Oblique interactions of weakly nonlinear long waves in dispersive systems

Fluid Dynamics Research 38 (2006) 868 – 898

Oblique interactions of weakly nonlinear long waves in dispersive systems Masayuki Oikawa∗ , Hidekazu Tsuji Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga-shi, Fukuoka-ken 816-8580, Japan Received 22 May 2006; received in revised form 15 June 2006; accepted 4 July 2006 Communicated by S. Kida

Abstract Studies on the oblique interactions of weakly nonlinear long waves in dispersive systems are surveyed. We focus mainly our concentration on the two-dimensional interaction between solitary waves. Two-dimensional Benjamin–Ono (2DBO) equation, modified Kadomtsev–Petviashvili (MKP) equation and extended Kadomtsev– Petviashvili (EKP) equation as well as the Kadomtsev–Petviashvili (KP) equation are treated. It turns out that a large-amplitude wave can be generated due to the oblique interaction of two identical solitary waves in the 2DBO and the MKP equations as well as in the KP-II equation. Recent studies on exact solutions of the KP equation are also surveyed briefly. © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Solitary waves; Soliton; Kadomtsev–Petviashvili equation; Oblique interaction; Mach reflection; Soliton resonance; Nonlinear wave propagation

1. Introduction The Korteweg–de Vries (KdV) equation ju jT

+ 6u

ju jX

+

j3 u jX 3

=0

∗ Corresponding author.

E-mail address: [email protected] (M. Oikawa). 0169-5983/$30.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2006.07.002

(1)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

869

is a typical soliton equation and has the solitary wave solution u = 2
2 sech2 [
(X − 4
2 T − X0 )],

(2)

where
and X0 are arbitrary constants. It was first derived by Korteweg and deVries (1895) as an evolution equation describing weakly nonlinear long water waves propagating in one direction. Their work settled the long-time controversy about the existence of the solitary water wave (see, for example, Miles, 1981). The KdV equation for water waves
 3 ∗ 
 j∗ j  3∗ j∗ c0 h2 3S + c0 1 + + = 0, c0 ≡ gh (3) 1− 2 ∗ ∗ ∗3 jt 2h jx 6 jx gh may be obtained from (1) through the transformation 3∗ u=± , 2h



∗

∗

X = ± (x − c0 t ), h

c0 ∗ T = t ,

6h

−1/2 
3S = ± 1− . gh2

(4)

Here, h is the undisturbed uniform water depth,  the density of water, S the surface tension, g the gravitational acceleration and h + ∗ the surface elevation. Further, t ∗ and x ∗ are the dimensional time and space variables, respectively, and the choice of ± in (4) follows the sign of 1 − 3S/gh2 . It is noted that Eq. (1) can be transformed into another KdV equation with arbitrarily given non-zero coefficients by a suitable scale transformation. Form (1) is often used because the solutions take simple forms. Zabusky and Kruskal (1965) found that the solitary wave solutions of the KdV equation are stable in interactions among them and named them “solitons”. The equation is also a universal equation in the sense that unidirectional, weakly nonlinear, long waves in dispersive systems are generally governed by the KdV equation (Taniuti and Wei, 1968; Su and Gardner, 1969). Further, Gardner et al. (1967, 1974) discovered a surprising method for solving initial-value problems of Eq. (1)—inverse scattering method. Since then the KdV equation was studied intensively and various properties of it were resolved (for example, Miura, 1976). Many other systems which have properties similar to the KdV equation were also found. They are called “soliton systems” or “integrable systems” and characterized by the properties that the system has an infinite number of conservation laws, it has solutions which represent the interactions of N solitons (N-soliton solution), it permits the inverse scattering formulation; it permits the Bäcklund transformation, it has the Painlevé property (Weiss et al., 1983), and so on. For soliton systems, refer to Ablowitz and Segur (1981) and Ablowitz and Clarkson (1991). The Kadomtsev–Petviashvili (KP) equation
 2 j ju ju j3 u 2j u + 6u + +  =0 (5) jX jT jX jX 3 jY 2 is one of the most important soliton systems. This is a weakly two-dimensional extension of the KdV equation and was first derived by Kadomtsev and Petviashvili (1970). Then, they investigated the stability of the KdV “line soliton” (2) to transverse perturbations by employing the KP equation to show that it is stable for 2 > 0 and unstable for 2 < 0. It is noted that the sign of 2 is of essential significance. Eq. (5) is referred to as KP-I if 2 < 0 and as KP-II if 2 > 0. Some authors refer to KP-I simply as KP and to KP-II as two-dimensional KdV. Sometimes KP-I is also called “KP with positive dispersion” and

