- Email: [email protected]
Internal solitary waves near a turning point
Volume 75A, number 5
PHYSICS LETTERS
4 February 1980
INTERNAL SOLITARY WAVES NEAR A TURNING POINT C.J. KNICKERBOCKER and Alan C. NEWELL Department of Mathematics, Clarkson College of Technology, Potsdam, NY 13676, USA Received 28 November 1979
We report an unexpected result and illustrate the mechanism by which a solitary wave propagating on a thermocline reverses its polarity as it passes through a turning point near the shoreline.
1. Introduction. Long solitary waves in lakes and estuaries, propagating on the thermocline separating two shallow layers of fluid of almost equal densities, are approximately described by the Korteweg—de Vries equation. The effect of the change in depth of the bottom layer, which the wave feels as it approaches the shore, results in the coefficient of the nonlinear term being a slowly varying function of position which has a zero at the point (the turning point) where the depths of the top and bottom layers are approximately equal. Since a solitary wave represents a balance between quadratic nonlinearity and linear dispersion, in the present context the sign of the nonlinear term determines that the wave must always face into the deeper layer. It has therefore been argued [1] that the solitary wave ceases to exist at the turning point and disintegrates into a train of dispersive nonlinear waves there. Although this does indeed occur, it is far from the whole story. What happens in addition is that the solitary wave, which initially faces into the bottom layer, develops in its wake a long shelf of the opposite polarity as it approaches the turning point. After the turning point, the original downward facing solitary wave disintegrates, but the upward facing shelf forms a new upward facing solitary wave which propagates through to the shoreline. 2. The model. Consider the situation as shown in fig. 1. Following the ideas of refs. [1,2],we find the equation which describes the nondirnensional elevation,
326
~.
hf
y
h
2+ N(z.t)
H,(z)
h2
Fig. 1. Physical system being considered, where N(x, t) is the dimensional elevation of the thermocline and the densities of the upper and lower fluids are P1
and P2~respectively.
V(X, 0) = D(X)JsJ(x, t)/ { [H2(X)/h2 1] (eh2)}, of the thermocline separating two shallow layers of fluid (the effects of surface waves are neglected) of slightly different densities p1 ~ (p1
V1 ÷A(X)VV0 -i-B(X) V008 + C(X)V = 0, 312x/h where I = e 2 measures position, 2(r)dr (ge/h 2t 0 = e~f D~ 2)~’ is the negative of the retarded time, —
(2.1)
Volume 75A, number 5
(1 =
PHYSICS LETTERS
p1 /p2) [1 H2(X)/h2] 1 + p1 [h2 H2(~]/p2h1 —
—
—
x and t are the dimensional space and time coordinates and 0 < ~ 1 is the ratio of wave amplitude to depth. The coefficients A(X), B(X) and C’(X) are defined by 12(lp A(X) = 3(1 p1l~/p2l~)/2D~ 1/p2)l2, 312(1 B(X) = 1112(12 + p111/p2)/6D 1 + p112/p2),
4 February 1980
analyses [2,3,6,8] which discuss a perturbation theory for it was only recently [4,5] the eq. role(2.4). of theHowever, shelf in creating or destroying massthat was appreciated. The global structure of the shelf, which is needed to show that f°°~ud0 is indeed a constant (in r) of the motion, was first given in ref. [7]. Here we follow the prescription given in refs.
