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Interaction of a solitary wave with an external force in the extended Korteweg-de Vries equation

Roger Grimshaw1 , Efim Pelinovsky2 ,

1

Department of Mathematical Sciences, Loughborough University, Loughborough, UK 2 Laboratory of Hydrophysics and Nonlinear Acoustics, Institute of Applied Physics, Nizhny Novgorod, Russia,

November 27, 2001

Abstract The interaction of a strongly nonlinear solitary wave with an external force is studied using the extended Korteweg-de Vries equation as a model. This equation has several different families of nonlinear wave solutions: solitons, the so-called ”thick” solitons, algebraic solitons, and breathers, depending upon the sign of the cubic nonlinear term. A simple nonlinear dynamical system of the second order for the amplitude and position of the solitary wave is dervived, and used to study the interaction. Its solutions are investigated in the phase plane. The conditions for the capture or reflection of a solitary wave by a single isolated external force are obtained, with an emphasis on the role of the cubic nonlinear term.

1

Introduction

The forced Korteweg-de Vries (KdV) equation is a canonical model for the resonant generation of solitary waves by an external force moving with a speed close to the speed of linearised long waves. In non-dimensional form it is, ∂u ∂u ∂ 3 u ∂f ∂u +∆ + 6u + = , 3 ∂t ∂x ∂x ∂x ∂x

(1)

where u(x, t) is the wave function, x is the spatial coordinate in the reference frame moving with an external force, t is the time, ∆ is the linear long-wave speed and the speed of the moving force), and f (x) is a representation of the external force. Explicit asymptotic derivations of the forced Korteweg-de Vries equation have been carried out for several situations in the ocean and atmosphere (see, for instance, the review papers by Grimshaw (2001a,b)). At the next order a perturbation theory in the wave amplitude the extended forced Korteweg-de Vries equation can be obtained, and in the general case it should include cubic nonlinearity, fifth-order linear dispersion, and nonlinear dispersion. The contribution of the various high-order terms depends on the parameters of the model (e.g. in the oceanic, or atmospheric, applications, on the density stratification and shear flow). For instance, for internal waves in a two-layer fluid, the quadratic nonlinear term vanishes when the pycnocline lies in the middle of the fluid (in the Boussinesq approximation), and the cubic nonlinear term is then the most important higher-order term; in this case, the extended Korteweg-de Vries equation coincides with the modified Korteweg-de Vries equation, which like the KdV equation, is integrable. More generally, when the quadratic nonlinear term is small, the cubic nonlinear term should also be taken into account, and the corresponding equation is, ∂u ∂u ∂u ∂ 3 u ∂f + 6u + βu2 + = . ∂t ∂x ∂x ∂x3 ∂x

(2)

In the absence of forcing, this is the extended Korteweg-de Vries (eKdV) equation, or Gardner equation. It is often used as a model for strongly nonlinear 1

internal waves in the ocean (Holloway et al, 2001). The coefficient β of the cubic nonlinear term may have either sign depending on the oceanic (or atmospheric) stratification. For the classical two-layer model it is always negative, but for certain three-layer models, it was recently shown that it may be as negative, or positive, (Grimshaw et al, 1997). The solutions of the extended Korteweg-de Vries equation describe various kinds of nonlinear waves (solitons, breathers, dissipationless shock waves) for various signs of the coefficients of the quadratic and cubic nonlinear terms. Further, although the unforced equation (2) is integrable, like the Korteweg-de Vries equation, the number of explicitly solved problems is quite small. In particular, soliton interactions have been investigated by Slunyaev & Pelinovsky (1999) and Slunyaev (2001) for both signs of the cubic nonlinear term. Instability of algebraic solitons in the framework of the extended Korteweg-de Vries equation has been studied by Pelinovsky & Grimshaw (1997), who showed that any small disturbance leads either to transformation to a ”normal” soliton, or to a breather. Interesting examples of the soliton transformation when the quadratic nonlinear term changes its sign (this situation may be realised for internal waves in the coastal zone with an inclined pycnocline) were given by Grimshaw et al (1999). The influence of all high-order terms on soliton dynamics was investigated numerically by Marchant & Smyth (1990, 1996). This paper is concerned with soliton dynamics in the framework of the forced extended Korteweg-de Vries equation (2) when the forcing has the form of a single impulse. Numerical simulations of this equation shown that the imposition of a forcing on an initially quiescent state generates various forms of nonlinear wave packets, as well as solitary waves (Choi et al, 1996, Grimshaw et al, 2001a). Here we suppose that the wave field consists of just one solitary wave, and we will study its interaction with an external force. Our analysis will use an asymptotic method for soliton perturbations, similar to that used by Grimshaw et al (1994) for the analogous problem for the forced KdV equation (1). Indeed, our main objective here is to determine how the cubic nonlinear term in equation (2) will affect the results obtained in Grimshaw et al (1994) in their study of an interaction of a solitary wave with an isolated external force in the KdV model (1).

