Wave Motion ! 5 (1992) 2 21 -__,
p27
Elsevier
221
Generation of flexural waves on a quasi-one-dimensional
KP soliton by a point-like dipole
Alexey V. Fedorov
Moscow Institute of Physics and Technology. Moscow. USSR
Boris A. Malomed
P,P. Shirshou institute for Oceanology of the USSR Academy of Sciences. 23 Krasikov Street. Moscow, GSP-7. USSR
Received !1 October 1990
Interaction of a quasi one-dimensional soliton of the Kadomtsev-Petviashvili (KP-II) equation with a moving local dipole,
described by an additional term in the equation, is investigated analytically by means of the Lagrangian-averaging (Whitham's)
technique. The main result of the interaction is the generation of left- and right-going flexural waves on the crest of the soliton.
The generated flexural waves are found ir~ an explicit form.
I. Introduction
The present work is devoted to the analysis of soliton dynamics tbr the Kadomtsc~ Petviashvili equation
(the so-called KP-II, see Ref. [1]) perturbed by a term describing a moving point-like dipolar source"
( Ut + 6UU,, + U,~,), + 3Uw = e 6 " ( x - vt)6( y - ut)
(!.!)
where ~ and 6" are, respectively, the delta function and its second derivative, (v, u) is the velocity vector
of the moving dipole, and e is its "power". Equation (1.1) is a natural two-dimensional generalization of
the well known perturbed Korteweg-de Vries (KdV) equation which describes long waves produced by a
small moving body in shallow water [2-6]. In Refs. [2-5], the generation of solitons by the dipole and
other dynamical effects has been studied numerically. Interaction of a soliton with the dipole has been
investigated in detail by means of the perturbation theory based on the Hamiltonian formalism and inverse
scattering transform [7] in Ref. [6]. As concerns the perturbed KP equation (1.1), the generation of quasilinear waves by the dipole (in the absence of solitons) has been recently considered by one of the present
authors [81.
The aim of the present work is to study the interaction of the dipole with the quasi one-dimensional
so!ito_n, which, in the case of e = 0, is given by the exact stable so!ut.io_n_of eq. (!. !) [9].
Usol = 2K "2 sech2(h'(x
- x(,(t)),
xo(t)=4~c2t,
(1.2)
(1.3)
where x is an arbitrary amplitude of the soliton. When the dipole crosses the sofiton's crest, one may expect
two effects: emission of radiation in the form of quasi-linear waves, and excitation of flexural waves on the
0923-5965/92/$05.00
1992 .... Elsevier Science Publishers B.V. All rights reserved
A. V. Fedorov, B.A. Malomcd / Flexural waves on soliton
222
crest. The former effect is also known in the one-dimensional situation [6]. In this work we will concentrate
on the latter effect which is a purely two-dimensional one.
In principle, one can try to attack this problem directly, looking for a perturbed solution of eq. (1.1)
close to the unperturbed one (1.2). However, we find it more convenient to solve the problem indirectly
by means of the Lagrangian-averaging technique similar to the well-known Whitham's method [ 10]. The
method is based upon assuming the soliton's amplitude ~c2, and velocity, ,~0, (see eq. (1.1)) to be slowly
varying functions of y and t. Next, one should insert the solution (1.2) with the slowly varying parameters
into a Lagrangian density of the perturbed KP system, integrate it over the rapid variable X, and find an
effective Lagrangian density for the slowly varying functions. In this manner, one arrives at an evolution
equation to govern low-frequency long flexural waves on the soliton.
Following this approach, we derive in Section 2 the equation tbr the long flexural waves, which proves
to be a forced D'Alembert equation. Its solutions are found in Section 3. At t --. oo, the flexural disturbance
produced by the dipole-soliton collision decays into two wave trains with permanent shapes travelling to
the left and to the right.
Finally, in concluding Section 4 we discuss the applicability conditions for the analysis that has been
developed. It is shown that the analysis can be warranted for two cases: if the y-component u of the dipole's
velocity (eq. ( 1.1 )) is large, u >> ~, u 2 >> v, or if its x component v is sufficiently close to the soliton's velocity,
Iv-41¢21 ,~ o. The latter (near-resonant) case is the most interesting one.