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M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

KP-II is called “KP with negative dispersion”. For water wave, 3S > gh2 and 3S < gh2 correspond to KP-I and KP-II, respectively. KP-II exhibits an intriguing phenomenon, i.e. “soliton resonance” in the interactions of line solitons (Miles, 1977a, b; Newell and Redekopp, 1977). On the other hand KP-I has “rational soliton” or “lump soliton” solutions which decay algebraically in all directions in the XY plane (Manakov et al., 1977; Johnson and Thompson, 1978; Ablowitz and Satsuma, 1978). There are many studies about the KP equation and the phenomena described approximately by it. Freeman (1980) reviewed the soliton interactions in two dimensions on the basis of the KP equation. Sections 7.9 and 7.10 of Infeld and Rowlands (1990) are also an excellent review of the similar subject including experimental and numerical studies. Using the KP model as the common link, Akylas (1994) reviewed three-dimensional, nonlinear long water-wave phenomena—oblique interactions of solitary waves and periodic waves, the propagation of long waves in channels of variable depth, the effect of rotation on the KdV solitons, and so on. Recently, there are some new developments in the solutions of the KP equation. Oblique interactions of solitary waves in the weakly two-dimensional extensions of some soliton systems—the Benjamin–Ono (BO) equation (Benjamin, 1967; Ono, 1975), the modified KdV (MKdV) equation (Zabusky, 1967; Watanabe, 1984) and the extended KdV (EKdV) equation (Kakutani and Yamasaki, 1978; Miles, 1979)—have been also investigated. These equations are soliton systems, but their weakly two-dimensional extensions are (probably) not integrable. So, the analytical solutions to them are not available. However, it is very interesting to study the oblique interactions of solitary waves in those equations. In the present article we review the recent advances in understanding the solutions of the KP equation and oblique interactions of solitary waves in the systems analogous to the KP equation avoiding unnecessary duplication with Freeman (1980), Infeld and Rowlands (1990), and Akylas (1994). In Section 2, we consider the weak interaction and the strong interaction of two interfacial solitary waves propagating in different directions in a two-layer fluid of infinite depth. For the strong interaction we obtain the weakly two-dimensional extension of the BO equation. In Section 3 the soliton resonance in the KP-II equation is introduced, then the results of oblique interactions of shallow-water solitons and those of ion-acoustic solitons are summarized. In Section 4 the numerical results of the oblique interaction (rather oblique reflection) for the two-dimensional BO (2DBO), the modified Kadomtsev–Petviashvili (MKP) and an extended Kadomtsev–Petviashvili (EKP) equations are surveyed. In Section 5 relatively recent development of study of the solutions to the KP equation are summarized. Section 6 is devoted to the concluding remarks.

2. Two-dimensional interaction of solitary waves In this section we describe a perturbation approach to the two-dimensional interactions of two weakly nonlinear solitary waves propagating in different directions. Here, we take up as a specific example the interfacial waves in a two-layer fluid of infinite depth (Oikawa, 1984). For similar interactions of shallow-water solitons, Benney and Luke (1964), Miles (1977a), and Freeman (1980) gave systematic expansion procedures. In particular, Miles (1977a) classified such interaction as “weak” if the propagation directions are not close to each other or “strong” if they are. The weak interaction is weak in the sense that the lowest-order approximation to the solution is described by the superposition of the solitary waves and the interaction is significant only near the intersection.

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z*

1

*1 z*= *(x* ; y* ; t*)

O 2 =1(1+∆)

*2

h

-h

Fig. 1. Two-layer fluid system. 1 and 2 = 1 (1 + ) are the densities of the upper and lower layers, respectively, and  > 0. ∗1 and ∗2 are the velocity potentials. h is the undisturbed uniform depth of the lower layer. ∗ is the interface displacement from the mean level.