2

Forced extended Korteweg - de Vries equation

First, we observe that equation (2) can be written in the Hamiltonian form ∂ δH ∂u =− , ∂t ∂x δu where the Hamiltonian H is š ›2 “ Z ’ β 1 ∂u 1 H= − uf dx. ∆u2 + u3 + u4 − 2 12 2 ∂x 2

(3)

(4)

Consequently equation (2) (or (3)) conserves both the mass Z M = udx and the Hamiltonian. However, the momentum Z 2 u dx, P = 2

(5)

(6)

is not conserved, although we note that equation (2) can be re-written as ∂ δP ∂ δH + = 0. ∂t δu ∂x δu

(7)

It is important to note that the mass is constant for any disturbance in the field of an isolated force f (x), where we assume that f → 0 as x → ±∞. We will study the case when the forcing is weak, and so, in the zero (leading order) approximation, the waves are free. Our focus is on solitary waves (solitons), and first we need to describe their properties. When f ≡ 0, it follows from equation (7) that a solitary wave of speed c satisfies the equation c

δH δP = . δu δu

(8)

The analytical expression for the soliton shape can be obtained for either sign of the coefficient of the cubic nonlinear term as u(x, t) =

γ2 , 1 + B cosh(γ(x − ct)

(9)

βγ 2 , c = ∆ + γ2, 6 where γ is a parameter characterising the inverse width of the solitary wave. The wave amplitude (peak value) is B2 = 1 +

a =

6 γ2 = (B − 1). 1+B β

(10)

There are three kinds of solitary waves. A. Negative cubic nonlinearity, β < 0. Here 0 < B < 1, and the solitary wave has positive polarity. Its shape for β = −0.5 is shown in Figure 1. For small wave amplitudes (γ → 0 or B → 1) the solitary wave (9) transforms into the Korteweg-de Vries soliton › šr a 2 (11) (x − ct) . u = a sech 2 3

As the wave amplitude increases, it approaches the limiting value alim =

6 . |β|

(12)

In this limit B → 0 and the width of the solitary wave increases to infinity, realising the so-called ”thick” soliton, see Figure 1. B. Positive cubic nonlinearity, β > 0. There are two families of solitary waves, both with B 2 > 1. The first has a positive polarity, with 1 < B < ∞. Here there is no limiting amplitude; for small amplitudes this solitary wave is again the KdV soliton (11) (B → 1), while for large amplitudes (as B → ∞) the solitary wave is a soliton of the modified Korteweg-de Vries equation # "r βa2 (x − ct) . u = a sech (13) 6 The other family of solitary waves has negative polarity, and is realised for −∞ < B < −1). For large amplitudes (B → −∞) this wave is also a soliton of the modified Korteweg-de Vries equation, and is given by (13), where a is now negative. The amplitude (in modulus) of these negative solitons should be greater than the critical value, acr = −

12 . β

(14)

As the amplitude tends to the critical value (B → −1), this wave becomes the algebraic soliton, with zero speed (γ → 0), u(x) =

acr . 1 + 6x2 /β

(15)

The shapes of the solitary wave of both polarities for the positive cubic nonlinearity (β = 0.5) are shown in Figure 2.

