It is analytically shown how competing nonlinearities yield multiscaled structures for internal solitary waves in stratified shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the Korteweg–de Vries (KdV) equation or its usual extensions. The multiscaling phenomenon exists or does not exist for almost identical density profiles. The trapped core inside the wave prevents the appearance of such multiple scales within the core area. The structural stability of waves of large amplitudes is briefly discussed. Waves of large amplitudes displaying quadratic, cubic and higher-order nonlinear terms have stable and unstable branches. Multiscaled waves without a vortex core are shown to be structurally unstable. It is anticipated that multiscaling phenomena will exist for solitary waves in various physical contexts.

The typical horizontal scale (or scales) is a major characteristic of a plane
disturbance propagating in a nonuniform medium. Usually, in an ideal
density-stratified shallow fluid, a wave of small albeit finite amplitude has
one typical scale resulting from the (local) balance between nonlinearity and
dispersion like in the realm of the Korteweg–de Vries (KdV) equation

Let us consider the two-dimensional steady motion of an ideal
density-stratified fluid in a framework of a reference moving with the phase
speed of wave

In Eq. (2)

In order to (locally) balance nonlinearity and dispersion, we have to require

The Weierstrass approximation theorem states that every continuous function
defined on a closed interval can be uniformly approximated as closely as
desired by a polynomial function. A recent account of the topic is reviewed
in

For the wave of amplitude

The function

Two-humped solitary waves for the stratification given by Eq. (17) exist in the domain shown in Fig. 1.

Existence domain for the two-humped solitary wave.

The two-humped solitary wave with amplitude

Amplitude function and surface displacement for the
two-humped solitary wave;

Streamlines for the two-humped solitary wave:

Indeed, the maximum derivative on

Amplitude function for the three-humped solitary wave.

For this case the stratification profile is

Further, we wish to examine the structure of solitary waves of a permanent
form for the stratification given by Eq. (16). We only consider the case

Phase velocity versus wave amplitude. Solid line:

Figure 5 shows that the phase velocity is an increasing function for

Profiles of stable solitary waves are shown by solid lines. Dashed
lines correspond to

Such waves are structurally stable according to

For

Profiles of unstable solitary waves are shown by solid lines. Dashed
lines:

For the particular stratification considered here, waves are structurally
unstable

Energy versus wave amplitude. The dotted line corresponds to

The theory described above is valid for wave amplitudes below

Multiscaling effects similar to those discussed above could be observed in
various physical media.

For the particular case of a nonlinear dispersive medium such as a
density-stratified fluid, we have addressed multiscaled solitary waves which
are predicted when there exists competition of several different types of
nonlinearity. The mechanism leading to these solutions differs from the
mechanism of multiscaling due to the competition of different types of
dispersion or effects due to the dissipation. We have shown that the length
used to scale the

No data sets were used in this article.

The author declares that he has no conflict of interest. Edited by: Roger Grimshaw Reviewed by: Tatiana Talipova and one anonymous referee