Wave group dynamics in weakly nonlinear long-wave models

Wave group dynamics is studied in the framework of the extended Korteweg-de Vries equation. The nonlinear Schrodinger equation is derived for weakly nonlinear wave packets, and 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 Schrodinger 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 analyzed. 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 Schrodinger equation and its modification as described above. Numerical simulations demonstrate the main features of wave group evolution and show some differences in the behavior of the solutions of the extended Korteweg-de Vries equation, compared with those of the nonlinear Schrodinger equation.