3

Dynamical equations for the solitary waveforce interaction

Considering the case of a weak force, the solution of the forced extended Kortewegde Vries equation (2) is sought as an asymptotic series u(x, t) = u0 (ξ, t) + u1 + ...,

ξ = x − X(t),

(16)

where  is measure of the forcing amplitude, and X(t) is the location of the solitary wave. The first term is the slowly-varying solitary wave, whose profile is given by (9), γ2 , (17) u0 (ξ, t) = 1 + B cosh(γξ) 4

dX = c = ∆ + γ2. (18) dt Due to the effect of the forcing, the solitary wave amplitude and position will change, so that the inverse-width parameter, γ is a function of time (and hence also the amplitude, etc.). We will not describe here the asymptotic method which allows us to obtain the equations for the solitary wave parameters, as the method is well-known, in this context see, for instance, Grimshaw et al (1994). At the first order of the asymptotic theory, the amplitude variation can be found from the momentum equation Z dP ∂f = u dx, (19) dt ∂x where

which is readily obtained from equation (2) and the definition (5) of P . Indeed, using the leading order term (17) of the asymptotic series (16) we get Z ∞ dP0 ∂f = u0 (x − X(t)) dx, (20) dt ∂x −∞ Z ∞ 1 2 P0 = where u0 dx. (21) −∞ 2 Since P0 is a function of γ(t), the system of ordinary differential equations (18) and (20) form a dynamical system for the wave amplitude a and the position X. To simplify the right-hand side of (20) we next suppose that the force has a large width compared to the solitary wave, the so-called “broad” forcing case of Grimshaw et al (1994). In this case (20) reduces to df (X) dP0 = M0 . dt dX Z ∞ where M0 = u0 dx,

(22) (23)

−∞

is the mass of the solitary wave. This reduced system has a Hamiltonian K0 given by ∂V c where K0 = V (P0 ) − f (X), = . (24) ∂P0 M0 This Hamiltonian is conserved, and hence all the orbits of the dynamical system are given by the curves K0 = constant in the X − a plane. Using equation (18) we see that the orbits are given by

where

f (X) = ∆F1 (a) + F2 (a) + constant, Z Z 2 dP0 γ dP0 F1 (a) = , F2 (a) = . and M0 M0 5

(25) (26)

Both functions, F1 (a) and F2 (a) are monotonic functions of the wave amplitude a. In general, when a solitary wave deforms there is the accompanying generation of a trailing shelf (see, for instance, Grimshaw and Mitsudera, 1993). However, in the present case of broad forcing it can be shown that these shelves are absent here to leading order. Indeed, their absence allows us to obatin the next-order correction to the solitary wave speed, and replace equation (18) with ∂M0 dX = ∆ + γ2 − f (X) . dt ∂P0

(27)

The simplest demonstration of this is to note that if no shelves are generated, the interaction of the solitary wave with the force retains a Hamiltonian character, and the Hamiltonian is just the expression (4) evaluated at leading order, denoted here by K and given by K = H0 (P0 ) − M0 f (X), where

∂H0 = c = ∆ + γ2. ∂P0

(28)

Here H0 is the Hamiltonian H evaluated for the solitary wave u0 and with f ≡ 0. Then, with the canonical variables X and P0 , we see that the dynamical system consisting of (22) and(27) is just ∂K dP0 = − , dt ∂X

∂K dX . = − dt ∂P0

(29)

The additional term in equation (27) can be shown to be significant only for small amplitudes, and since our main interest here is in large-amplitude solitary waves, we shall not consider the extended equation (27) further in this paper, and instead use only the reduced equation (18). We also note that the Hamiltonians K and K0 are proportional, K = K0 M0 .

