## Abstract

It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg–de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg–de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg–de Vries solitary wave.

## 1. Introduction

The Ostrovsky equation

In the majority of cases, the dispersive coefficients of this equation are such that *βγ*>0, and we refer to these cases as the normal case. However, our concern here is with the *anomalous* case when *βγ*<0 which is also realizable in physical systems [6–8]. It is known that in the former case, equation (1.1) does not support stationary steady solitary waves [9–11], whereas in the latter case a family of steady envelope solitary waves exists [6]. This situation is similar to that for the Kadomtsev–Petviashvili (KP) equation which has different properties depending on the sign of the dispersion coefficient. In the case of positive dispersion equation, KPI possesses solitary wave solutions in the form of two-dimensional lumps, whereas in the case of negative dispersion equation, KPII does not possess such solutions [12].

Without loss of generality, we may choose *α*>0 and *β*>0, and then the properties of the Ostrovsky equation will be determined by the sign of the rotation coefficient *γ*. The normal case is when *γ*>0, and the anomalous case is when *γ*<0. Note that the nonlinear coefficient *α* can be absorbed into the dependent variable *u* by replacing it with *αu*. Then the case *β*<0 is recovered by replacing *u*,*x* with −*u*,−*x* which leaves the left-hand side of equation (1.1) unchanged, whereas the right-hand side changes its sign.

The Ostrovsky equation (1.1) has two important conservation laws, for localized or periodic solutions,
*M* expresses conservation of mass and holds when *γ*≠0 (in the case of the KdV equation, when *γ*=0, *M* is a constant which is determined by the initial condition). The equation for *E* expresses conservation of wave action flux [1,13]; as will be shown below, this equation plays an important role in the asymptotic solutions of the Ostrovsky equation.

Equation (1.1) has the linear dispersion relation
*k*, frequency *ω* and phase velocity *c*=*ω*/*k*. When *γ*<0, the phase velocity has a maximum *c*=*c*_{m}=−2(*β*|*γ*|)^{1/2} at *k*=*k*_{m}=(|*γ*|/*β*)^{1/4}, and the corresponding group velocity is
*k* increases and *c*_{g}=*c*_{m} at the critical wavenumber *k*=*k*_{m} (figure 1). This can be contrasted with the case when *γ*>0 as then it is the group velocity which has a maximum, while the phase velocity is monotonically decreasing.

It is precisely this significant difference in the linear dispersion relation which leads to the existence or otherwise of a steady solitary wave. When *γ*>0, there is no gap in the linear spectrum of the phase velocity, hence implying the non-existence of a steady solitary wave. Of course, this is only suggestive, but is confirmed by rigorous proofs [9–11]. It is now known that when the initial condition for the normal Ostrovsky equation (1.1) is a KdV solitary wave, it decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities, the phase and group velocities, respectively, corresponding to that wavenumber where the group velocity has a maximum. Such a wave packet was first seen in experiments with an electromagnetic transmission line aimed to model waves in a rotating ocean (see the review [14]) and then in a laboratory experiment [15], but misinterpreted by the authors.

Then Ostrovsky & Stepanyants [16] and later Grimshaw *et al.* [13] studied theoretically and numerically the early stage of envelope wave packet formation from an initial KdV solitary wave. Although the adiabatic decay theory was correctly done there, the outcome was misinterpreted as a quasi-recurrence phenomenon, essentially because these simulations were not long enough to see the envelope wave packet emerging. Helfrich [17] also saw the recurrence in numerical simulations of a fully nonlinear two-fluid model (the MCC system), but integrated long enough to see the envelope wave packet emerging. It was that result which prompted the study by Grimshaw & Helfrich [18] which was the first to clearly state that the key feature of the emerging envelope wave packet was that its speed was close to the maximum in the linear group velocity, and hence could be interpreted as connected with a nonlinear Schrödinger (NLS) equation. Subsequently, laboratory experiments carried out on the Coriolis platform [11] confirmed the emergence of the envelope wave packet with a speed related to the maximum group velocity. Recently, Whitfield & Johnson [19,20] have proposed that modulational instability is the mechanism through which energy is transferred from the initial KdV solitary wave to the envelope wave packet,

In this paper, we address the same issue, namely the evolution of a wave packet from a KdV solitary wave initial condition, but now for the anomalous case when *γ*<0 in the Ostrovsky equation (1.1). The principal difference is that now the emerging envelope wave packet can be steady. In §2, we present a theory for these steady wave packets, which bifurcate from *k*=*k*_{m}. Then in §3, we present an asymptotic theory to describe how initial KdV solitary wave may decay adiabatically due to radiation of long waves. In both these sections although our focus is on the case when *γ*<0, we retain *γ* as a parameter in order to bring out the differences between the normal and anomalous cases. In §4, we present numerical results simulating the decay of an initial KdV solitary wave and its long-time conversion to a steady envelope solitary wave. We conclude in §5.

