Bilinear-form and similarity-reduction visit to a variable-coefficient generalized dispersive water-wave system concerning Acta Mech. 233, 2527 and 233, 2415

Abstract

Getting enthusiastic from two Acta Mech. water-wave advances, this Letter proposes to visit a variable-coefficient generalized dispersive water-wave system for certain long-weakly nonlinear and weakly-dispersive surface waves of variable depth in a shallow water. In regard to the height modeling the deviation from the equilibrium position of the water and to the surface velocity of water waves along a horizontal direction, we symbolically compute out the following: (1) bilinear forms, the same as those published but by way of a different method; (2) similarity reductions, each of which to a known ordinary differential equation. Results are related to the fluid density, velocity and viscosity.

Appendix: Remark 3 in Ref. [44]

In line with Assumptions (19), we brief Remark 3 in Ref. [44] as follows:

Without loss of generality, there exist 3 freedoms in the resolution of $\theta$, $\omega$, z and r, which could help people arrange such a procedure:

• If $\theta (x,t)$ possesses the manner $\theta (x,t)=\theta _0(x,t)+\omega (x,t)\Upsilon (z)$, people could adopt $\Upsilon \equiv 0$, by replacing $r(z)\rightarrow r(z)-\Upsilon (z)$;

• If $\omega (x,t)$ possesses the manner $\omega (x,t)=\omega _0(x,t)\Upsilon (z)$, people could adopt $\Upsilon \equiv 1$, by replacing $r(z)\rightarrow r(z)/\Upsilon (z)$;

• If z(x, t) is decided by an equation of the manner $\Upsilon (z)=z_0(x,t)$ with $\Upsilon (z)$ as an invertible function, people could adopt $\Upsilon (z)=z$, by replacing $z\rightarrow \Upsilon ^{-1}(z)$.

Details can be seen in Ref. [44] and references therein.