Wave group dynamics in weakly nonlinear long-wave models

The dynamics of wave groups is studied for long waves, using the framework of the extended Korteweg–de Vries equation. It is shown that the dynamics is much richer than the corresponding results obtained just from the Korteweg–de Vries equation. First, a reduction to a nonlinear Schrödinger equation is obtained for weakly nonlinear wave packets, and it is demonstrated that either the focussing or the defocussing case can be obtained. This is in contrast to the corresponding reduction for the Korteweg–de Vries equation, where only the defocussing case is obtained. Next, the condition for modulational instability is obtained. It is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term in the extended Korteweg–de Vries equation, and for a high carrier frequency. At the boundary of this parameter space, a modified nonlinear Schrödinger equation is derived, and its steady-state solutions, including an algebraic soliton, are found. The exact breather solution of the extended Korteweg–de Vries equation is analysed. It is shown that in the limit of weak nonlinearity it transforms to a wave group with an envelope described by soliton solutions of the nonlinear Schrödinger equation and its modification as described above. Numerical simulations demonstrate the main features of wave group evolution and show some differences in the behaviour of the solutions of the extended Korteweg–de Vries equation, compared with those of the nonlinear Schrödinger equation.