## 2. Nonlinear Schrödinger equation

In a wide variety of physical systems, the usual NLS equation is
*A*(*x*,*t*) of the weakly nonlinear solution
*ω*=*ω*(*k*) satisfies the linear dispersion relation and at leading order the envelope moves with the linear group velocity *c*_{g}=d*ω*/d*k*. It is understood that *A*(*x*,*t*), *A*_{2}(*x*,*t*), *A*_{0}(*x*,*t*), etc. are slowly varying, while we expect that in general the second harmonic *A*_{2} and the mean term *A*_{0} are *O*(|*A*|^{2}), where |*A*|≪1. The dispersive term in (2.1) generically has the coefficient λ=(1/2)(d*c*_{g}/d*k*), but the coefficient *μ* of the cubic nonlinear term is system-dependent. For the Ostrovsky equation (1.1)

When *γ*<0, λ is negative for all values of *k*>0 and increases from negative infinity to a maximum value of 4*γ*/*k*^{3}_{m} at *k*=*k*_{m} as *k* increases from zero, and then decreases to negative infinity as *γ*>0 when λ decreases from positive infinity to negative infinity as *k* increases and is zero at *k*=(*γ*/3*β*)^{1/4} [18]. Note, however, that the Ostrovsky equation is valid only within the limited range of wavenumbers [1,3]
*c*_{0} is the wave speed of long linear waves (here equation (1.1) is written in the coordinate frame moving with the velocity *c*_{0}).

When *μλ*>0, the NLS equation (2.1) is modulationally unstable and has the well-known solitary wave solution

In general, *K*,*κ*,*V* are *O*(*a*) and *σ* is *O*(*a*^{2}). Of special interest here is that when *k*=*k*_{m}, *c*=*c*_{g}=*c*_{m} and so the solution (2.5) combined with (2.2) is an asymptotic representation of a steady solitary envelope wave solution of the Ostrovsky equation (1.1). Note that by choosing *κ* so that the nonlinear speed correction (*k*+*κ*)*V* =*σ* ensures that the envelope and phase speeds coincide up to second order in the wave amplitude. Within this NLS framework, the sign of the amplitude *a* can be either positive or negative. Further, although the asymptotic solution (2.5) allows an arbitrary phase to be added to *X*, this is not allowed in the full equation when symmetry under the transformation *x*→−*x* for a steady solution is imposed. Such symmetry is required when a rigorous existence proof for stationary solutions of (1.1) is to be constructed (see [21] for instance).

The next step is to derive the envelope equation (2.1) from the Ostrovsky equation (1.1), and hence obtain the envelope solitary wave solution (2.5) in terms of the coefficients of equation (1.1). This is described in §2a,b where we use the NLS equation to describe the modulation instability of a plane wave.

### (a) Derivation of the nonlinear Schrödinger equation

To find the value of the nonlinear coefficient *μ* in (2.1), an asymptotic derivation is needed. This follows the same lines as that described in [18] when *γ*>0 and hence here we present only the necessary minimal details. For the Ostrovsky equation, we substitute the expansion (2.2) into (1.1). The leading-order terms yield the dispersion relation,

Then the coefficient of the first harmonic yields
*N* is given by
*D*/∂*ω*=*k* gives
*N*.

First, we note that to leading order
*A*_{0} is two orders of magnitude smaller than |*A*|^{2}, provided that *γ*≠0, and so can be neglected henceforth. But note that if *γ*=0, then *A*_{0}=*α*|*A*|^{2}/*c*_{g} and is then of the same order as |*A*|^{2}. Also, if *γ*≪1, then the solution for *A*_{0} is again of the same order as |*A*|^{2}, but is non-local. Henceforth, we assume that *γ*≠0 and is *O*(1), since then *k*_{m} is also *O*(1).

Next to leading order,

Importantly, note that *D*_{2}>0 for all *k* when *γ*>0, but when *γ*<0, *D*_{2}=0 when

Hence, on substituting (2.12) into (2.9), we find that

Thus when *γ*<0, *μ*>0 for *k*<*k*_{c} and *μ*<0 for *k*>*k*_{c}. Recalling that λ<0 for *γ*<0 (see (2.3)), we conclude that the NLS equation (2.1) is modulationally stable for *k*<*k*_{c} and unstable for *k*>*k*_{c} (see the next subsection for details). Importantly, as *k* near *k*_{m}. On the other hand, when *γ*>0, *μ*<0 for all *k*, and λ>0(<0) when *k*<(>)(*γ*/3*β*)^{1/4}, so that the NLS equation (2.1) is modulationally stable (unstable) accordingly.