—
—
CLX) = —D1/4D
+
121/212,
where 1~= h1/h2 and 12 = 1 H2(X)/h2 are considered to=be0(a)), slowly (i.e. H~Hm. sovarying that thefunctions change inofI the bottom slope occurs over many solitary wave widths; however, the terms reflecting changes in A, B and Care more important than the next terms (higher order non linearity and dispersion) which would appear in eq. (2.1) (i.e. ~2 ca ~ e). Note that the coefficient A(X) —
~
2_
[6,7] and show how, by a judicious use of the conservation laws of eq. (2.4), we can obtain all the leading order behavior for distances r and retarded times 0 of order 1/a. We begin at the point r = 0 when the perturbation is switched on with the pure soliton state, 2sech2~i(0 0), 0~.= 4~2 ~= no. q5(r, 0) =calculate 2i~ First we the slow change in the parameter n(r) induced by the p~rturbation.By using the conservation of energy —
j
a ar
-_
q~dO
=
—~
_OO
of the nonlinear term changes sign where P211 p 1l~,which when p2 Pi = O.O6p1 occurs when Ii and’1 2 are approximately equal. Using the transformation 0) = A(X)V(X, 0)! 1B(r)dr weW(r, obtain B(~,where r = f W~+ WW 0 + W000 + (C + Br/B A ~/A)W = 0. (2.2)
f
2f~
q~dO,
—=
—
—
—
T,
Once H2(X) begins change, the terms A~A B1BT and Care allto important; however, as the
turning point (r = 1/a) is approached, the term A 1A 7 is dominant. This allows us to simplify our model further by neglecting C’(r), changes in B(r) and writing A(r) = oK(r 1/a), where —
112(r)B(r),
aK = —3pil2~(r)/l~(p2 p1)D evaluated at r = 1/a. Setting u(r, 0) = KW(r, O)/A(r), we obtain what we shall call the TKdV (transitional Korteweg—de Vries) equation, —
213(r). Second, from the conservawe obtain n(r) = ~0f tion of center of gravity ~ Oq dO = 3 q2 dO +~ Oq dO,
J
f
f
we fifld that to leading order, the :olitary wave velocity 2(r)[the true velocity, in the physical coorO~.is 4~x, t is dinates 0t/(0x O~) —
=
~~~ (gDh
1~’2B2(X)KD3I2(X)N(x,t) 2)~~ + 2(gDh2) A(X)[H 2(X)—h2} which can be integrated to give 0 as a function of r: 0(r) = 1 2n~{ 1 [—f(r)]713} /7o. —
Next, we calculate the shape of the secondary strucu~.+f(r)uu 0
+
u000
0,
(2.3)
where f(r) —1 + or. We exploit the fact that f(r) changes slowly with r by setting u(r, 0) = 6q(r, 0)! f(r), whence eq. (2.3) becomes +
6qq0
+
q090
(f~/f)q.
(2.4)
3. Perturbation theory. There have been many
ture. Because the solitary wave parameter has been already modulated to satisfy the conservation of energy requirement (the only consistent choice), the local mass conservation relation cannot be satisfied. Only two-thirds of the extra mass created (depleted) per unit distance by the perturbation can be absorbed (lost) by the solitary wave. Therefore a shelf q~(r,0) is created between 0 = 0 and 0 = 0, the present posi327
Volume 75A, number 5
PHYSICS LETTERS
4 February 1980
tion of the solitary wave. Its height upon creation can be found by balancing the expression for the local conservation of mass,
universally and results from other important examples are listed in the appendix.
a
4. Numerical results. Here we focus on fig. 2 which gives the results of a numerical integration of eq. (2.3) ufrom r = 0, with a = 1/20 2p and initial conditions 5(O, 0) = —12n~sech 0O.
a
=
~
dO —=
+ ~— fqc dO 2(r) 0yields
f
f
=
~
q~dO.