2. The Lagrangian-averaging technique
To introduce the Lagrangian description, one needs to rewrite eq. (1.1) in terms of the potential function
[10], ep(x, y, t), where U= ¢,:
q~,, + 6q~,q~,,. + ¢,,.,, + 3~b,.y= e 8 ' ( x - v t ) 8 ( y - u t ) .
(2.1)
As a matter of fact, eq. (2.1) has been obtained from eq. (1.1) by integrating it once over x with zero
boundary conditions at Ixl = ~ . The Lagrangian density for eq. (2.1) is
1
I
2
3
"~
L = 2¢,~,- + ~b~~- 2,6,.,.,. + ~.~by- ~:~.,.t~(x- v t ) t ~ ( y - u t ) ,
(2.2)
where the last term accounts for the perturbation. In terms of the unperturbed soliton solution, (1.2) takes
the form
~.,ol = 2x t a n h [ x ( x - x0(t)].
(2.3)
Let us proceed to the description of the long flexural waves. Following the steps of Whitham's method
[!0], we 't t., o.~., t.t ' '~"
J .... ". . . .~.z...z j , assum ing that r and (X0), are s iowiy
t a l ~. . .v.v.a v. ~.,. ,-..._.
tOl|l!
~,-~
. . 2 ] ~, :
l t.l t.O. . t i L_,
ll;:; L~li:lll~li:lll :
ucn~lty
varying functions of), and t, and that the unperturbed relation X o ( t ) = 4~¢2t (see eq. (1.2)) is valid only as
the zeroth approximation, lntegratiqg the resultant Lagrangian density over the rapid coordinate x, one
arrives at the following effective (averaged, in terms of the Whitham's technique [10]), Lagrangian"
3
£ = 16
2,
"
[-(Xo),+.~(Xo)~]+~ -5
--
--
2epc26(v
ut) sech2[~c(xo-ut)].
(2.4)
A.V. Fedorov, B.A. Malomed / Flexural waves on soliton
223
The effective Lagrangian (2.4) is written in the lowest approximation. We have omitted the terms r.~.(Xo)~.
and toy,
2 which would give rise to higher-order terms in the resultant long-wave equation, see below.
However, the coefficient in front of the term r~,2 in the full Lagrangian density is equal to 6 tanh2[K-(x _ xo)] .
Integrating this term over x results in a divergence. As it will be demonstrated below, this divergence
imposes a limitation on a time scale over which our approach is valid. For the time being, we will introduce
a large limit R for the integration over x, Ixl ~<R. Thus, the additional term in the effective Lagrangian is
A/~= 12R~'~.
(2.5)
The effective Lagrangian (2.4) yields the equations of motion
½~c2[- (Xo), + 3(X0)y21+ 2~c4 - ~ 6 3 ( y - u t ) {
- ~a¢2~¢,+ [(Xo)yrc3ly+ ~ 6 ~ ( y -
tc2 seeh2[ ~c(xo- vt)]} ".= 0
ut){ r 2 seth2[ ~C(Xo- vt)] }~.0= 0.
(2.6)
(2.7)
It is natural to linearize eqs. (2.6) and (2.7), assuming
~c = Xo + ~:t(y, t),
(2.8)
where a¢1 is a small variable part, and ~Cois a large constant part of the soliton's amplitude. Finally, the
quantity r~ may be excluded in favor of Xo, and the linearized equation that we seek takes the form:
(Xo), - 16x'o2(Xo)yy= -'a6rc~2{6(y-ut)[~CZosech2(~,'o(Xo- vt))],:o,;~'
+ 2 6~'ot~( y - ut){sech2[tCo(Xo - vt)]j~'.~o.
(2.9)
A solution to eq. (2.9) will be looked for in the form t,-,.t"¢eq. ~,..ujj~"
Q~I
Xo = 4K'o2t + Z ( Y , t).
(2.10)
The first term in eq. (2.10) describes the undisturbed uniform motion of the soliton, while the second one
accounts for the flexural waves propagating along its crest. In the lowest approximation, it is sufficient to
substitute Xo on the right-hand side of eq. (2.9) (after the differentiations) by 4~C2ot.Thus, insertion of eq.
(2.10) into eq. (2.9) brings the latter equation into the form of the forced D'Alembert equation"
, ~ , - 16x~Zy,, = 6 { - ~ t c o 2 [ $ ( y - u t ) ( I ¢ 2 o
sech2(tco(Xo-Vt)))'~o]~
P
"t
+ 2K'oS(y - ut)( seth 2( ~Co(Xo - vt) ) ):,o }Ixo=,,-~t.
