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R. Grimshaw D. Pelinovsky, E. Pelinovsky and T. Talipova

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Wave group dynamics in weakly nonlinear long-wave models

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Physica D, 159, 35-57 (2001)

**Abstract:**

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.

**Keywords:**

NONLINEAR SCHRODINGER EQUATION, EXTENDED KORTEWEG-DE VRIES
EQUATION, WAVE GROUP DYNAMICS, ALGEBRAIC SOLITON, MODULATIONAL INSTABILITY