UsingO~= 4n
[f~I3p(r)f(r)]I~=~_,
0) =
~
We used a modification of the Vliegenthart scheme
—
which may be written as a function of 0 using 317,tthe he last relation f(fl = aIr 1 = —(1 7aO/12n~) expression obtained isbysmall integrating O~it= varies 4n~.Sslowly ince the shelf amplitude and since with respect to 0, its subsequent evolution is given by balancing q~and f~q/f.Thus, when the solitary wave is at at retarded time 0, the height of the shelf at 0 is
It is important to stress that while the shelf amplitude is order a, its mass content is order one. Therefore the shelf plays a crucial role in the leading order description of the system. For example, from our results we note that solitary wave for TKdV is given by
[9] conservation and checked of accuracy by continuously monitoring mass (fudO = —24r~ 0)andfixed energy 2dO = 192~~). In fig. 2, we drew, at each (Ju position r, the negative (and retarded; the phase depends on location) time history 0 of the pulse. For locations between r = 0 and 15 (r = 16 is the approximate breakdown position), the results of perturbation theory and numerical experiment were compared and are in close agreement. After the breakdown of the perturbation theory and before the turning point 15
u s’~O r~ 2 Ir_f( ~ —1/3 I = 12 fl() J ~TIJ
ing gain of (positive) the begins shelf takes place at over smaller intervals mass and aofpeak to appear
—
—
~,
—
-
—
—
—
q~(°, r) = qc(O, r)f(r)/f(r) 2 8/7
—
o(ar—l)/3nn(l---7a0/l2nn)
—
,
0<0<0.
‘
X sech2
{~0[—f(r)] 2/3 (0
— —
0)}
.
the front of the shelf. At the turning point r = 20, the peak is quite pronounced. As the turnmg point is passed, the peak and the remaining shelf breaks up into a train of pulses. These pulses, if they are to be .
its mass content is ~—24n 3theshelf uc(O, r) 0(-—f)~’ is 2a/?lo(l aO/l2p~)8/7, 0<0<0, and zero elsewhere and its mass content is 24n 13 24n 0(—J)~ 0. Note the conservation of total mass f°°,,,u dO to leading order depends crucially on the existence of the shelf. We also observe that while the solitary wave decreases in amplitude, its mass content and consequently that of the shelf increases without bound as the turning point is approached. Indeed, the perturbation theory breaks down at a point where the amplitudes of the solitary wave and the shelf are of the same order, when r = (l/a)[l 0(01/3)]. Nevertheless, as we see from the numerical results which follow, the perturbation theory gives us qualitatively accurate results beyond the breakdown point; the solitary wave (and consequently the shelf) continues to gain negative (positive) mass. We remark that the perturbation procedure shown here can be used quite 328 —
.
—
—
solitary waves, must still pass through the valley created by the remains of the original solitary wave. The criterion (determined numerically; one can also make a plausible analytical argument using inverse seattering theory) that a solitary wave forms is that for some r, the amplitude of the leading pulse is at least twice that of the valley. At r = 30, we can see that, in this case, the leading pulse satisfies this critenon and is beginning to separate from the others which form part of a dispersive wave train. At r = 40, at which point we take f(r) = 1, the new upward facing solitary wave is about to emerge from the effects of the disintegrating original solitary wave. We continued the numerical calculation till r = 200 in order to determine that the emerging solitary wave has indeed 2 the character Korteweg—de X sech2k(Oof a O~ 4k2r). Vries soliton, 12k —
—
Volume 75A, number 5
PHYSICS LETTERS
4 February 1980
2 >100
2’
•
40
a -j
Q.
2 4
w
V• 0
~\~fc0
20
—I-
0
Fig. 2. Numerical integration of eq. (2.3) with
15
40
DISTANCE
-
n~=
1 and a
=
1/20. The turning point occurs at r
20.
2 0 I0 -j
4
z
I Ui I-
•
7 V 7
7
7
V
V —.——
/
‘II
/
/ .5,
/ I‘
1.0
~ —
I
1.5
INITIAL SOLITON
Fig. 3. This graph shows (—) the relationship between i~( r~)of the initial soliton and k of the terminal soliton for various values of a (a 1/40, 1/20, 1/5, 1/2 reading from left to right). The line (— — —) represents the relation k = 2~o/3.The lines (- - -) appearing at the lower portion of the graph are due to the inability of the numerical scheme to give precise information for small k.