### (b) Modulational instability

Recently, Whitfield & Johnson [19,20] have proposed for the normal Ostrovsky equation when *γ*>0 that modulation instability is a mechanism whereby energy is transferred from the radiation field of a decaying KdV solitary wave to the emerging envelope wave packet. We examine the same concept here, but importantly note that for the normal Ostrovsky equation there is a splitting of modulation stability to instability at the critical wavenumber, whereas here for the anomalous Ostrovsky equation there is no such splitting and it is modulationally unstable for all *k* mear *k*_{m}.

First, we recall the pertinent details about modulation instability. The NLS equation (2.1) has the exact plane wave solution

Perturbing this solution with *θ*(*x*,*t*)|≪1, we obtain the dispersion relation for small perturbations *λμ*>0. The maximum growth rate is |*μ*(*k*)||*A*_{0}|^{2} when λ(*k*)*K*^{2}=*μ*(*k*)|*A*_{0}|^{2}. For the case of interest here when *γ*<0, for a fixed *A*_{0}, the maximum growth rate decreases from infinity to zero as *k* increases from *k*_{c} to infinity and the corresponding *K* also decreases from infinity to zero.

For a general wave field, a useful measure of the likelihood of modulation instability is the Benjamin–Feir index (BFI) which can be defined as (see [22] for instance)
*A* is the wave amplitude and Δ*k* is the perturbation to the basic wavenumber *k*. For the plane wave (2.15) with the maximum growth rate, Δ*k*=*K* and *BFI*=1. As pointed out by Whitfield & Johnson [19,20], the BFI index needs modifying for a decaying solitary wave and they proposed instead a time-dependent criterion. Here, we use a rather simpler criterion based on the initial solitary wave (2.5), namely Δ*k*=*K* and

### (c) Steady envelope solitary wave

As discussed above, for the anomalous Ostrovsky equation, the steady envelope solitary wave of (1.1) can be approximated by (2.5) when *k*=*k*_{m} so that *c*=*c*_{g}=*c*_{m}. It is given by
*K* and *σ* are defined by equation (2.6).

In order that this solution represent a steady wave, the speed of the envelope must equal the speed of the carrier wave, that is

Using the expressions (2.6) for *σ*,*V* , we get an equation for *κ*
*K* is *O*(*a*), it follows that *κ* is here *O*(*a*^{2}), and hence we find that
*μ*,λ at *k*=*k*_{m}. With the same accuracy up to *O*(*a*^{2}),

In the limit *a*→0, we obtain essentially just a linear wave train, whereas for small, but finite *a* we have a steady envelope solitary wave. Figure 2 illustrates a series of such solutions for different values of *V*_{s}, using three typical values of wave amplitude, *a*=0.01, 0.025 and 0.05; then the corresponding wave speeds from equation (2.23) for *α*=*β*=1 and *γ*=−10^{−4} are *V*_{s}=−0.019, −0.013 and 0.0078 (the limiting value of wave speed at *a*=0 is *c*_{m}=−0.02). These values for the speeds were then used as the input parameters for the numerical code based on the Petviashvili method (for details, see [6,23] and the references therein) to find the steady wave solutions. The obtained numerical results are shown in figure 2 by dots.

As can be seen, there is a good qualitative agreement between the approximate theoretical results and the numerical solutions. The number and positions of the extrema coincide, but their values at crests and troughs differ. The results shown in figure 2*c* are beyond the range of applicability of the NLS equation, but even in this case, when the width of the envelope is comparable with wavelength of the carrier wave, there is a reasonable qualitative agreement between the theoretical and numerical profiles. In this case, note that the speed correction term *V* is sufficiently large relative to the linear speed *c*_{m} that the direction of the wave is reversed.

At a certain amplitude, *viz* *V*_{s} completely compensates the linear speed *c*_{m}, so that the total wave speed *V* =0 (we recall that equation (1.1) is written in the coordinate frame moving with the linear wave speed *c*_{0}, therefore the wave speed relative to the ‘immovable observer’ is *c*_{0}). The shape of such particular wave is similar to that shown in figure 2*c*.

## 3. Decaying solitary wave

Our interest here is when the initial condition for the Ostrovsky equation (1.1) is the KdV solitary wave
*γ*|≪1). Both these analysis have been examined for the normal Ostrovsky equation [18–20], but here we retain *γ* as an available free parameter, so that the similarities and differences between *γ*>0 and *γ*<0 can be brought out.