2.1. Weak interaction Consider a two-layer fluid system of infinite depth as shown in Fig. 1. We take the coordinate system in which the origin is in the mean level of the interface, the (x ∗ , y ∗ ) plane horizontal, and the z∗ -axis vertically upward. It is assumed that the fluids are inviscid and incompressible and their motion is irrotational. Then, the fundamental equations and boundary conditions for the velocity potentials ∗1 , ∗2 and the interface displacement ∗ from the mean level are given by ∇ ∗2 ∗1 = 0

for z∗ > ∗ ,

(6a)

∇ ∗2 ∗2 = 0

for − h < z∗ < ∗ ,

(6b)

j∗ jt ∗ j∗ jt ∗

+ ∇2∗ ∗ · ∇2∗ ∗1 = + ∇2∗ ∗ · ∇2∗ ∗2 =

j∗1 jz∗

j∗2 jz∗

on z∗ = ∗ ,

(6c)

on z∗ = ∗ ,

(6d)


∗ j2 1 ∗ ∗ 2 1 ∗ ∗ 2 (∇ (∇ +
g  ∗ +  ) = (1 +  ) +  ) 1 2 jt ∗ 2 jt ∗ 2

j∗1

|∇ ∗ ∗1 | → 0 j∗2 jz∗

=0

as z∗ → ∞,

on z∗ = ∗ ,

(6e) (6f)

on z∗ = −h.

(6g)

Here ∇ ∗ ≡ (j/jx ∗ , j/jy ∗ , j/jz∗ ) is the gradient operator and ∇2∗ ≡ (j/jx ∗ , j/jy ∗ ) the gradient operator in the (x ∗ , y ∗ ) plane. g denotes the gravitational acceleration. Linearization of system (6) yields the dispersion relation ∗2 =


g|k∗ |

1 + (1 + ) coth(|k∗ |h)

,

(7)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

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where k∗ is the wavenumber and ∗ the angular frequency. For |k∗ |h>1,
 |k∗ |h ∗2 2 ∗2 1− =V k + ··· . 1+ √ Here V ≡ gh/(1 + ) is the phase speed of linear long waves. Further, for |k∗ |h>1, ∗1 = O(V ∗ ),

∗2 = O(V ∗
/ h),

(8)

(9)

where
is the characteristic horizontal length of waves. In order to consider the weak nonlinear long waves, we introduce the following non-dimensionalization: ∗ = a ,

x∗ = (x ∗ , y ∗ ) =
x =
(x, y),

t ∗ = (
/V )t,

∗1 = aV 1 ,

z∗ = hz,

∗2 = (aV
/ h)2 ,

(10a) (10b)

where a is a characteristic wave amplitude. While the scale of variation in the vertical direction of 2 is h, that of 1 is anticipated to be
. So, we assume that 2 is a function of x, y, z, t but 1 a function of x, y, z˜ , t. Here, z˜ ≡ z∗ /
= z, ≡ h/
. The non-dimensional form of system (6) involves two dimensionless parameters  ≡ a/ h and . We assume

= O(),

>1.

(11)

The equation for 2 gives together with the bottom boundary condition

2

2 = f (x, t) −

2!

(∇22 f )(z + 1)2 +

4

4!

(∇24 f )(z + 1)4 − · · · .

(12)

Here, f (x, t) is the velocity potential at the bottom and ∇2 = (j/jx, j/jy). Then, up to O(, ), the equation for 1 and boundary conditions are reduced to j2 1

∇22 1 +

j1

|∇2 1 |, 

j1

j jt





jz˜



jz˜

=0

jz˜ 2

z˜ =0

=

for z˜ > 0,

→0

j jt

as z˜ → ∞,

,

jt




 z˜ =0

= (1 + )

(13b) (13c)

+ ∇2  · ∇2 f = −(∇22 f )(1 + ), j1

(13a)

 1 2 + (∇2 f ) +  . jt 2

jf

Solving (13a) under conditions (13b) and (13c), we obtain  ∞ ∞ 1 jˆ 1 1 (x, z˜ , t) = − exp[ik · x − |k|˜z] dk + O(). (2 )2 −∞ −∞ |k| jt

(13d) (13e)

(14)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

873

Here, ˆ denotes the Fourier transform of , and if we write k = (kx , ky ), dk denotes dkx dky . From this, 

j1



jt

1 =− 2 z˜ =0



∞



∞

−∞ −∞

1 j2   (x , t) dx + O(), |x − x| jt 2

(15)

where dx denotes dx  dy  for x = (x  , y  ). Using this for (13e), we obtain the system for f and : j jt

+ ∇22 f + ∇2 · (∇2 f ) = 0,

(16a)

jf

1 +  + (∇2 f )2 + jt 2 2 (1 + )



∞



∞

−∞ −∞

1 j2   (x , t) dx = 0. |x − x| jt 2

(16b)

In order to investigate the interactions of two waves propagating in different directions nj =(cos j , sin j ) ( j : the angle between the positive x-axis and nj ; j = 1, 2), we introduce the new coordinates j = nj · x − t + j (x, t),

 = t

(j = 1, 2),

(17)

where we introduce also the phase variables j to take the phase shifts into account (Oikawa and Yajima, 1973). Further, we expand  and f as  = (0) (1 , 2 , ) + (1) (1 , 2 , ) + · · · ,

(18a)

f = f (0) (1 , 2 , ) + f (1) (1 , 2 , ) + · · · .