4

Interaction of weak-amplitude solitary waves with an external force

First, let us demonstrate the main features of the dynamics of weak-amplitude solitary wave (i.e. the KdV approximation). In this case all integrals can be obtained in explicit form, √ √ 1/2 2 2 3/2 (30) M = 2 2a , γ 2 = 2a, P = a , 3 and

F1 (a) =

a , 2

6

F2 (a) =

a2 . 2

These expressions were obtained by Grimshaw et al (1994). We recall that the force is isolated, that is, f (x) → 0 as x → ±∞. The soliton dynamics now depends on the sign of ∆. If ∆ > 0 (”subcritical” forcing motion) the solitary wave has a speed greater than the speed of the external force for all amplitudes. In this case an exact resonance between the force and the solitary wave is impossible. The phase portrait for this case is shown in Figure 3 for a positive disturbance (f > 0). The parameters of are f (x) = b exp(−x2 ),

b = 0.5,

∆ = 0.1.

(31)

The main regime here is the passage of fast-moving solitary wave through the external force. The solitary wave takes momentum from the force (it is evident from equation (22)), and its amplitude increases as r ∆ ∆2 a(x) = + 2f (x) + ∆a∞ + a2∞ − , (32) 4 2 (a∞ is the amplitude of the ”free” soliton) when the solitary wave passes through the forcing area. There is no net change in the solitary wave amplitude during passage, but the solitary wave has an additional phase shift due to the interaction. In addition to this passage regime we have also the regime of birth/decay. A weak solitary wave is generated behind the force, increases up to the moment of overtaking and then decays to zero in front of the force. Maximal amplitude of this virtual wave can be found from (32) at a∞ = 0 r ∆2 ∆ amax = + 2fmax − . (33) 4 2 We should mention that the stage of birth/decay cannot be described by (18), (22), because the weak solitary wave has a very large width, and the approximation of broad forcing is not valid in this case. A more detailed analysis of all possible wave regimes was done by Grimshaw et al (1994). For instance, if the force has negative polarity, (f (x) < 0), the solitary waves pass through the forcing area, but the trajectories on the phase plane are shifted in the direction of weak amplitudes. A more interesting situation is realised for ∆ < 0 (”supercritical” forcing motion), when the solitary wave can be in resonance with the force. The exact resonant condition is 2a = |∆|, (34) obtained from (18). The system (18), (22) has an equilibrium state X = 0,

2a0 + ∆ = 0,

7

(35)

where we are assuming that f (x) has a single stationary point at X = 0. The solitary wave with amplitude a0 located at the the centre of the the force will be in synchronism with the force, and may propagate together with it indefinitely. The character of this equilibrium point depends on the polarity of the force. If f (x) > 0, the equilibrium point is a centre, and near this point the solitary wave amplitude oscillates, so that a positive force can capture solitary waves. This regime, called trapping, is shown in the phase portrait for b = 0.2 and ∆ = −2 (Figure 4). Large-amplitude waves cannot be captured due to the large difference in the speeds, and they pass through forcing area with a modulation in amplitude. This passage regime is separated from the trapping regime by the ”homoclinic” orbit p a(x) = a0 + 2f (x). (36) Another ”homoclinic” orbit

a(x) = a0 −

p 2f (x)

(37)

separates the periodic orbits from a family of non-periodic orbits. If 2fmax < a20

(38)

this lowest ”homoclinic” orbit does not touch the axis a = 0, and weak-amplitude waves with a small speed cannot be captured. From (35) and (38) this case is realised if |∆| is large enough, and is presented in Figure 4. The regime birth/decay here is the same as in Figure 3. If the sign in (38) is opposite, the ”homoclinic” orbit is split into two curves ending on the axis a = 0. In this case some solitary waves generated behind the force will amplify significantly (their amplitude will exceed a0 more than twice), and then damp and decay (this case is not shown in Figure 4). But as we pointed out before, the analysis of very weak-amplitude solitary waves violates the broad-forcing hypothesis. A more detailed analysis can be found in the paper by Grimshaw et al (1994). It is important to mention that the maximal variation of the amplitude of a captured solitary wave an be found from (36) and (37) amax + amin = 2a0 .