### (a) Linear initial value problem

This linear initial value problem for *u*, when *α*=0 in equation (1.1), is readily solved in a standard manner using Fourier transforms. The outcome is
*ω*=*kc*(*k*), *c*(*k*) satisfies the linear dispersion relation (1.3), and *F*(*k*) is known from (3.1). The long-time limit is now found using the method of stationary phase which shows that the solution disperses as *t*^{−1/2}. Specifically,
*k* propagates with the group velocity *c*_{g}(*k*), and this relation defines *k* as a function of *x*/*t*. For the normal Ostrovsky equation when *γ*>0, there is a focusing of energy towards the critical wavenumber where λ=0. In contrast for the anomalous Ostrovsky equation when *γ*<0, the dispersion relation (1.3) and group velocity expression (1.4) (figure 1) show that long waves (*k*<*k*_{0}=(|*γ*|/3*β*)^{1/4}=*k*_{m}/3^{1/4}) propagate in the positive *x*-direction and short waves (*k*>*k*_{0}) propagate in the negative *x*-direction; note that *k*_{c}<*k*_{0}<*k*_{m}.

It is pertinent to recall that λ(*k*) from equation (2.3) tends to negative infinity as *k*=*k*_{m}. By itself, this indicates a tendency for the wave energy to be enhanced around the critical wavenumber *k*_{m}. For the initial condition (3.1)
*k* and *F*(0)=*a*_{s}*D*_{s}/*π*; as expected, the solitary wave energy is concentrated at low wavenumbers. Hence the expression (3.3) indicates that on this linear theory, most of the solitary wave energy will decay by radiation in the positive *x*-direction.

In order to estimate the BFI (2.17) for this dispersing wavetrain, we adapt the criterion introduced by Whitfield & Johnson [19] and estimate Δ*k* for each fixed *t* by
*k*>0 and decreases monotonically from 0 to −*πD*_{s}/2 as *k* increases from zero to infinity. In contrast, −(1/2λ)(d*λ*/d*k*) varies from approximately 3/2*k* to 0 for 0<*k*≤*k*_{m} and then is negative for *k*>*k*_{m} and tends to zero as *k*>*k*_{c} is relevant here.

It is useful here to adopt the normalization *α*=*β*=−*γ*=1 when *k*_{m}=1 and then the regime of weak to moderate large-scale dispersion corresponds to the regime where the initial amplitude *a*_{s} of the pulse (3.1) is large (see [18] and below in §4). The combination forming Δ*k* depends on the parameter *σ*=*πD*_{s}*k*_{m}/2. In the regime of interest, choose *a*_{s} =3 and *D*_{s}=2 as a typical example, so that *σ*=*π*. A plot of (1/*A*)(d*A*/d*x*) is shown in figure 3 indicating that Δ*k*=0.3/*t*, achieved at *k*=1.24*k*_{m}; the corresponding location in space is *x*=*c*_{g}(1.38*k*_{m})*t*. This plot also shows that for these same parameter values, the amplitude |*A*(*x*)| has a maximum of 1.11*t*^{−1/2} at *k*=0.59*k*_{m}. Thus the BFI (2.17) is estimated as 1.29*t*^{1/2}. Thus the BFI will exceed unity on timescales of order unity, centred around the critical wavenumber *k*_{m}, indicating the occurrence of modulation instability and the possible eventual formation of a nonlinear wave packet. Note that for larger amplitudes, the BFI increases but remains proportional to *t*^{1/2}.

### (b) Slowly varying solitary wave

Next, we consider the slowly varying solitary wave asymptotic theory based on the premise that |*γ*| is a small parameter. To do this, we adapt the argument of Grimshaw *et al*. [13] (see also [24]) and assume the following asymptotic form for a decaying solitary wave solution of equation (1.1),
*a*(*t*) varies slowly with time, while *u*^{(1)} is a first-order correction term. Using the initial condition (3.1), *a*(0)=*a*_{s}, the determination of how *a*(*t*) varies in time can be found from the conservation law (1.2) for the energy flux *E* using the asymptotic theory described in [13] (see also [24]) for the case when *γ*>0. That theory is briefly described here as there some differences when *γ*<0. Substitution of (3.7) into (1.1) yields that at the leading order, *u*^{(0)} satisfies the KdV equation. Then at the next order, we find that
*u*^{(1)} and this needs the formulation of suitable boundary conditions as

When *γ*>0, the group velocity is negative for all wavenumbers (see equation (1.4)) and we deduce that all radiation is in the negative *x*-direction. Hence we can impose the condition that *u*^{(1)}→0 and also *v*^{(0)}→0 as *γ*<0, the group velocity is positive for long waves and negative for short waves (figure 1). Since this asymptotic expansion essentially requires that the correction term is small and slowly varying, we make the assumption that it is the long-wave radiation which is dominant, and hence impose the boundary condition when *γ*<0 that *u*^{(1)}→0 and also *v*^{(0)}→0 as

