The global solution of an initial boundary value problem for the damped Boussinesq equation

On Compact and Noncompact Structures for the Improved Boussinesq Water Equations

“…Remark 3. From the formulae (37) to (46), we can conclude that the exponent , the wave numbers and , and the coefficients 1 , 2 , 1 , and 2 appearing in Boussinesq equation (5) or (6) are the main factors which determine the change of physical structures of solutions. More precisely, exponent and number ( 1 2 + 2 2 )/( 1 2 + 2 2 ), positive or negative, directly result in the qualitative changes in the physical structures of solutions.…”

“…(II) Variant of the (2+1) dimensional improved Boussinesq equation with positive exponents (some typing's errors occurred for (5) and (6) in [12]. This can be found through the reading of the whole paper) is given by…”

“…where 1 + 2 ̸ = 0 and 1 + 2 ̸ = 0. Wazwaz [12] assumed that the solutions of (3), (4), (5), and (6) take the forms in terms of sine or cosine; namely,…”

“…[1] discussed and analyzed new exact traveling wave solutions for the two-dimensional KdV-Burgers and Boussinesq equations. Lai and Wu [4] and Lai et al [5] proposed an approach for constructing asymptotic solution of the Boussinesq equations. Rosenau [6] proved that the vibration of the anharmonic mass-spring chain leads to a new Boussinesq equation admitting compactons and compact breathers.…”

“…One can find that many meaningful achievements that outline the structure of compactons and the collision of two compactons were reported in [6][7][8]. For more details about the compactons and soliton solutions for nonlinear evolution equations, the reader is referred to [4,5,[9][10][11] in which many analytical and numerical methods such as the pseudospectral methods, the Galerkin method, the finite differences method, the sinecosine ansatz, and the tanh method have been presented.…”