(39)

The appearance and disappearance of large-amplitude waves in the process of the interaction with an external force may be important for the explanation of the nature of the rogue (freak) waves in the ocean; existing models are presently usually based only on the ”free” evolution equations (Osborne et al, 2000; Pelinovsky et al, 2000, Onate et al, 2001; Kharif et al, 2001). If the force has a negative polarity, the equilibrium point is a saddle point, and a typical phase portrait is presented in Figure 5 for b = −0.2 and ∆ = −2. Solitary waves reflect from the forcing area with a strong variation in the wave amplitude, the relation between the initial value a(∓∞) and final value a(±∞) being a(∓∞) + a(±∞) = 2a0 . (40) 8

The forcing thus introduces an anisotropy into the solitary wave field: solitary waves behind the force have weak amplitudes, and the waves in front of the force have large amplitudes. Additionally to the repulsion regime there are again two passage regimes, large-amplitude solitary waves overtake the force, and weak-amplitude waves become detached from the force. In the area of very small amplitudes there is again a regime birth/decay which as we mentioned cannot be described correctly here.

5

Influence of a negative cubic nonlinearity on the solitary wave-force interaction

When the cubic nonlinear term in the forced extended Korteweg-de Vries equation (2) is taken into account, we should consider the more general case for equations (18) and (22). Qualitatively, the phase portrait is the same as for weak-amplitude solitary waves described above, with the one important remark that wave amplitude is bounded by the limiting value (12). This affects the resonant condition obtained from (18) ∆ + γ 2 = 0.

(41)

Due to this amplitude limitation, the parameter γ is bounded (γ 2 ≤ 6/|β|), and resonance is possible only when 6/β ≤ ∆ ≤ 0.

(42)

To plot the phase portrait, we need to evaluate the integrals (21) and (23). The mass and momentum of the solitary wave are given by the parametric curves s r 6 6 1−B M0 = 4 atanh , a= (1 + B), |β| 1+B |β|

r 3 βγ 2 B = 1+ P0 = (M0 − 2γ), . (43) |β| 6 p The functions M0 (a), normalised by 4 6/|β|; P0 (a), normalised on (6/|β|)3/2 ; F1 (a), normalised by 3/2|β|, and F2 (a), normalised by 9/β 2 are displayed in Figure 6. For very small wave amplitudes these expressions coincide with the Korteweg-de Vries limit (30). In the vicinity of the limiting solitary wave amplitude, all these functions tend to infinity, having the asymptotic behaviour, M0 ∼



alim |ln(1 − a/alim )|;

F1 ∼

alim lnM ; 2

F2 ∼ 9

P0 ∼

alim M; 2

a2lim lnM. 2

(44)

The infinite limit of the functions F1 and F2 leads to a ”thickening” of the trajectories near the upper limit of the wave amplitude. Such trajectories are almost straight lines described by the expressions, obtained from the system (18), (22) (45) X(t) ∼ X0 + (∆ + a2lim )t, ln(M0 (t)/M00 ) ∼

∆ + a2lim f (x(t)). alim /2

(46)

Note that the mass of the solitary wave during the interaction varies significantly, see (46). In the case of resonance the character of the equilibrium point is the same as for the Korteweg-de Vries limit: it is a centre for a positive force, and it is a saddle point for a negative force. When the solitary wave can be captured the characteristic frequency of the variations in the wave amplitude near the centre is 2M0 (1 − a/alim ) ∂ 2 f ω2 = − (X = 0). (47) dP0 /da ∂x2 When a → alim , the frequency is decreased. Physically it is evident because it is difficult to change solitary wave parameters when its mass (and momentum) is large. The phase portrait for a ”supercritical” positive force is presented in Figure 7, when the resonant wave is close to the ”thick” solitary wave. The parameters are: b = 0.3 and ∆ = −0.95. As a result, the captured solitary wave periodically changes its form from a ”thin” to a ”thick” solitary wave. All trajectories in the upper part of the phase portrait press towards the limiting line corresponding to the solitary wave of limiting amplitude. It is important to note that the variation in the wave amplitude is less (in this general case) than 2a0 (the Korteweg-de Vries limit).