In order to solve equation (3.8) for *u*^{(1)} with this boundary condition, we need to consider the adjoint equation for the homogeneous part (left-hand side) of equation (3.8),
*u*^{(1)}≡const., *u*^{(1)}≡*u*^{(0)} and a third solution, which is unbounded at infinity. Only the second solution *u*^{(1)}≡*u*^{(0)} satisfies the condition that *u*^{(1)}→0 as *u*^{(0)}. This leads to the compatibility condition that
*E*). For the full Ostrovsky equation (1.1), the right-hand side would be zero, as the equation carries zero ‘mass’ [1,13]. But here the asymptotic expansion shows that it is applied locally to the KdV solitary wave solitary wave, with the balance being carried away in radiated waves.

Substituting for *u*^{(0)} from equation (3.7), we find that
*t*_{e}. Note that this is the same expression when *γ*>0, although the derivation is slightly different since then the radiation is in the negative *x*-direction. When we use the normalization *α*=*β*=−*γ*=1, the extinction time is *a*_{s}=3, the extinction time is *t*_{e}=0.5 which is comparable with the time 0.6 for the BFI to exceed unity at this amplitude.

The decay of the solitary wave is accompanied by the formation of a shelf, described to the leading asymptotic order as a linear long wave. This is needed to conserve the total mass. When *γ*<0, long waves propagate in the positive *x*-direction, so the shelf appears ahead of the decaying solitary wave. In detail, it follows from equation (3.8) that
*θ*∼−*γ*^{−1}. The remedy is that this inner solution needs to be matched with an outer solution which consists of the shelf. To leading order, the shelf is a linear long wave, which using the theory of wave kinematics, can be locally represented for *x*≥*P*(*t*) in the form
*k* scales with |*γ*|^{1/2}. But unlike the case *γ*>0, the phases of this shelf and the solitary wave cannot be matched when *γ*<0, as this linear long wave has a phase speed *c*=*γ*/*k*^{2}<0, whereas the solitary wave speed *V* >0. Nevertheless, we could assume that the group velocity *c*_{g}=−*γ*/*k*^{2} can be matched with the solitary wave speed, so that *V* *k*^{2}=−*γ*. In this event, as the solitary wave decays, the wavenumber *k* increases. Because phase matching is not possible, the amplitude of the shelf at the location of the solitary wave *a*_{shelf}=*u*_{shelf}(*x*=*P*(*t*),*t*) is estimated using conservation of total mass.
*t* and taking the long wave limit *k*→0 yields

Thus, according to this asymptotic theory, the shelf forms instantaneously, is always negative and decreases to negative infinity as *t*→*t*_{e}. Also, pertinently, note that this asymptotic theory requires that |*a*_{shelf}|≪*a*, and hence requires that *a*_{s}≫(72*β*|*γ*|)^{1/2}/*α*. Hence, this slowly varying solitary wave theory is only valid for nonlinear waves and is quite distinct from the linear theory in §3a. For the normalization *α*=*β*=−*γ*=1, it is required that *a*_{shelf} becomes comparable with the solitary wave amplitude *a* at a time less than *t*_{e} and so this asymptotic theory then fails. This breakdown time *t*_{b} can by estimated by equating (3.12) and (3.17):
*α*=*β*=−*γ*=1 and *a*_{s}=12, the right-hand side is *t*_{b}=0.16*t*_{e}. As the initial amplitude increases, so does the breakdown time.

## 4. Numerical Results

Two sets of numerical simulations were performed. One was to examine the radiative decay of an initial KdV solitary wave for both cases *γ*>0 and *γ*<0 in relation to the asymptotic theory for the adiabatic decay presented in §3b. The other was to examine the long-time outcome for the case *γ*<0 in relation to the analogous long-time outcome found by Grimshaw & Helfrich [18] for the case *γ*>0.

### (a) Radiative decay

For these numerical calculations, it is convenient to present equation (1.1) with the initial condition in a dimensionless form. To this end we assume that the initial condition is *u*(*x*,*t*=0)=*U*_{0}*f*(*x*/*D*_{0}), where *f*(*ξ*) is a dimensionless function of unit height with the characteristic width 1, and *U*_{0} is the ‘amplitude’ of the initial perturbation. Then in the new variables *ξ*=*x*/*D*_{0}, *τ*=*αU*_{0}*t*/*D*_{0} and *υ*=*u*/*U*_{0}, equation (1.1) and the initial condition become, in a dimensionless form,

Since the numerical simulations are of necessity on a finite interval of length *L*, the initial condition (3.1) must be supplemented by a negative pedestal *d* to ensure that the total initial mass is zero. Thus, assuming that *D*_{0}≪*L*, the initial condition becomes
*σ*^{2}=12 and *S*=12*βγ*/(*αa*_{s})^{2} on setting *U*_{0}=*a*_{s},*D*_{0}=*D*_{s}. Note that for an infinite domain, *d*→0.