329
Volume 75A, number 5
Fig. 3 is a graph of k versus
PHYSICS LETTERS
for various a. Note
that as long as n0 is large enough, the amplitude of the first upward facing pulse will be sufficient to create a solitary wave. For larger values of ~0, we note the remarkable result, for which we have 2n no analytical explanation at the present time, that k = 0/3, independent of a. Appendix. Perturbation results for various equations. qt +6q’q1 +q~~1 =F, 2~n(X = a sech ar (r+ l)(r+2)~2/3r2. —
I),
Exam
4n 1leand1: r = Xt ~
1, F = aq; i~=
exp(2at/3),
514, for 0< I < I. 0(l+ aX/3~~) Example 2: r = 1,F aq 1~p = p0(l + l6p~at/ 2 and 15)1/2, ~ 4p = (8p 0a/15)exp(—2aX/15), for 0<1
qc
=
a exp (at)/3~
n0
~
330
Example 3: r =
4 February 1980 2,
F = oq; n =
n0 exp(2at), X~~
and =
ira exp(2at)/n~(l+ 4aX/n~), for 0 <1<1.
References [1] V.D. Djordjevic and L.G. Redekopp, J. Phys. Ocean. 8 (1978) 1016. [2] R.S. Johnson, Proc. Camb. Phios. Soc. 73 (1973) 183. [3] 5. Leibovich and J.D. Randall, J. Fluid Mech. 53 (1973) 481. [41 307. V.1. Karpman and E.M. Maslov, Phys. Lett. 60A (1977) [5] D.J. Kaup and A.C. Newell, Proc. Roy. Soc. London A361 (1978) 413. [6] CJ. Knickerbocker and A.C. Newell, Shelves and the Korteweg—de be published. Vries equation, J. Fluid Mech. (1979), to [7] A.C. Newell, Soliton perturbation and nonlinear focussing, Symp. on Nonlinear structure and dynamics in condensed matter (Oxford Univ., Solid State Physics), Vol. 8,Phys. pp. 52—68. [8] (Springer, E. Ott and1978) R.N. Sudan, Fluids 13 (1970) 1432. [9] A.C. Yliegenthart, J. Eng. Math. 5 (1971) 137.
PHYSICS LETTERS
4 February 1980
INTERNAL SOLITARY WAVES NEAR A TURNING POINT C.J. KNICKERBOCKER and Alan C. NEWELL Department of Mathematics, Clarkson College of Technology, Potsdam, NY 13676, USA Received 28 November 1979
We report an unexpected result and illustrate the mechanism by which a solitary wave propagating on a thermocline reverses its polarity as it passes through a turning point near the shoreline.
1. Introduction. Long solitary waves in lakes and estuaries, propagating on the thermocline separating two shallow layers of fluid of almost equal densities, are approximately described by the Korteweg—de Vries equation. The effect of the change in depth of the bottom layer, which the wave feels as it approaches the shore, results in the coefficient of the nonlinear term being a slowly varying function of position which has a zero at the point (the turning point) where the depths of the top and bottom layers are approximately equal. Since a solitary wave represents a balance between quadratic nonlinearity and linear dispersion, in the present context the sign of the nonlinear term determines that the wave must always face into the deeper layer. It has therefore been argued [1] that the solitary wave ceases to exist at the turning point and disintegrates into a train of dispersive nonlinear waves there. Although this does indeed occur, it is far from the whole story. What happens in addition is that the solitary wave, which initially faces into the bottom layer, develops in its wake a long shelf of the opposite polarity as it approaches the turning point. After the turning point, the original downward facing solitary wave disintegrates, but the upward facing shelf forms a new upward facing solitary wave which propagates through to the shoreline. 2. The model. Consider the situation as shown in fig. 1. Following the ideas of refs. [1,2],we find the equation which describes the nondirnensional elevation,
326
~.
hf
y
h
2+ N(z.t)
H,(z)
h2
Fig. 1. Physical system being considered, where N(x, t) is the dimensional elevation of the thermocline and the densities of the upper and lower fluids are P1
and P2~respectively.
