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The periodic soliton resonance: Solutions to the Kadomtsev-Petviashvili equation with positive dispersion
Volume 143, number 4,5
PHYSICS LETTERS A
15 January 1990
THE PERIODIC SOLITON RESONANCE: SOLUTIONS TO THE KADOMTSEV-PETVIASHVILI EQUATION WITH POSITIVE DISPERSION Masayoshi TAJIRI and Youichi MURAKAMI Department ofMathematical Sciences, College ofEngineering, University ofOsaka Prefecture, 4-804 Mozu-Umemachi, Sakai, Osaka 591, Japan Received2 August 1989; revised manuscript received 2 November 1989; accepted for publication 2 November 1989 Communicated by D.D. HoIm
The two periodic soliton solution to the Kadomtsev—Petviashvili equation with positive dispersion is analyzed to show that the soliton resonance between them exists. It is shown that there is a parameter regime where the interactions result in a center shift of solitons not only in the direction ofpropagation but also in the transverse direction. The resonant interactions are associated with the parametric points on the boundary between the regime for transverse center shift and that for no transverse center shift ofsolitons.
We consider the Kadomtsev—Petviashvili (KP) equation [1] ~
s=±1,
(1)
which corresponds to the cases of negative and positive dispersion when s = + 1 and s = 1, respectively. The interaction between obliquely moving solitons has been studied by Miles [2]. He has shown that when the relative inclination between wave normals is at a certain small critical angle, two solitons interact strongly in the case of negative dispersion to make a resonant soliton from the point at which the two incident solitons meet. On the other hand in positive dispersion media, plane solitons never satisfy the resonant condition. In addition to this they are unstable to transverse perturbation of their fronts [1], but instead a two-dimensional localized soliton exists, which takes the form of a rational function [31.Zaitsev [4] first obtained another type of soliton, that is, the periodic soliton that is formed by superposition of rational solitons in positive dispersion media. Abramyan and Stepanyants [5] rediscovered the same solution using another method. Recently Tajiri and Murakami [6] obtained more general periodic soliton solutions, which represent the interaction among arbitrary numbers of periodic solitons that are sequences of localized solitons in arbitrary directions. It is known that the interactions among rational solitons result in no center shift of solitons at large times [3]. On the other hand the properties of the interaction between periodic solitons are not known to us. In this Letter we investigate the interaction between the y-periodic solitons, that is, an array of localized solitons in the y-direction, to show that resonant interactions between them exist in the positive dispersion case. The one periodic soliton solution that has a periodic structure in the y-direction is given as [4—6] 2c 2M[l_(l/,~/A~) cosh(ax—Qt±o)cos(öy+O)] 2 —
~—
—
[~/A~cosh(ax_Qt+c)_cos(öy+O)]2
‘
(
with Q=a3+3d2/a, M=ô2/(d2—a4) where the soliton frequency Q is determined by the soliton dispersion relation,
(3)
0375-9601 /90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
217
,
Volume 143, number 4,5
PHYSICS LETTERS A
15 January 1990
(4) D(Q,K,L)=KQ+3L2—K4, for given K= a and L = iö and a and 9 arbitrary constants. The existence condition for the nonsingular solution (2) is given by M> 1, i.e. ô2> a4. This solution represents a stationary wave periodic in y and exponentially decaying along the direction of propagation x [4—6].The propagation speed of the periodic soliton solution (2) is given by a2 + 3~52/a2 Hence the propagation speed is a decreasing function of a due to the existence condition. This peculiar propagation property seems to be related to the possibility of resonant interaction of periodic solitons. Thus we examine the interactions of two periodic solitons to the KP equation with positive dispersion. A convenient form of this solution [6] is given using the bilinear form [7] u=2(IogJ)~~,
(5)
with f=1+ -~exp(2~ 1)+ ~-~gexp(2~2)+ M~I~~1I~2 exp[2(~1
—
+~2)]
_-~exp(~i)(l+ M2NIN2 exp(2~2))cos(ni)_-~~~exp(~2)(l+ MININ2 exp(2~i))cos(?72)
+
2exp(~1+~)[N1cos(~1+,~2)+N2cos(i~1—,~2)], a1a2
where M ~
c5~—a~’ ‘
N
—
‘
~~=a
2— (~ 2 (a1 —a2) 1/a1—d2/a2) (a 2 1 +a2) —(ô1/a1 —d2/a2)
1x—Q~t+a~, ~=d~y+O~,
N
(6)
(a 2=
2— (d 2 1 —a2) 1/a1 +ô2/a2) (a 2—(5 2 1 +a2) 1/a1 +d2/a2)
Q1=a]+3c~/a~,j=1, 2.