As already mentioned in §3b, the asymptotic theory for the adiabatic decay of a solitary wave is based on the premise that the right-hand side of equation (4.1) is small in comparison with the terms on the left-hand side. This requires that |*S*|≪1 or *τ*_{e}=1/|*S*|. The requirement for |*S*|≪1 restricts the range of validity of the adiabatic theory. From the definition of *S*, it follows that in the process of wave decay this parameter increases. Using equation (3.12) for the wave amplitude, we obtain the formal restriction for the time interval when the adiabatic theory is valid
^{1/4}≈1.57, which arises from the more rigorous requirement that the generated tail be at least smaller than the amplitude of the decaying solitary wave.

These numerical simulations were performed using the finite-difference numerical scheme described in [23] with periodic boundary conditions on the interval of length *L*. In the first run which can be treated as the reference case studied in [23], we put the following coefficients in equation (1.1): *α*=1, *β*=1 and *γ*=10^{−4} (this corresponds to the ‘normal’ Ostrovsky equation). Choosing then the initial condition in the form of KdV solitary wave on a pedestal (4.2) with *a*_{s}=1 on the interval *L*=5000, we obtain a width *d*=−1.39×10^{−3}. In the dimensionless variables, this corresponds to equation (4.1) with *σ*^{2}=12 and *S*=1.2×10^{−3}.

For this case *γ*>0, we recall the well-established results from [3,13,23]. The solitary wave in the course of propagation experiences terminal decay at the early stage of evolution in accordance with equation (3.12). The dependence of pulse amplitude on time is shown in figure 4 by the dotted curve. The solitary wave moving to the right radiates long waves only in the negative *x*-direction, as shown in figure 5*a*. This adiabatic stage formally is valid until *t*/*t*_{e}≪1−*S*^{1/4}≈0.81, but in fact formula (3.12) agrees well with the numerical data up to this limiting value and even further until almost the complete extinction of the initial KdV solitary wave at *t*=*t*_{e}≈2.89×10^{3}. However, as described in §3b, in the process of pulse evolution, it generates a tail which grows in amplitude and evolves into another solitary-like pulse as shown in figure 5*b*. This secondary pulse also experiences terminal decay. The process repeats several times, but at each time a certain portion of energy leeks to negative infinity. Eventually, the initial KdV solitary wave evolves into an unsteady wave packet with a steady envelope [3,11,13,18].

Next, the same parameters and initial conditions were used, but with *γ*=−10^{−4}, corresponding to the anomalous Ostrovsky equation. In this case, the solitary wave did not experience terminal decay, but instead smoothly transferred into a steady envelope solitary wave (Ostrovsky soliton) [6]. Its amplitude changed very little in time as shown in figure 4 by rhombuses. The solitary wave radiated both long waves in the positive *x*-direction and short waves in the negative *x*-direction, and then, omitting the stage of extinction and resurrection, evolved instead almost directly into the steady envelope wave packet as shown in figure 6*a*. Figure 6*b* represents the comparison of the pulse shape at *t*=540 with the steady envelope solution of equation (1.1) with *γ*=−10^{−4}. The latter solution was obtained numerically by means of the Petviashvili method (see [6,23] and references therein).

We also studied the cases when the parameter *γ* in equation (1.1) is not so small, and then it emerges that the asymptotic expansion is not valid. Figure 7 shows the time dependence of pulse amplitudes when *γ*=0.1 (dots) and *γ*=−0.1 (rhombuses), whereas the solid line represents the theoretical dependence from equation (3.12). As can be seen, the asymptotic theory fails to approximate the numerical data even in the very early stage of wave evolution. We have found, however, that at the very early stage an empirical formula similar to equation (3.12), but with *t*_{e} two times greater can be used for data fitting. The corresponding fitting line 4 is shown in figure 7. The values of other parameters of equation (1.1) were as follows: *α*=1 and *a*_{s}=*D*_{s}=1 on the interval *L*=2000, so that the pedestal was *d*=−10^{−3}. In such cases, equation (1.1) coincides with its dimensionless form (4.1) with *β*=*σ*^{−2} and *γ*=*S*.