6

Influence of a positive cubic nonlinearity on the solitary wave-force interaction

For a positive cubic nonlinear term there are two families of solitary waves of opposite polarities. First, the positive solitary waves will be analysed. There is no limit for the wave amplitude and it can vary widely. Both integrals, (21) and (23) can again be presented in parametric form, r r 6 B−1 6 atan , a = (B − 1), M0 = 4 β 1+B β r 3 βγ 2 B = 1+ P0 = (2γ − M ), . (48) β 6 10

These functions (with the same normalisation as in Figure 6) are displayed in Figure 8. For weak amplitudes all expressions coincide with Korteweg de Vries limit (30), while for the large amplitudes they coincide with the expressions obtained for the extended modified Korteweg-de Vries equation (Pelinovsky, 2000) r r 6 6 βa2 , P0 = a , γ2 = . (49) M0 = π β β 6 βa3 a , F2 (a) = (50) . π 18π Phase portraits of these positive solitary wave dynamics in the field of an external force are similar to the Korteweg-de Vries limit, see Figures (3) - (5), and will not discussed here in detail. The maximal variation of the solitary wave amplitude in the process of the interaction with an external ”super-critical” force (when the solitary wave can be captured for a positive force, or reflected for negative force) in this general case is less than the Korteweg - de Vries limit (39). In particular in the limit of the modified Korteweg-de Vries solitary waves, the maximal and minimal values of the wave amplitude satisfy to equation r 3a2 amin (51) + 3 − min , amax = − 2 4 √ and the sum amin + amax varies from 3 to 2. The next case is the interaction of a negative solitary wave with an external force. All integrals can be evaluated in the parametric form s r 6 6 |B| + 1 atan M0 = −4 , a = (B − 1), β |B| − 1 β F1 (a) =

r

3 P0 = (2γ − M0 ), β

B=−

1+

βγ 2 . 6

(52)

and are displayed in Figure 9. The main feature of this solitary wave is its negative mass. Also its mass (in modulus) decreases an increase of the wave amplitude (in modulus). Formally, changing the sign of the mass in the system (18), (22) is equivalent to changing the polarity of the external force. As a result, the phase portrait in the case of a resonance with an ”supercritical” external force is changed significantly. For instance, a negative solitary wave cannot be captured by a positive ”supercritical” forcing (as in the previous cases), and the equilibrium point is a saddle point. A phase portrait for this situation is shown in Figure 10 for b = 80 and ∆ = −30. The next important comment here relates to the limitation of the solitary wave amplitude, that it should be more (in modulus) than acr , see (14). All

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functions calculated in (52) are finite at this critical amplitude, and correspond to the values for the algebraic solitary wave (15) r p |alim |3 (53) M0 = −π 2|alim |, P0 = π . 2 It means that the solitary wave interaction with an external force is not stopped when the solitary wave transforms to the algebraic solitary wave. For amplitude less than this critical value,the solitary wave is unstable and transforms into a breather (Pelinovsky and Grimshaw, 1997); this domain is shown in Figure 10. The transformation of a solitary wave into a breather, as well as the transformation of a breather into a solitary wave was investigated by Grimshaw et al (1999) for the extended Korteweg-de Vries equation with a variable coefficient of the quadratic nonlinear term. The same transformation can be expected for the present forced extended Korteweg-de Vries equation. The study of this interesting case is postponed for further study.

