1. Introduction
It is well known that modulation instability, that is, the exponential growth of long wave perturbations to a periodic plane wave, leads to the formation of nonlinear wave packets, and sometimes to rogue waves. This process is often modelled by the nonlinear Schrödinger equation (NLS), and then the nonlinear wave packets can be represented by the soliton and breather solutions of the NLS, while rogue waves are often modelled by the Peregrine breather (see for instance Kharif et al. [
1], Osborne [
2], Grimshaw and Tovbis [
3]) in the deep-water wave context. The process occurs in many other physical contexts (see Grimshaw et al. [
4], Chow et al. [
5]) for internal wave applications, and the related articles in that special issue for other cases. In this paper, we develop the formulation in the water wave context to be specific, but the outcome can be applied to many other physical contexts.
In the pioneering work of Benney and Newell [
6], Zakharov [
7], Hasimoto and Ono [
8], the NLS equation for water waves is derived by using a multi-scale asymptotic expansion, in which the leading order nonlinear terms are balanced by weak linear dispersion around the dominant carrier wavenumber of the wave packet (see the review by Grimshaw [
9]). A wave packet in one horizontal space dimension is given by
Here
is the water surface elevation above the undisturbed depth
h, and
k is the carrier wavenumber, while the wave frequency
satisfies the linear dispersion relation.
is the slowly varying wave amplitude, and at the leading order the wave packet moves with the group velocity
.
,
is a small, dimensionless parameter measuring the wave amplitude and dispersion about the dominant wavenumber
k. The leading order omitted terms in (
1) are
second harmonic and mean flow terms.
A multi-scale asymptotic expansion in
in which the linear dispersive effects are scaled to balance the leading order nonlinear effects leads to the NLS equation,
The coefficient
of the nonlinear term is given by
Note that due to the balance of terms, the small parameter
can be omitted in (
4). In deep water (
), the second term vanishes, and the coefficient
. In general,
according as
, where
. Modulation instability occurs when
. For water waves
and so modulation instability occurs for waves in deep water when
,
. Similar expansions apply in many other physical systems, again leading to the NLS Equation (
4). The main difference is the linear dispersion relation (
3) and in the expressions for the coefficients
(see Grimshaw [
9], Akhmediev and Pelinovsky [
10] for instance). We note that in this water wave context, the wave amplitude from (
1) is
and is required to be small since
but
itself does not appear explicitly in (
4).
In this paper, we are concerned with the effect of forcing on modulation instability. We model this by extending the NLS Equation (
4) to a forced NLS equation (fNLS) by the addition of a linear forcing term (see for instance Leblanc [
11], Touboul et al. [
12], Montalvo et al. [
13], Brunetti et al. [
14], Slunyaev et al. [
15], Grimshaw [
16], Grimshaw [
17], Grimshaw [
18]) in the wind wave context,
The forcing is modelled by the linear growth rate term with coefficient
. Various expressions can be found in the literature, the most well-known being that originally derived by Miles [
19] and subsequently adapted and modified in various ways (see for instance Grimshaw [
16], Miles [
20], Morland and Saffman [
21], Janssen [
22], Stiassnie et al. [
23], Sajjadi et al. [
24], Zakharov et al. [
25]). Here our concern is with the effect of
on modulation instability and wave packet, or breather, formation. The effect of forcing on modulation instability has been examined in the present one space-dimension framework for deep water waves by Leblanc [
11], Touboul et al. [
12], Brunetti et al. [
14], Slunyaev et al. [
15], Grimshaw [
16]. Here we extend these studies, which were mostly concerned with the evolution of wave spectra, by focusing on the development of wave packets through comprehensive numerical simulations of the fNLS Equation (
6). Validation of the NLS simulations concerning modulation instability and the formation of rogue waves in the water wave tank, is described by Chabchoub et al. [
26]. A similar study of the Peregrine breather over zero background was recently given by Chabchoub et al. [
27]. Some numerical simulations based on a higher order NLS equation were conducted and compared with experimental studies to validate the numerical simulations of modulation instability and rogue waves (see Onorato et al. [
28] and León and Osborne [
29]). Here we have performed numerical simulations by adding a forcing term to model the formations of solitons and breathers under external forcing. The formulation of the problem is presented in
Section 2. In
Section 3, we present these numerical simulations and some accompanying analysis. We conclude in
Section 4.