V(X, 0) = D(X)JsJ(x, t)/ { [H2(X)/h2 1] (eh2)}, of the thermocline separating two shallow layers of fluid (the effects of surface waves are neglected) of slightly different densities p1 ~ (p1
V1 ÷A(X)VV0 -i-B(X) V008 + C(X)V = 0, 312x/h where I = e 2 measures position, 2(r)dr (ge/h 2t 0 = e~f D~ 2)~’ is the negative of the retarded time, —
(2.1)
Volume 75A, number 5
(1 =
PHYSICS LETTERS
p1 /p2) [1 H2(X)/h2] 1 + p1 [h2 H2(~]/p2h1 —
—
—
x and t are the dimensional space and time coordinates and 0 < ~ 1 is the ratio of wave amplitude to depth. The coefficients A(X), B(X) and C’(X) are defined by 12(lp A(X) = 3(1 p1l~/p2l~)/2D~ 1/p2)l2, 312(1 B(X) = 1112(12 + p111/p2)/6D 1 + p112/p2),
4 February 1980
analyses [2,3,6,8] which discuss a perturbation theory for it was only recently [4,5] the eq. role(2.4). of theHowever, shelf in creating or destroying massthat was appreciated. The global structure of the shelf, which is needed to show that f°°~ud0 is indeed a constant (in r) of the motion, was first given in ref. [7]. Here we follow the prescription given in refs.
—
—
CLX) = —D1/4D
+
121/212,
where 1~= h1/h2 and 12 = 1 H2(X)/h2 are considered to=be0(a)), slowly (i.e. H~Hm. sovarying that thefunctions change inofI the bottom slope occurs over many solitary wave widths; however, the terms reflecting changes in A, B and Care more important than the next terms (higher order non linearity and dispersion) which would appear in eq. (2.1) (i.e. ~2 ca ~ e). Note that the coefficient A(X) —
~
2_
[6,7] and show how, by a judicious use of the conservation laws of eq. (2.4), we can obtain all the leading order behavior for distances r and retarded times 0 of order 1/a. We begin at the point r = 0 when the perturbation is switched on with the pure soliton state, 2sech2~i(0 0), 0~.= 4~2 ~= no. q5(r, 0) =calculate 2i~ First we the slow change in the parameter n(r) induced by the p~rturbation.By using the conservation of energy —
j
a ar
-_
q~dO
=
—~
_OO
of the nonlinear term changes sign where P211 p 1l~,which when p2 Pi = O.O6p1 occurs when Ii and’1 2 are approximately equal. Using the transformation 0) = A(X)V(X, 0)! 1B(r)dr weW(r, obtain B(~,where r = f W~+ WW 0 + W000 + (C + Br/B A ~/A)W = 0. (2.2)
f
2f~
q~dO,
—=
—
—
—
T,
Once H2(X) begins change, the terms A~A B1BT and Care allto important; however, as the
turning point (r = 1/a) is approached, the term A 1A 7 is dominant. This allows us to simplify our model further by neglecting C’(r), changes in B(r) and writing A(r) = oK(r 1/a), where —
112(r)B(r),
aK = —3pil2~(r)/l~(p2 p1)D evaluated at r = 1/a. Setting u(r, 0) = KW(r, O)/A(r), we obtain what we shall call the TKdV (transitional Korteweg—de Vries) equation, —
213(r). Second, from the conservawe obtain n(r) = ~0f tion of center of gravity ~ Oq dO = 3 q2 dO +~ Oq dO,
J
f
f
we fifld that to leading order, the :olitary wave velocity 2(r)[the true velocity, in the physical coorO~.is 4~x, t is dinates 0t/(0x O~) —
=
~~~ (gDh
1~’2B2(X)KD3I2(X)N(x,t) 2)~~ + 2(gDh2) A(X)[H 2(X)—h2} which can be integrated to give 0 as a function of r: 0(r) = 1 2n~{ 1 [—f(r)]713} /7o. —
Next, we calculate the shape of the secondary strucu~.+f(r)uu 0
+
u000
0,
(2.3)
where f(r) —1 + or. We exploit the fact that f(r) changes slowly with r by setting u(r, 0) = 6q(r, 0)! f(r), whence eq. (2.3) becomes +
6qq0
+
q090
(f~/f)q.