7

Conclusions

The extended Korteweg-de Vries equation is commonly used as a model for strongly nonlinear internal waves in the ocean. The solutions for different signs of the cubic nonlinearity have a wide variety: solitary waves of positive and negative polarities, ”thick” solitary waves, algebraic solitary waves, breathers. The interaction of these solitary waves with an external force is investigated here in the framework of the forced extended Korteweg-de Vries equation. The equation for the solitary wave parameters (amplitude and position) are derived in the approximation of a weak isolated broad force. This simplified dynamic system is analysed in the phase plane. For solitary waves of weak amplitude the results coincide with the known results obtained earlier in the framework of the forced Korteweg-de Vries equation (Grimshaw et al, 1994). The influence of the cubic nonlinearity on the process of the wave-force interaction is studied for both signs of the cubic nonlinear term. The main new results here are: - the existence of a finite domain for the resonant interaction when there is negative cubic nonlinearity, - the transition of a solitary wave into a breather for the case of positive cubic nonlinearity. Acknowledgements This study was particularly supported for EP from the INTAS (99-1068).

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References [1] Choi, J.W., Sun, S.M., and Shen, M.C. Internal capillary-gravity waves of a two-layer fluid with free surface over an obstruction - Forced extended KdV equation. Physics Fluids, 1996, v. 397 - 404. [2] Grimshaw, R. Internal solitary waves. In “Environmental Stratified Flows”, ed. R. Grimshaw, Kluwer, Boston, 2001a, Chapter 1, 1-28. [3] Grimshaw, R Nonlinear effects in wave scattering and generation. In: Proceedings of the IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity, Manchester, 2001b, (to appear). [4] Grimshaw, R., Chan, K.H., and Chow K.W. Transcritical flow of a stratified fluid: the forced extended Korteweg - de Vries model. Department of Mathematical Sciences, Phys. Fluids. (to appear). [5] Grimshaw, R. and Mitsudera, H. Slowly-varying solitary wave solutions of the perturbed Korteweg-de Vries equation revisited. Stud. Appl. Math., 1993, v. 90, 75-86. [6] Grimshaw, R., Pelinovsky, E., and Bezen, A. Hysteresis phenomenon in the interaction of a damped solitary wave with an external force. Wave Motion, 1997, v. 26, 253 - 274. [7] Grimshaw, R., Pelinovsky, E., and Sakov, P. Interaction of a solitary wave with an external force moving with variable speed. Stud. Appl. Math. , 1996, v. 97, 235 - 276. [8] Grimshaw, R., Pelinovsky, E. and Talipova,T. The modified Korteweg-de Vries equation in the theory of the large-amplitude internal waves. Nonlinear Processes in Geophysics, 1997, v. 4, 237 - 250. [9] Grimshaw, R., Pelinovsky, E., and Talipova, T. Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity. Physica D, 1999, v. 132, 40 - 62. [10] Grimshaw, R., Pelinovsky, E., and Tian, X. Interaction of a solitary wave with an external force. Physica D, 1994, v. 77, 405 - 433. [11] Holloway, P., Pelinovsky, E., and Talipova, T. Internal tide transformation and oceanic internal solitary waves. In “Environmental Stratified Flows”, ed. R. Grimshaw, Kluwer, Boston, 2001, Chapter 2, 29-60. [12] Kharif, C., Pelinovsky, E., Talipova, T., and Slunyaev, A. Focusing of nonlinear wave groups in deep water. JETP Letters, 2001, v. 73, N. 4, 170 - 175.