Figure 8 shows profiles of the evolving wave for different values of *γ*. Figure 8*a* illustrates the wave profile (line 2) at *t*=3.25 when *γ*=0.1 and the initial KdV solitary wave of unit amplitude (line 1). Figure 8*b* illustrates the wave profile (line 3) at *t*=8 when *γ*=−0.1 and a stationary envelope solitary wave of the same amplitude (line 4). We see from figure 8*b* that the process of formation of a steady envelope wave packet is not yet completed. Instead, the pulse continues to radiate waves of different wavelengths to the right (longer waves) and to the left (shorter waves) in agreement with the theoretical prediction.

### (b) Long-time evolution

In order to study the long-time evolution, it is convenient to use the normalized Ostrovsky equation (1.1) when *α*=*β*=−*γ*=1, as then the timescale which scales with *γ*^{−1} is greatly reduced, Grimshaw & Helfrich [18] where the analogous long-time simulations are described for the case when *γ*>0. This is achieved with the transformation *L*^{4}=*β*/|*γ*|, *T*=*L*^{3}/*β*, *P*=*β*/*αL*^{2}. That is, we can replace (1.1), after omitting the ‘tilde’ symbol, by

The initial condition is again the KdV solitary wave (4.2) on a negative pedestal *d*(*x*), given here by
*t*≫*t*_{e}. The numerical simulations now use a pseudo-spectral method in a periodic domain of length *L*, similar to that described in [26]. At each end of the periodic domain, there is a linear damping region to prevent the possibility of radiated waves re-entering the region and disturbing the main wave structure, and de-aliasing is used on the nonlinear term to remove high-frequency components. The damping term, *r*(*x*) is inserted on the left-hand side of (4.4) in the form *r*_{x} where
*ν* and *κ*. For instance, *κL*=24 and the value of *ν* is chosen so that the damping occurs quickly. Note that because of this damping term, the pedestal *d*(*x*) is truncated back to zero at each end of the domain where *r*(*x*) is non-zero, and hence inherits an *x*-dependence there.

The results are shown in figure 9 for initial amplitudes *a*_{s}=8,10,12,14,16 in (4.5). In each case, we see the emergence of a steady envelope solitary wave packet. In the initial stages, there is some radiation both in the positive and negative *x*-directions, but relatively quickly this envelope solitary wave has separated from this radiation and continues to propagate apparently without any decay.

Figure 10 shows a cross-section at *t*=60 for each case and a summary of the results is shown in table 1. A striking feature is that for these simulations, the envelope wave packets are strongly nonlinear, and their amplitudes correlate quite well with the amplitudes of initial KdV solitary waves. Although these wave packets are strongly nonlinear, their speeds increase with the amplitude, as predicted by the weakly nonlinear NLS theory described in §2c. In particular, the expression (2.23) predicts a speed *V* =−2+*a*^{2}_{env}/9 for the present parameters, where *a*_{env} is the amplitude of the wave packet, defined in equation (2.18). Using the numerically found amplitude *a*_{n} at *t*=60 (table 1) to estimate that *a*_{env}=*a*_{n}/2, this expression predicts speeds for the envelope solitary waves of *V*_{s}=−1.73,−1.11,−0.31,0.86,2.81,7.23 and for the initial conditions *a*_{s}=6,8,10,12,14,16, respectively.

Simulations with smaller amplitudes (not shown here) do not show the emergence of wave packet, and would seem to be essentially linear, although possibly very long-time simulations may show otherwise. There is good agreement for the smallest initial amplitude, but although the trend is the same, not surprisingly any quantitative agreement falls off as the initial amplitude is increased, since then the emerging wave packets are strongly nonlinear.

In order to compare these results with those in the previous §4a, we note that in the scaling used to convert (1.1) to (4.1), for the present parameters *σ*^{2}=12 and *S*=−12/*a*^{2}_{s}; thus the set of initial conditions used here correspond to *S*=−0.19,−0.12,−0.083,−0.061,−0.047.

Figure 11 shows a comparison between the numerical simulations at *t*=60 for initial amplitudes *a*_{s}=6 and *a*_{s}=12 and the steady envelope solitary wave solution of the same amplitude found using the Petviashvili numerical method [6,23]. The agreement is very good in both cases.

Then in figure 12, we show the corresponding Fourier spectra of solutions. In both cases, the Fourier spectrum shifts from the origin where the initial spectrum of the KdV solitary wave is localized to the vicinity of the finite wavenumber *k*_{m}=1. Note that the emerging peak in the Fourier spectrum is slightly below *k*_{m} as the weakly nonlinear theory in §2c predicts. In the small amplitude case, we see some evidence of fluctuations about the theoretical spectrum, and the growth of sidebands as predicted by modulation instability theory; but these are barely visible in the large amplitude case.