(7)
When we assume that a1 >0, a2>0 and Q1 /a1 > Q2/a2, at large times the two separated periodic solitons before and after the interaction are given by M1 (8a) f2(~2,~2,NiN2)=~exP(2~i)(i+ M2N~N~ exp(2~)—~~2exp(~)cos(~2)),
(8b)
and ~ ~
(8c)
-~a2
a2
(8d)
,
respectively. Taking into account that u is unchanged even if f is multiplied by exp (ax+ b) with a and b independent of x, we find that the interactions are of the form [tj(~~, r,~),f2(~~2+1, ,2)]—~[j(~1 +1’, ‘ii),f2(’~, 12)],
forN1N2>0, forN1N2<0,
(9)
where r= log N1N2 I. It is shown that the interaction between periodic solitons yields the after-interaction so218
Volume 143, number 4,5
PHYSICS LETTERS A
15 January 1990
litons with phase shift T’, and if N1N2 <0, the solitons are also shifted by the respective half wave-length to the transverse direction and the sum of the phases of the before-interaction solitons is equal to that of the afterinteraction solitons. This is cruciallydifferent from the interaction between two rational solitons which results in no phase shift of solitons. We can see that the phase shift of~becomesinfinite, F—, +oo or when 1N1N21 +~ or 0. In the case ~a2> 0, the limit F-~ ~ means that the length ofthe intermediate region, where the solitons propagate together with a mean velocity, becomes infinite. This may be thought of as periodic soliton resonance, the conditions of which are obtained by setting the denominator of N1 and N2 to zero: —~,
c51/a1=d2/a2±(a1+a2),
—~
(lOa) (lOb)
respectively. These conditions (1 Oa) and (lOb) correspond to D(Q1+Q2,a1+a2,id1±id2)=0,
(11)
which is the dispersion relation of the resonant one periodic soliton. It should be noted that the resonant periodic soliton Ur satisfies one of the conservation laws: JUrdX=
Jui~+
Ju2dx=4(ai+a2),
(12)
where u1 and u2 are two periodic solitons before interaction. In factwhen the parameters satisfy the condition (lOa) or (lOb), two periodic solitons (7) interact so as to make a new one periodic soliton, ~_2a2M[l_(h/~j~M)c05h(~_Qt+a)c05(ôy+0)] [~,/AMcosh(ax_Qt+a)_cos(öy+O)]2
(13)
—
where 2—(a M=M1M2N~=
(d +d2)
4’ 1+a2)
a=a 1 +a2,
ÔÔ1
+ô2, Q=Q1 +Q~.
(14)
We note here that 5~+d2 in eq. (14) is replaced by d~—ô2 for the case d1ô2<0. It should be stressed that the condition (10) corresponds to the boundaries between the parametric region where the transverse center shift of the solitons occurs when N11’12 <0 and where no transverse center shift occurs when N1N2>0. On the other hand, the case N1N2=0 is not satisfied with this condition (12) [8]. The detailed structures of interaction of two periodic solitons having parameters near the resonant condition are shown in fig. 1. We choose these parameters in order to avoid the infinite values of the coefficients in (6) due to the exact resonant condition. Initially they are separated well enough to look like two independent solitons in fig. 1 a. Then they collide (fig. lb) and begin to separate transversely in fig. 1 c, which indicates that the interaction yields a new resonant periodic soliton as if each hump forms into a line. We can see that the intermediate one periodic soliton is generated in fig. ld as predicted by (13) and its propagation persists over a comparatively long time interval. After sufficiently long time the intermediate soliton begins to separate into two periodic solitons, the figures ofwhich are omitted, for we choose the parameters so that they do not satisfy the resonant condition strictly. In conclusion we have shown that resonant interactions can occur which are associated with the parametric points on the boundary between the regime for transverse center shift and for no transverse center shift of solitons. This is crucially different from the resonant interactions between two plane solitons to the KP equation with negative dispersion which are associated with the parametric end points of the regime of singular interactions [2]. In this Letter we have paid attention to the resonant interaction between y-periodic solitons. 219
Volume 143, number 4,5
U
PHYSICS LETTERS A
15 January 1990
UI
~
Q~JQ~
~
Fig. 1. The interaction of two periodic solitons having parameters near the resonant condition. Parameters: (a
1, a2, c5~,ô2) = (0.5, 0.41, 2/~J~, l/~/~).Initial constants: (a~,o~,O~,02)= (0, —20/a2, 0,0). (a) t=0, (b) t=0.6, (c) t=0.8, (d) t= 1.8.