(2.11)
Note that in the unperturbed case (6 = 0) we recover the homogeneous D'Alembert equation for the long
flexural waves derived first by Kadomtsev and Petviashvili [9].
3. Solving the forced wave equation
To solve eq. (2.11 ), we will apply the well-known Green's function G for the D'Alembert equation, which
satisfies the condition G = 0 at t = - ooG( y - y', t - t') = ( 8~Co) - ~Q( 4sCo( t - t') - Iy - .v' I),
(3.1)
224
A.V. Fedorov. B.A. Malomed / Flexural waves on soliton
Q(=) being the step function: Q(-) = 0 if - < 0 , and Q(z)= 1 if .z>_.0. A convolution of the right-hand side
ofeq. (2.11) with the Green's function (3.1) yields a solution in explicit form. For the case u<4xo we have
(for the sake of definiteness, we set u ~ 0 ) :
Z ( Y , t) = - ~tQ( .v - ut){[(tCo(4~Co-U)) -I + ( o - 4 x ' ~ )
-I] sech 2
- (~Co(4~¢o-u))- ~ sinh ~ sech 3 ~ } - ~ O ( u t - y , , { [ ( ~ C o ( 4 ~ C o + U ) ) -~
+ (v-4~-o) -t] sech 2 r / - (X'o(4Xo+U))-t r/sinh
1/sech3 q},
(3.2)
where
,~-= ~¢o(O - 4~co~(4~,-o- U)- I ( y - 4tCot)
(3.3)
1/= tCo(V- 4K'0)(4X'o+ u) - i(), + 4tOot~.
(3.4)
In the opposite case, u > 4~Co, we obtain:
Z(Y, t ) = ~ e Q ( u t - y ) { [ ( v - 4 ~ C o )
-I -(Xo(u-4~¢o)) -~] sech"
+ (~Co(U-4~Co))-I~ sinh ~ sech 3 ~-[(SCo(U +4~Co)) -t + (v-4 Co) -']sech" I/
+ (Xo(U+4Xo))- ~r/sinh 1/sech 3 1/}.
(3.5)
Straightforward consideration of eqs. (3.2) to (3.5) demonstrates that at t ~ r, where the transient time
r may be estimated as
r ~ t,%-IIv-4~'ol -I
(3.6)
the flexura! wave ~...,,..,.,,.,.,',,-,,~,",',~,~
r,,,,..,,r,,~,.,,.,--v--,,-'4;'"1"decays into ,,.,,.i,~',and right-going pulses of a permanent shape. If
u = 0, the two pulses are symmetric, and they take the form shown in Fig. 1(a) if
(4x~) ~ + ( v - 4 r c ~ ) ~ > 0
(3.7)
or either form shown in Figs. l(b) and l(c) in the opposite case. If u # 0 , the pulses are asymmetric (they
differ in size and amplitude). In particular, the right-going pulse has a iazger amplitude and a smaller size
as compared to the left-going one (recall that we have set u~>0, i.e., the pulse with the larger amplitude
and smaller size travels in the same direction as the dipole). Finally, if the asymmetry is sufficiently strong,
the two pulses may belong to different types (see Figs. i(a)-(c)).
4. Applicability conditions
As was mentioned in Section 2, the higher-order correction (2.5) to the effective Lagrangian contains
t,c u~vc~gc,~t coc~nc~ent R (recall we have imposed the limitation i.x'i< R on the coordinate x). it is straightforward to see that this term gives rise to the additional term
6 r o t RZ,.,.,.., ,
(4.1)
on the left-hand side of eq. (2.11).
Evidently, Aae analysis developed in the preceding section remains relevant as long as the term (4.1) may
be neglected ~ compared to the basic term )~',, on the left-hand side of eq. (2.11). The explicit solutions
A.V. Fedoroo. B.A. Malomed / Flexural waves on soliton
225
Z
I
Fig. !. Three possible shapes of the left- and right going flexural pulses on the soliton's crest created by the collision with the dipole.
