Abstract
The completely integrable case of the nonlinear Bousinesq abcd system describing 1+1-dimensional weakly nonlinear and weakly nondispersive model for gravity and capillary waves is developed. The case of gravity and surface-tension solutions for the wave elevation and transverse parcel velocity on the surface of the fluid is examined in detail. The total wave energy of the system is written in Hamiltonian canonical variables and can be used to generate the dynamics in this model from a Poisson bracket. The Poisson bracket and dynamics is defined explicitly using the calculus of variations. The energy is the first conservation law obtained immediately from the Hamiltonian formulation. An infinite number of additional conservation laws are derived using a matrix Lax pair formulation of the system. Analytical soliton solutions (a kind of travelling wave) are derived and are combined in nonlinear superposition via the nonlocal dbar dressing method. The forward and inverse scattering transform (IST) for this system is developed and applied to the calculations of analytical multi-soliton solutions, periodic solutions, and undular bore like solutions. The capabilities of this method are demonstrated via various numerical solutions with a variety of boundary conditions. This nonlinear system and other well-known nonlinear wave models such as the KdV equation are compared. Validation of this system for the description of head on collision of solitons in an experimental flume was conducted in 2019 by Redor et. al. is discussed. Potential practical engineering applications of this system are examined and discussed.