According to the parameters and the positions of peaks, various types of interactions between them are classified, which will be reported [8]. Our solutions [6] can represent other interactions, e.g. between inclined periodic solitons, between periodic solitons and conventional plane solitons, and between periodic solitons and rational solitons, etc. These matters are now under study [9]. We wish to express our cordial thanks to Professor K. Gotoh for his encouragement during the work. Thanks are also due to the anonymous referees for useful comments.
References [1] [2] [3] [4] [5] [6]
B.B. Kadomtsev and M.I. Petviashvili, Soy. Phys. DokI. 15 (1970) 539. J.W. Miles, J. Fluid Mech. 79 (1977)157, 171. S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its and V.B. Matveev, Phys. Lett. A 63 (1977) 205. A.A. Zaitsev, Soy. Phys. DokI. 28 (1983) 720. L.A. Abramyan and Yu.A. Stepanyants, Radiophys. Quantum Electron. 28 (1985) 26. M. Tajiri and Y. Murakami, J. Phys. Soc. Japan 58 (1989) 3029. 17] J. Satsuma, J. Phys. Soc. Japan 40 (1976) 286. [8) Y. Murakami and M. Tajiri, submitted for publication. [9] M. Tajiri and Y. Murakami, in preparation.
220
PHYSICS LETTERS A
15 January 1990
THE PERIODIC SOLITON RESONANCE: SOLUTIONS TO THE KADOMTSEV-PETVIASHVILI EQUATION WITH POSITIVE DISPERSION Masayoshi TAJIRI and Youichi MURAKAMI Department ofMathematical Sciences, College ofEngineering, University ofOsaka Prefecture, 4-804 Mozu-Umemachi, Sakai, Osaka 591, Japan Received2 August 1989; revised manuscript received 2 November 1989; accepted for publication 2 November 1989 Communicated by D.D. HoIm
The two periodic soliton solution to the Kadomtsev—Petviashvili equation with positive dispersion is analyzed to show that the soliton resonance between them exists. It is shown that there is a parameter regime where the interactions result in a center shift of solitons not only in the direction ofpropagation but also in the transverse direction. The resonant interactions are associated with the parametric points on the boundary between the regime for transverse center shift and that for no transverse center shift ofsolitons.