The traveling coordinate : is ~ or q, eqs. (3.3) and (3.4). The shape (a) takes place if the inequality (3.7) holds, and (b) and (c) are
possible in the opposite case.
specified by eqs. (3.2) to (3.5) give rise to the following estimate for a characteristic size A.of the generated
wave trains:
A ~- ~r{T'l(u - 4K'o)/(v - 41co)1.
(4.2)
The required inequality Rg,.yyy ,~, g.,.y corresponds to R ,~ A2. Finally, eq. (4.2) yields the following limitation
on the overall size R:
(4.3)
R ,~ K'o(U- 4 ~,'o)2/ ( o - 4 K'o)".
The distance R is traveled by the soliton during the time T-,, R/~co. Thus, it follows from eq. (4.3) that the
time of applicability of our analysis is limited as follows:
t ~ R/~c~"~. ~Co~(U -
4~-o)-/(v
"
-
4~'g) 2.
(4.4)
Finally, the fundamental condition under which the solutions obtained in the preceding section are relevant
is that the transient time given by eq. (3.6) must satisfy the inequality (4.4):
lu - 4K'~l~ (u - 4 ~:o)'.
(4.5)
First of all, the inequality (4.5) implies that the quantity (u-4~'o) may not be too small. Note that,
at (u - 4~Co)--, 0, the solutions (3.2) and (3.5) become singular. Thus, the inequality (4.5) precludes the
singularity.
A.V. Fedorov. B.A. Malomed / Flexurai waves on so!iton
226
Next, a straightforward consideration demonstrates that the strong inequality (4.5) can be satisfied either
if
u-"~ h'~,
u-'>>v
(4.6)
or
Iv-4~col ,~ v.
(4.7)
The former case (eq. (4.6)) is not of great interest. The latter one (eq. (4.7)), which can naturally be called
the near-resonant case, seems much more interesting.
The inequality (4.7) (or (4.6)) is the fundamental condition for the applicability of our approach based
on the Lagrangian-averaging technique. Note that there is another natural condition which must be satisfied: the characteristic size Z of the flexural wave train must be much greater than the squared size too 2 of
the soliton (1.2). Using the estimate (4.2) for Z, it is easy to see that the inequality (4.7) implies that this
condition is satisfied as well.
To conclude the discussion of the role of the additional term (4.1) in eq. (2.11 ), let us note that this term
has the physical meaning of spatial dispersion, which, at very large times, must result in dispersive spreading
out of the flexural wave trains.
Another substantial approximation employed in our analysis is linearization of eqs. (2.6) and (2.7). A
maximum amplitude A of the wave trains specified by eqs. (3.2) and (3.5) may be estimated, with regard
to the underlying inequality (4.7), as follows:
(4.8)
A "- e,/Iv-4t¢ol.
Comparing the basic terms in the linear equation (2.11) to nonlinear corrections and making use of eq.
(4.8), one can find that the condition necessary for the linearization takes quite a natural form: e,~ K'o.
Finally, it is noteworthy that in the limit Iv- 4K'oZl,~ v which is crucial for our analysis, the energy of the
generated flexural wave trains can grow beyond bounds. Indeed, in this limit the energy of the two pulses,
calculated as
I
"~
E= 2
--
is
"~
-~
[(Xo)7 + 16tc~(Xo);] dy
;(
equal:
Eright = ~8
2 ~ c3o l ( 4 1 ¢ o - U ) ( V - 4 ~ ' o ) l
El~ft =~e
s 2~ c3o ( 4 ~ : ~ + u ) I v - 4 , ' c a ) l
i
,
- '
And as for the case >>tc~, u2>>v the energy carried by each of the pulses becomes proportional to u-3
and vanishes if the value of u increases.
In conclusion, let us note that the Lagrangian technique makes it possible to solve analogous problems
for other s~stems supporting stable quasi one-dimensional solitons; for example, for the two-dimensional
sine-Gordon system, where the soliton is the topological kink [11]. In Ref. [11], more general problems
have also been solved" Generation of flexural waves by a periodic chain of point-like defects or by a
rectangular lattice of the defects. Another generalization developed for the two-dimensional sine-Gordon
model in Ref. [l I] is generation of compressionai waves in a quasi one-dimensional periodic array of
solitoi~s (cnoidal wave). It would be interesting to perform a similar analysis for the KP model.
A. V. Fedorov. B.A. Malomed +/Fiexural waves on soliton
227
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