(2.4)
3. Perturbation theory. There have been many
ture. Because the solitary wave parameter has been already modulated to satisfy the conservation of energy requirement (the only consistent choice), the local mass conservation relation cannot be satisfied. Only two-thirds of the extra mass created (depleted) per unit distance by the perturbation can be absorbed (lost) by the solitary wave. Therefore a shelf q~(r,0) is created between 0 = 0 and 0 = 0, the present posi327
Volume 75A, number 5
PHYSICS LETTERS
4 February 1980
tion of the solitary wave. Its height upon creation can be found by balancing the expression for the local conservation of mass,
universally and results from other important examples are listed in the appendix.
a
4. Numerical results. Here we focus on fig. 2 which gives the results of a numerical integration of eq. (2.3) ufrom r = 0, with a = 1/20 2p and initial conditions 5(O, 0) = —12n~sech 0O.
a
=
~
dO —=
+ ~— fqc dO 2(r) 0yields
f
f
=
~
q~dO.
UsingO~= 4n
[f~I3p(r)f(r)]I~=~_,
0) =
~
We used a modification of the Vliegenthart scheme
—
which may be written as a function of 0 using 317,tthe he last relation f(fl = aIr 1 = —(1 7aO/12n~) expression obtained isbysmall integrating O~it= varies 4n~.Sslowly ince the shelf amplitude and since with respect to 0, its subsequent evolution is given by balancing q~and f~q/f.Thus, when the solitary wave is at at retarded time 0, the height of the shelf at 0 is
It is important to stress that while the shelf amplitude is order a, its mass content is order one. Therefore the shelf plays a crucial role in the leading order description of the system. For example, from our results we note that solitary wave for TKdV is given by
[9] conservation and checked of accuracy by continuously monitoring mass (fudO = —24r~ 0)andfixed energy 2dO = 192~~). In fig. 2, we drew, at each (Ju position r, the negative (and retarded; the phase depends on location) time history 0 of the pulse. For locations between r = 0 and 15 (r = 16 is the approximate breakdown position), the results of perturbation theory and numerical experiment were compared and are in close agreement. After the breakdown of the perturbation theory and before the turning point 15
u s’~O r~ 2 Ir_f( ~ —1/3 I = 12 fl() J ~TIJ
ing gain of (positive) the begins shelf takes place at over smaller intervals mass and aofpeak to appear
—
—
~,
—
-
—
—
—
q~(°, r) = qc(O, r)f(r)/f(r) 2 8/7
—
o(ar—l)/3nn(l---7a0/l2nn)
—
,
0<0<0.
‘
X sech2
{~0[—f(r)] 2/3 (0
— —
0)}
.
the front of the shelf. At the turning point r = 20, the peak is quite pronounced. As the turnmg point is passed, the peak and the remaining shelf breaks up into a train of pulses. These pulses, if they are to be .
its mass content is ~—24n 3theshelf uc(O, r) 0(-—f)~’ is 2a/?lo(l aO/l2p~)8/7, 0<0<0, and zero elsewhere and its mass content is 24n 13 24n 0(—J)~ 0. Note the conservation of total mass f°°,,,u dO to leading order depends crucially on the existence of the shelf. We also observe that while the solitary wave decreases in amplitude, its mass content and consequently that of the shelf increases without bound as the turning point is approached. Indeed, the perturbation theory breaks down at a point where the amplitudes of the solitary wave and the shelf are of the same order, when r = (l/a)[l 0(01/3)]. Nevertheless, as we see from the numerical results which follow, the perturbation theory gives us qualitatively accurate results beyond the breakdown point; the solitary wave (and consequently the shelf) continues to gain negative (positive) mass. We remark that the perturbation procedure shown here can be used quite 328 —
.