We consider the Kadomtsev—Petviashvili (KP) equation [1] ~
s=±1,
(1)
which corresponds to the cases of negative and positive dispersion when s = + 1 and s = 1, respectively. The interaction between obliquely moving solitons has been studied by Miles [2]. He has shown that when the relative inclination between wave normals is at a certain small critical angle, two solitons interact strongly in the case of negative dispersion to make a resonant soliton from the point at which the two incident solitons meet. On the other hand in positive dispersion media, plane solitons never satisfy the resonant condition. In addition to this they are unstable to transverse perturbation of their fronts [1], but instead a two-dimensional localized soliton exists, which takes the form of a rational function [31.Zaitsev [4] first obtained another type of soliton, that is, the periodic soliton that is formed by superposition of rational solitons in positive dispersion media. Abramyan and Stepanyants [5] rediscovered the same solution using another method. Recently Tajiri and Murakami [6] obtained more general periodic soliton solutions, which represent the interaction among arbitrary numbers of periodic solitons that are sequences of localized solitons in arbitrary directions. It is known that the interactions among rational solitons result in no center shift of solitons at large times [3]. On the other hand the properties of the interaction between periodic solitons are not known to us. In this Letter we investigate the interaction between the y-periodic solitons, that is, an array of localized solitons in the y-direction, to show that resonant interactions between them exist in the positive dispersion case. The one periodic soliton solution that has a periodic structure in the y-direction is given as [4—6] 2c 2M[l_(l/,~/A~) cosh(ax—Qt±o)cos(öy+O)] 2 —
~—
—
[~/A~cosh(ax_Qt+c)_cos(öy+O)]2
‘
(
with Q=a3+3d2/a, M=ô2/(d2—a4) where the soliton frequency Q is determined by the soliton dispersion relation,
(3)
0375-9601 /90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
217
,
Volume 143, number 4,5
PHYSICS LETTERS A
15 January 1990
(4) D(Q,K,L)=KQ+3L2—K4, for given K= a and L = iö and a and 9 arbitrary constants. The existence condition for the nonsingular solution (2) is given by M> 1, i.e. ô2> a4. This solution represents a stationary wave periodic in y and exponentially decaying along the direction of propagation x [4—6].The propagation speed of the periodic soliton solution (2) is given by a2 + 3~52/a2 Hence the propagation speed is a decreasing function of a due to the existence condition. This peculiar propagation property seems to be related to the possibility of resonant interaction of periodic solitons. Thus we examine the interactions of two periodic solitons to the KP equation with positive dispersion. A convenient form of this solution [6] is given using the bilinear form [7] u=2(IogJ)~~,
(5)
with f=1+ -~exp(2~ 1)+ ~-~gexp(2~2)+ M~I~~1I~2 exp[2(~1
—
+~2)]
_-~exp(~i)(l+ M2NIN2 exp(2~2))cos(ni)_-~~~exp(~2)(l+ MININ2 exp(2~i))cos(?72)
+
2exp(~1+~)[N1cos(~1+,~2)+N2cos(i~1—,~2)], a1a2
where M ~
c5~—a~’ ‘
N
—
‘
~~=a
2— (~ 2 (a1 —a2) 1/a1—d2/a2) (a 2 1 +a2) —(ô1/a1 —d2/a2)
1x—Q~t+a~, ~=d~y+O~,
N
(6)
(a 2=
2— (d 2 1 —a2) 1/a1 +ô2/a2) (a 2—(5 2 1 +a2) 1/a1 +d2/a2)
Q1=a]+3c~/a~,j=1, 2.
(7)
When we assume that a1 >0, a2>0 and Q1 /a1 > Q2/a2, at large times the two separated periodic solitons before and after the interaction are given by M1 (8a) f2(~2,~2,NiN2)=~exP(2~i)(i+ M2N~N~ exp(2~)—~~2exp(~)cos(~2)),
(8b)
and ~ ~
(8c)
-~a2
a2
(8d)
,
respectively. Taking into account that u is unchanged even if f is multiplied by exp (ax+ b) with a and b independent of x, we find that the interactions are of the form [tj(~~, r,~),f2(~~2+1, ,2)]—~[j(~1 +1’, ‘ii),f2(’~, 12)],
forN1N2>0, forN1N2<0,
(9)
where r= log N1N2 I. It is shown that the interaction between periodic solitons yields the after-interaction so218
Volume 143, number 4,5
PHYSICS LETTERS A
15 January 1990
litons with phase shift T’, and if N1N2 <0, the solitons are also shifted by the respective half wave-length to the transverse direction and the sum of the phases of the before-interaction solitons is equal to that of the afterinteraction solitons. This is cruciallydifferent from the interaction between two rational solitons which results in no phase shift of solitons. We can see that the phase shift of~becomesinfinite, F—, +oo or when 1N1N21 +~ or 0. In the case ~a2> 0, the limit F-~ ~ means that the length ofthe intermediate region, where the solitons propagate together with a mean velocity, becomes infinite. This may be thought of as periodic soliton resonance, the conditions of which are obtained by setting the denominator of N1 and N2 to zero: —~,
c51/a1=d2/a2±(a1+a2),
—~
(lOa) (lOb)
respectively. These conditions (1 Oa) and (lOb) correspond to D(Q1+Q2,a1+a2,id1±id2)=0,
(11)
which is the dispersion relation of the resonant one periodic soliton. It should be noted that the resonant periodic soliton Ur satisfies one of the conservation laws: JUrdX=
Jui~+
Ju2dx=4(ai+a2),
(12)
where u1 and u2 are two periodic solitons before interaction. In factwhen the parameters satisfy the condition (lOa) or (lOb), two periodic solitons (7) interact so as to make a new one periodic soliton, ~_2a2M[l_(h/~j~M)c05h(~_Qt+a)c05(ôy+0)] [~,/AMcosh(ax_Qt+a)_cos(öy+O)]2
(13)
—
where 2—(a M=M1M2N~=
(d +d2)
4’ 1+a2)
a=a 1 +a2,
ÔÔ1
+ô2, Q=Q1 +Q~.