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[13] Marchant, T.R. and Smyth, N.F. The extended Korteweg-de Vries equation and the resonant flow over topography. J. Fluid Mech. 1990, v. 221, 263 - 288. [14] Marchant, T.R. and Smyth, N.F. Soliton interaction for the extended Korteweg - de Vries equation. J. Appl. Math., 1996, v. 56, 157 - 176. [15] Onarato, M., Osborne, A.R., Serio, M., and Bertone, S. Freak waves in random oceanic sea states. Physical Review Letters, 2001, v. 86, 5831 - 5834 [16] Osborne, A.R., Onorato, M., and Serio, M. The nonlinear dynamics of rogue waves and holes in deep-water gravity wave train. Phys. Letters A, 2000, v. 275, 386-393. [17] Pelinovsky, D., and Grimshaw, R. Structural transformation of eigenvalues for a perturbed algebraic soliton potential. Phys. Lett., 1997, v. A229, 165 - 172 [18] Pelinovsky, E.N. Autoresonance processes under interaction of solitary waves with external fields. Applied Hydromechanics (Ukraine), 2000, v. 2 (74), 67 - 72. [19] Pelinovsky, E., Talipova, T., and Kharif, C. Nonlinear dispersive mechanism of the freak wave formation in shallow water. Physica D, 2000, v. 147, 83-94. [20] Slyunyaev, A., Pelinovskii, E. Dynamics of large-amplitude solitons. JETP, 1999, v. 89, N. 1, 173 - 181. [21] Slyunyaev, A.V. Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity. JETP, 2001, v. 119, 606-612.

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Figure Captions Figure 1. Solitary wave shape for negative cubic nonlinearity. Figure 2. Solitary wave shape for positive cubic nonlinearity. Figure 3. Phase portrait for interaction of the weak amplitude solitary wave with an external positive force, ”sub-critical” case. Figure 4. Phase portrait for interaction of the weak amplitude solitary wave with an external positive force, ”super-critical” case. Figure 5. Phase portrait for interaction of the weak amplitude solitary wave with an external negative force, ”sub-critical” case. Figure 6. Functions, M0 , P0 , F1 and F2 for the negative cubic term. Figure 7. Phase portrait for interaction of the eKdV solitary wave with the positive force (negative cubic nonlinear term). Figure 8. Functions, M0 , P0 , F1 and F2 for the positive solitary waves (positive cubic term). Figure 9. Functions, M0 , P0 , F1 and F2 for the negative solitary waves (positive cubic term). Figure 10. Phase portrait for interaction of the negative eKdV solitary wave with the positive force (positive cubic nonlinear term).

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wave shape

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8

4

0 -6

-3

0

3

6

coordinate Figure 1. Solitary wave shape for negative cubic nonlinearity. 60

40

wave shape

20 0 -20 -40 -60 -80 -0.8

-0.4

0

coordinate Figure 2. Solitary wave shape for positive cubic nonlinearity.

16

0.4

0.8

amplitude

2

1

0

G

-3

0

3

phase Figure 3. Phase portrait for interaction of the weak amplitude solitary wave with an external positive force, ''sub-critical'' case.

amplitude

2

1

0

G

-3

0

3

phase Figure 4. Phase portrait for interaction of the weak amplitude solitary wave with an external positive force, ''super-critical'' case.

17

amplitude

2

1

0

G

-3

0

3

phase

Figure 5. Phase portrait for interaction of the weak amplitude solitary wave with an external negative force, ''sub-critical'' case.

18

P0 4

0

0

0.5

1

a/alim 3

M0

2

1

0

0

F(a)

0.5

1

a/alim

4

F1

2

F2 0

0

0.5

a/alim Figure 6. Functions, M0, P0, F1 and F2 for the negative cubic term. 19

1

amplitude

alim

0

phase

Figure 7. Phase portrait for interaction of the eKdV solitary wave with the positive force (negative cubic nonlinear term).

20

P0 10

0

0

5

10

aβ/6

M0 0.8

0.4

0

0

5

10

aβ/6

F(a) 200

F2 100

F1

0

0

5

aβ/6 Figure 8. Functions, M0, P0, F1 and F2 for the positive solitary waves (positive cubic term).

21

10

P0 15 10 5 0

-15

-10

-5

0

-5

0

-5

0

a(β/6)1/2 -15

-10

0

-1

-2

M0

a(β/6)1/2 -15

-10

0

F2 F1

-400

F(a)

Figure 9. Functions, M0, P0, F1 and F2 for the negative solitary waves (positive cubic term). 22

amplitude |acr|

G

breather zone phase

Figure 10. Phase portrait for interaction of the negative eKdV solitary wave with the positive force (positive cubic nonlinear term).

23