2009, background): “Madsen et al. [14], Nwogu [15], Sch¨aer and Madsen [16] or Bona et al. [ 17 , 18]).”
Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory (2004, methodology): “This paper is a continuation of our earlier work [11], where we put forward a formal derivation of the four-parameter system ηt + ux + (uη)x + auxxx − bηxxt = 0, ut + ηx + uux + cηxxx − duxxt = 0 (1.”