—
—
solitary waves, must still pass through the valley created by the remains of the original solitary wave. The criterion (determined numerically; one can also make a plausible analytical argument using inverse seattering theory) that a solitary wave forms is that for some r, the amplitude of the leading pulse is at least twice that of the valley. At r = 30, we can see that, in this case, the leading pulse satisfies this critenon and is beginning to separate from the others which form part of a dispersive wave train. At r = 40, at which point we take f(r) = 1, the new upward facing solitary wave is about to emerge from the effects of the disintegrating original solitary wave. We continued the numerical calculation till r = 200 in order to determine that the emerging solitary wave has indeed 2 the character Korteweg—de X sech2k(Oof a O~ 4k2r). Vries soliton, 12k —
—
Volume 75A, number 5
PHYSICS LETTERS
4 February 1980
2 >100
2’
•
40
a -j
Q.
2 4
w
V• 0
~\~fc0
20
—I-
0
Fig. 2. Numerical integration of eq. (2.3) with
15
40
DISTANCE
-
n~=
1 and a
=
1/20. The turning point occurs at r
20.
2 0 I0 -j
4
z
I Ui I-
•
7 V 7
7
7
V
V —.——
/
‘II
/
/ .5,
/ I‘
1.0
~ —
I
1.5
INITIAL SOLITON
Fig. 3. This graph shows (—) the relationship between i~( r~)of the initial soliton and k of the terminal soliton for various values of a (a 1/40, 1/20, 1/5, 1/2 reading from left to right). The line (— — —) represents the relation k = 2~o/3.The lines (- - -) appearing at the lower portion of the graph are due to the inability of the numerical scheme to give precise information for small k.
329
Volume 75A, number 5
Fig. 3 is a graph of k versus
PHYSICS LETTERS
for various a. Note
that as long as n0 is large enough, the amplitude of the first upward facing pulse will be sufficient to create a solitary wave. For larger values of ~0, we note the remarkable result, for which we have 2n no analytical explanation at the present time, that k = 0/3, independent of a. Appendix. Perturbation results for various equations. qt +6q’q1 +q~~1 =F, 2~n(X = a sech ar (r+ l)(r+2)~2/3r2. —
I),
Exam
4n 1leand1: r = Xt ~
1, F = aq; i~=
exp(2at/3),
514, for 0< I < I. 0(l+ aX/3~~) Example 2: r = 1,F aq 1~p = p0(l + l6p~at/ 2 and 15)1/2, ~ 4p = (8p 0a/15)exp(—2aX/15), for 0<1
qc
=
a exp (at)/3~
n0
~
330
Example 3: r =
4 February 1980 2,
F = oq; n =
n0 exp(2at), X~~
and =
ira exp(2at)/n~(l+ 4aX/n~), for 0 <1<1.
References [1] V.D. Djordjevic and L.G. Redekopp, J. Phys. Ocean. 8 (1978) 1016. [2] R.S. Johnson, Proc. Camb. Phios. Soc. 73 (1973) 183. [3] 5. Leibovich and J.D. Randall, J. Fluid Mech. 53 (1973) 481. [41 307. V.1. Karpman and E.M. Maslov, Phys. Lett. 60A (1977) [5] D.J. Kaup and A.C. Newell, Proc. Roy. Soc. London A361 (1978) 413. [6] CJ. Knickerbocker and A.C. Newell, Shelves and the Korteweg—de be published. Vries equation, J. Fluid Mech. (1979), to [7] A.C. Newell, Soliton perturbation and nonlinear focussing, Symp. on Nonlinear structure and dynamics in condensed matter (Oxford Univ., Solid State Physics), Vol. 8,Phys. pp. 52—68. [8] (Springer, E. Ott and1978) R.N. Sudan, Fluids 13 (1970) 1432. [9] A.C. Yliegenthart, J. Eng. Math. 5 (1971) 137.