2. Formulation
We consider the case when there is modulation instability, so that
(
). Then fNLS (
6) can be expressed in canonical form
This canonical form is achieved through the change of variables
Here we have introduced the free parameter
as it is useful to represent the scaling properties of the NLS equation. In the small
limit, an asymptotic procedure can be used to describe the generation of a family of Peregrine breathers from a modulated plane periodic wave (see [
3] for an application to water waves).
The fNLS Equation (
7) has the energy law
Here, if
in an infinite domain, then
must decay sufficiently fast at infinity; otherwise in a finite domain periodic boundary conditions are applied at
. The expression (
9) can be used to estimate the growth of the wave amplitude as explained in our previous work (see Grimshaw et al. [
30]). Briefly, if the absence of forcing the solution is
where
M is a free amplitude parameter, then substitution into (
9) yields an estimate for the growth
M under forcing. This is used here as a guide to interpreting each of the cases we consider. If the absence of forcing modulation instability can be measured by the Benjamin–Feir index (BFI), the ratio of wave steepness (nonlinearity) to spectral bandwidth (dispersion) and in the absence of forcing
(see Grimshaw and Tovbis [
3]). By using the change of variables
, it is readily shown that in the presence of forcing this becomes
.
The forced NLS Equation (
7) is solved numerically on the periodic domain
by using a Fourier spectral method in space and a Runge–Kutta approximation in time. More details can be found in Grimshaw and Maleewong [
31] that use a similar approach for numerically solving the forced Koreteweg–de Vries equation. Here we set
to minimise boundary truncation effects. With this periodic boundary condition, we choose modulation scales so that the solutions decay to the initial background at both ends of the domain well within numerical error. In most cases of the numerical simulations, we set the number of mesh points as 4096 and
= 5 × 10
, which satisfies numerical stability condition in the Fourier and time domains.
4. Discussion and Summary
In this paper, motivated by an application to wind waves, we have used the forced NLS Equation (
7) expressed in canonical form to model the generation of wave packets and breathers by adding a linear growth term to the usual NLS equation. In the absence of such forcing, the principal solutions of the NLS equation are solitons and breathers, representing wave packets and possibly rogue waves (see Kharif et al. [
1], Osborne [
2], Grimshaw and Tovbis [
3] for instance). In the forced NLS equation, the forcing is represented by a linear growth term with a rate parameter
so that
and
represents cases with and without forcing respectively (see Leblanc [
11], Touboul et al. [
12], Montalvo et al. [
13], Brunetti et al. [
14], Slunyaev et al. [
15], Grimshaw [
16] for the context of the generation of water waves by wind). In this context, the non-dimensional growth rate parameter
depends on several physical factors, especially the wind shear, the surface roughness, and the initial water wave wavelength. It can range from
for weak winds to
for strong winds (see Leblanc [
11], Touboul et al. [
12], Slunyaev et al. [
15] for instance). Here we have varied
over the range from zero to order unity, covering the range of weak to moderate forcing appropriate for our weakly nonlinear model.
Four scenarios are investigated through an appropriate choice of initial condition. These scenarios are (1) an initial condition which in the unforced case would generate a Peregrine breather (
10); (2) an initial condition which in the unforced case would generate a moving soliton (
12); (3) a slowly varying long wave perturbation which in the unforced case would generate either a few solitons for
of order unity, or a family of Peregrine breathers when
becomes very small; (4) a long-wave periodic perturbation which in the unforced case would generate modulation instability and the formation of both solitons and breathers.