(14)
We note here that 5~+d2 in eq. (14) is replaced by d~—ô2 for the case d1ô2<0. It should be stressed that the condition (10) corresponds to the boundaries between the parametric region where the transverse center shift of the solitons occurs when N11’12 <0 and where no transverse center shift occurs when N1N2>0. On the other hand, the case N1N2=0 is not satisfied with this condition (12) [8]. The detailed structures of interaction of two periodic solitons having parameters near the resonant condition are shown in fig. 1. We choose these parameters in order to avoid the infinite values of the coefficients in (6) due to the exact resonant condition. Initially they are separated well enough to look like two independent solitons in fig. 1 a. Then they collide (fig. lb) and begin to separate transversely in fig. 1 c, which indicates that the interaction yields a new resonant periodic soliton as if each hump forms into a line. We can see that the intermediate one periodic soliton is generated in fig. ld as predicted by (13) and its propagation persists over a comparatively long time interval. After sufficiently long time the intermediate soliton begins to separate into two periodic solitons, the figures ofwhich are omitted, for we choose the parameters so that they do not satisfy the resonant condition strictly. In conclusion we have shown that resonant interactions can occur which are associated with the parametric points on the boundary between the regime for transverse center shift and for no transverse center shift of solitons. This is crucially different from the resonant interactions between two plane solitons to the KP equation with negative dispersion which are associated with the parametric end points of the regime of singular interactions [2]. In this Letter we have paid attention to the resonant interaction between y-periodic solitons. 219
Volume 143, number 4,5
U
PHYSICS LETTERS A
15 January 1990
UI
~
Q~JQ~
~
Fig. 1. The interaction of two periodic solitons having parameters near the resonant condition. Parameters: (a
1, a2, c5~,ô2) = (0.5, 0.41, 2/~J~, l/~/~).Initial constants: (a~,o~,O~,02)= (0, —20/a2, 0,0). (a) t=0, (b) t=0.6, (c) t=0.8, (d) t= 1.8.
According to the parameters and the positions of peaks, various types of interactions between them are classified, which will be reported [8]. Our solutions [6] can represent other interactions, e.g. between inclined periodic solitons, between periodic solitons and conventional plane solitons, and between periodic solitons and rational solitons, etc. These matters are now under study [9]. We wish to express our cordial thanks to Professor K. Gotoh for his encouragement during the work. Thanks are also due to the anonymous referees for useful comments.
References [1] [2] [3] [4] [5] [6]
B.B. Kadomtsev and M.I. Petviashvili, Soy. Phys. DokI. 15 (1970) 539. J.W. Miles, J. Fluid Mech. 79 (1977)157, 171. S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its and V.B. Matveev, Phys. Lett. A 63 (1977) 205. A.A. Zaitsev, Soy. Phys. DokI. 28 (1983) 720. L.A. Abramyan and Yu.A. Stepanyants, Radiophys. Quantum Electron. 28 (1985) 26. M. Tajiri and Y. Murakami, J. Phys. Soc. Japan 58 (1989) 3029. 17] J. Satsuma, J. Phys. Soc. Japan 40 (1976) 286. [8) Y. Murakami and M. Tajiri, submitted for publication. [9] M. Tajiri and Y. Murakami, in preparation.
220