The numerical simulations show that there is a strong correlation between the amplitudes of the initial KdV solitary wave *a*_{s} and the final wave packet amplitude *a* recorded in table 1. This correlation is plotted in figure 13. For relatively small amplitudes, the correlation can be approximated by the straight line *a*_{n}≈1.164*a*_{s}−3.796 which suggests that the nonlinear wave packets can emerge from the KdV solitary waves only when *a*_{s}>3.2618. This is in broad agreement with the failure of the numerical simulations to find an emerging wave packet when *a*_{s}=4.

## 5. Conclusion

In this paper, we have examined the long-time formation of envelope wave packets in the Ostrovsky equation (1.1) from a KdV solitary wave initial condition (3.1) for both normal dispersion when *β*>0, *γ*>0 and anomalous dispersion *β*>0, *γ*<0, with an emphasis on the limit when |*γ*|, which measures the effect of large-scale dispersion, is relatively very small. In the former case of normal dispersion, it is known that there are no steady solitary wave solutions, and the initial KdV solitary wave decays adiabatically through the radiation of long waves in the negative *x*-direction (see §3) and is eventually replaced by an envelope wave packet propagating with a speed very close to the maximum allowed group velocity (which is negative) (see [18] and the references therein). Importantly, although this wave packet propagates with a constant speed, it is intrinsically unsteady as the carrier waves inside the envelope propagate at a different speed close to the linear phase speed (which is positive). The numerical experiments of Grimshaw & Helfrich [18] and the laboratory experiments of Grimshaw *et al.* [24] were unable to find a clear relationship between the initial KdV solitary wave amplitude and that of the emerging envelope wave packet. However, recently Whitfield & Johnson [19,20] found such a connection for small initial amplitudes based on the concept that the energy transfer is due to modulation instability.

In contrast, for anomalous dispersion, there exists a steady envelope solitary wave solution, which for small amplitudes bifurcates from the point in wavenumber space where the phase velocity has a maximum, and where the phase and group velocities (both negative) are equal. In the small amplitude limit, this can be described asymptotically through the solitary wave solution of the NLS equation (see §2), and then propagates in the negative *x*-direction, but for large amplitudes it becomes strongly nonlinear with only a few carrier waves inside the envelope and then propagates in the positive *x*-direction. For this case, the asymptotic theory described in §3b, again predicts that an initial KdV solitary wave decays adiabatically through the radiation of long waves, now in the positive *x*-direction. However, our early-time numerical simulations reported in §4a show that this slow adiabatic decay does not occur. Instead, after a very short interval, the initial KdV solitary wave is replaced by the formation of a steady envelope solitary wave. Then our long-time numerical simulations for weak large-scale dispersion, reported in §4b, show that the emergence of this steady envelope solitary wave is a very robust feature. Further, unlike the case of normal dispersion, there is a very clear strong positive correlation between the amplitude of the initial KdV solitary wave and the amplitude of the steady envelope solitary wave. Indeed as the initial amplitude increases, so does that of the envelope solitary wave to an extent that it can propagate in the positive *x*-direction.

It is not clear why the adiabatic theory for the decaying KdV solitary wave is valid for the case of normal dispersion, but fails completely in the case of anomalous dispersion. But it may be relevant to note that in the former case, the decaying solitary wave has a positive speed and can generate linear long waves with the same positive speed; this process can be described using linear wave kinematics [13,18] and is a relatively slow process. In contrast in the case of anomalous dispersion, the decaying solitary wave with its positive speed cannot match with any phase speed of a linear long wave, and hence its initial decay is inhibited. In §3b, we attempted to describe the decay process through the matching with the positive group velocity of linear long waves, but the numerical evidence suggests that this is not an effective process. Instead, it seems that the initial decay of the KdV solitary wave generates waves in both the positive and negative *x*-directions (figure 8*b*) and this enables the rather rapid formation of the envelope wave packet which eventually will be symmetric about the wave crest.

## Authors' contributions

R.G. and Y.S. contributed to the major part of the theory and development of the mathematical model. Y.S. implemented and performed the numerical simulations shown in figures 2, 4–8 and 11, A.A. implemented and performed the numerical calculations presented in figures 9 and 10 and in table 1. Figures 12 and 13 were produced by Y.S. on the basis of numerical data obtained by A.A. All authors contributed to the writing of the paper and gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

R.G. was supported by the Leverhulme Trust through the award of a Leverhulme Emeritus Fellowship; Y.S. was supported by the State Project of the Russian Federation in the field of scientific activity, Task no. 5.30.2014/K and by the grant of London Mathematical Society (LMS) for visits to UK, Scheme 2, in 2015.

## Acknowledgements

Y.S. is indebted to Dr Karima Khusnutdinova for the invitation, LMS for providing funding and the Department of Mathematical Sciences at Loughborough University for the hospitality.

- Received June 19, 2015.
- Accepted December 3, 2015.

- © 2016 The Author(s)