In case 1, a Peregrine breather is formed when
and agrees with the well-known exact solution. When
, a forced Peregrine breather initially develops with an increased amplitude, growing at twice the linear growth rate, but instead of decreasing to zero, the amplitude continues to grow and oscillates with increasing frequency. In case 2, with
a steadily moving soliton with a constant amplitude forms. When
, the soliton amplitude grows at the rate
, twice the linear growth rate while continuing to move with a constant speed. In case 3, with
and with a very small dispersion parameter
, there is a gradient catastrophe followed by the formation of a family of Peregrine breathers as expected (see Grimshaw and Tovbis [
3]). When
, in contrast to the unforced case the Peregrine breathers are replaced by a mixture of breathers and solitons. Three scenarios were found, the generation of mainly moving solitons with increasing amplitudes, nearly stationary solitons with increasing amplitude, and a combination of both breathers and increasing amplitude solitons. In case 4, a periodic long-wave perturbation with wavenumber
K is imposed as the initial condition. Modulation instability with wavenumber
K occurs when
where
M is the initial amplitude of the periodic long wave. When
, a mixture of solitons and breathers form as is well-known (see Osborne [
2] for instance). However, as the forcing parameter
increases, the breathers begin to be eliminated and are replaced by solitons with growing amplitudes, progressively fewer forming as the forcing increases.
For each case (1–4) of these initial conditions, we investigated the effect of initial random noise. The case of an initial moving soliton is unchanged except that some small solitons are generated downstream () due to the initial random noise perturbation. The maximum growth rate can still be used to make an accurate prediction of the amplitude growth. For cases when breathers form, initial random noise shifts the locations of the unforced solutions. When forcing is involved, the maximum growth rate of the breathers increases and oscillates with a mean growth rate . For the case of an initial periodic plane wave, initial random noise changes the modulation pattern from deterministic to chaotic with the implication that the location of the maximum amplitude cannot be determined exactly. However, the growth of the maximum amplitude is still approximately . Overall, the predicted growth rate is robust for these initial value problems with and without an initial random noise effect.
Modulation instability and the subsequent formation of small amplitude waves that generate large amplitude waves or sometimes rogue waves has been studied experimentally for water waves by many authors (see for instance [
28]). This large wave is generally unstable due to its growing nonlinear wave packet amplitude when the Benjamin–Feir index (BFI) increases. In this ocean application, large-amplitude waves are generally unstable due to the growing nonlinear wave packet amplitude and modulation instability, measured by the BFI, the ratio of wave steepness (nonlinearity) to spectral bandwidth (dispersion). In this work,
(see Grimshaw and Tovbis [
3]). Even for
which is a moderate value, external forcing that can be viewed as wind blowing over the sea causes modulation instability, and large-amplitude waves are developed. For instance, see
Figure 18 for initial periodic plane wave with wind effects
. When there is a wind effect
, we characterise the development of large-amplitude waves into two stages: the first stage is the development of breathers which could be interpreted as a random sea state as time evolves, and then a second stage forms with large-amplitude waves. The large waves in the second stage collect energy from neighbouring small waves with different wave frequencies. Rogue waves are observed when the BFI is large with amplitudes three or four times the background sea state during their evolution (see for instance [
2,
28]). In our present work even for
modulation instability occurs in the predicted long-wave perturbation range (
) in the first stage, but then large amplitude waves develop due to the wind effect. The larger the value of the wind forcing coefficient, the larger are the waves in the second stage, and they become unstable. Instead of using the periodic wave plane as an initial condition, in case (3) the initial condition of a slowly varying long wave perturbation with a sech-profile also develops into a modulation instability region.
Recent work by [
29] shows the region of high and low wave frequency nonlinear wave interaction where a nonlinear wave component can grow exponentially, leading to rogue wave packets. Outside this region, the small waves are stable. This situation is comparable with our results shown in
Figure 11 with and without wind effects. It can be seen from the case without wind effect that a sequence of breathers is generated, but introducing a wind effect can generate large waves growing in wave amplitude and stationary. An explicit formula that expresses rogue wave formation under wind effect and nonlinear wave packet interaction remains a challenge for further studies.