ABSTRACT
In 1895, Korteweg and de Vries introduced the famous third-order KdV equation of shallow water waves
vt = vx x x + 2vvx in IR × IR (4.1) and its explicit soliton
, (4.2)
which is the traveling wave solution moving to the left with unit velocity.∗ This and other multi-soliton solutions play a determining role in general water waves theory and theory of integrable PDEs. By scaling x → λx , t → λ3t , and v → Cv, (4.1) reduces to vt = vx x x + 2Cλ2vvx and takes standard forms
vt = vx x x + vvx for C = 12λ2 , or vt = vx x x + 6vvx for C = 3λ2 .