(18b)

Then, we obtain in O(0 ), (0) =

jf (0) j1

+

jf (0) j2

,

2(1 − p)

j2 f (0) j1 j2

= 0,

p ≡ n1 · n2 .

(19)

Accordingly, if 1 − p  = 0, f (0) = F1 (1 , ) + F2 (2 , ),

(0) =

jF1 j1

+

jF2 j2

.

(20)

F1 (1 , ) and F2 (2 , ) are determined in the next order. Next, in O() we obtain (1)



 
 j3 F1 1 jF1 2 = + − − − 2q L j1 j2 j 2 j1 j31  
 jF2 j3 F2 1 jF2 2 − − 2q L − j 2 j2 j32  

j1 j1 jF1 j2 j2 jF2 jF1 jF2 + + + −p , + jf (1)

j1

jf (1)

j2

jF1

j1

j1

j2

j2

j1 j2

(21a)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

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 j3 F1 3 jF1 2 2(1 − p) = + + 2q L 2 j1 j2 j1 j 2 j1 j31  


 j jF2 j3 F2 3 jF2 2 2 + + 2q L + j2 j 2 j2 j32  j2 jF1 ((1 + 2p)F2 − 2(1 − p)1 ) + j2 f (1)

j



jF1

j1 j2

+ ((1 + 2p)F1 − 2(1 − p)2 ) q≡



2(1 + )

L(F (1 , 2 ))

1 ≡ 2



∞

jF2 j2

j1

 ,

,



(21b) (21c)

∞

−∞ −∞

F (1 , 2 ) d1 d2 .  (1 − 1 )2 + (2 − 2 )2 − 2p(1 − 1 )(2 − 2 )

We can show that if F is independent of 2 , the relation
  ∞ F (1 ) jF 1 = P L d1 = H(F )  j1 −∞ 1 − 1

(21d)

(22)

holds. Here, P denotes the principal value and H(F ) is the Hilbert transform of F. L(j3 F1 /j31 ) = H(j2 F1 /j21 ) and L(j3 F2 /j32 ) = H(j2 F2 /j22 ) do not, therefore, depend on 2 and 1 , respectively. From the condition that f (1) does not involve secular terms, we obtain  
 jFj j2 Fj 3 jFj 2 = 0 (j = 1, 2), (23) + + qH j 4 jj j2j which is the equation to determine Fj (j , ). Differentiating this with respect to j , this is reduced to the BO equation   juj j2 uj jFj 3 juj = 0, uj ≡ + uj + qH (j = 1, 2). (24) 2 j 2 jj jj jj So, we see that in the lowest order of , the interface displacement can be represented by the superposition of two waves governed by the respective BO equations. In addition, if we choose the phase variables as 1 =

1 + 2p F2 , 2(1 − p)

2 =

1 + 2p F1 , 2(1 − p)

(25)

then f (1) = G1 (1 , ) + G2 (2 , ). G1 (1 , ) and G2 (2 , ) are determined in the next order.

(26)

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Up to O(), the result is summarized as follows: f = F1 (1 , ) + F2 (2 , ) + [G1 (1 , ) + G2 (2 , )],


 u21 ju1 jG1 + − qH  = u1 (1 , ) + u2 (2 , ) +  4 j1 j1


 u22 1 + p + p2 ju 2 jG2 + − qH u1 u2 . + + 4 j2 j2 1−p

(27)

(28)

It is noted that this approximation is valid only for 1 − p = 2 sin2 (( 2 − 1 )/2)?. Further, note that it is only phase terms 1 and 2 and the last term of (28) that result from the interaction of two waves. The former is non-local effect and the latter is local effect. The BO Eq. (24) is one of the typical soliton equations (Matsuno, 1979, 1984), and the soliton solutions of (24) and (23) are given by uj =

aj bj2 (j

− (3/8)aj  − j 0 ) + 1 2

,

Fj = 83 q tan−1 [bj (j − 38 aj  − j 0 )]

bj =

3aj (j = 1, 2), 8q

(j = 1, 2),

(29a) (29b)

where we can add an appropriate constant to Fj according to a given condition. The magnitude of the phase shifts in the interaction of solitons is given by   2  1 + 2p 

. (30) 3 1−p 1+ It is noted that this does not depend on the amplitude of another soliton. Matsuno (1998) gave a higherorder approximation to the above problem to show that the next order term of the phase shift of each soliton is proportional to the amplitude of another soliton. 2.2. Strong interaction—2DBO equation For 1 − p = O(), that is, | 2 − 1 | = O(1/2 ), the above perturbation approach is not valid. Instead, in this case the waves propagate in almost the same direction. If we take the main propagation direction as the positive x-axis, the waves may vary slowly with length scale of O(−1/2 ) in the y direction. So, introducing the new coordinates  = x − t,

 = 1/2 y,

 = t, (0)

(31) (1)

and expanding 1 , , and f as 1 = 1 (, , , z˜ )+ 1 (, , , z˜ )+· · ·, we obtain the following equations from (13): in the order of 0 , (0)

j2 1 j2

(0)

+

j2 1 jz˜ 2

=0

for z˜ > 0,

(32a)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

876 (0)

j1



(0)

j1

,

j

(0)

j1



=−

jz˜

−

→0

jz˜

j(0)

z˜ =0

jf (0) j

as z˜ → ∞,

j

(32b)

,

(32c)

+ (0) = 0.

(32d)

From these we have jf (0)

(0)

1 |z˜ =0 = H((0) ),

j

= (0) .

(33)

Eliminating (1) and f (1) from the relations in the order of , − −

j(1) j jf (1) j

+

j2 f (1)

+

j2 (1)

+

j(0) j

+

j(0) jf (0) j

j

+ (0)

j2 f (0)

+

j2

2  (0)

j1 1 jf (0) + + + 2q j 2 j j

j2 f (0)

jf (0)

j2

= 0,

= 0,

(34a)

(34b)

z˜ =0

and using (33), we have

  j2 (0) 3 (0) j(0) 1 j2 f (0) +  + qH = 0, + j 2 j 2 j2 j2

j(0)

jf (0) j

or j j



= (0)

(35a)

(35b)

  1 j2 (0) j2 (0) 3 (0) j(0) + +  + qH = 0. j 2 j 2 j2 j2

j(0)

(36)

This is a weakly two-dimensional extension of the BO equation and describes the interactions of weakly nonlinear waves propagating nearly in the x direction. Here we call them 2DBO equation. This type of equation was also derived by Ablowitz and Segur (1980) for a case of continuous density stratification. The linearized version of (36) can be obtained from the dispersion relation (8). After the normalization (10), it is written as 2 = k2 (1 − 2q|k| + · · ·),

k = (kx , ky ).

For kx = O(1) and ky = O(1/2 ), this becomes   2 k y 2 = kx2 1 + 2 − 2q|kx | + o(), kx

(37)

(38)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

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and for waves propagating in one direction kx = kx2 − qk 2x |kx | + 21 ky2 + o().

(39)

Disregarding o() terms and making the replacement kx → −ij/jx, ky → −ij/jy, → ij/jt, sgn(kx ) → −iH in (39) and the change of variables (31), we have the linear version of (36).

3. Oblique interactions of solitary waves in the KP systems Here we call the systems “KP systems” which can be reduced to the KP equation under the assumption of weakly nonlinear and nearly unidirectional waves. Many systems including shallow-water waves and ion-acoustic waves are KP systems. Weak interactions of solitary waves with small but finite amplitudes in KP systems can be treated by a perturbation method similar to that used in Section 2.1 (see Miles, 1977a; Freeman, 1980). On the other hand, strong interactions can be described approximately by the KP equation for weakly nonlinear cases. For shallow-water waves, the weakly two-dimensional extension of (3) can be obtained in consideration of the linear dispersion relation like the above: j




jx ∗

j∗ jt ∗


+ c0

3 ∗ 1+ 2h



j∗

c0 h2 + jx ∗ 6




3S 1− gh2



j3 ∗



jx ∗3

+

c0 j2 ∗ = 0. 2 jy ∗2

(40)

By the transformation (4) together with 

Y = y ∗, h

(41)

Eq. (40) reduces to (5) with 2 ≡ 32 /2 . Zakharov and Shabat (1974) and Satsuma (1976) gave the N-soliton solutions to the KP-II equation. Miles (1977a, b) investigated oblique interactions of two gravity wave solitons in shallow water to find singular solutions and resonant (or phase-locked) soliton solutions which they appear to overlook. Through the transformation u=2

j2 jX 2

log f¯(X, Y, T ),

(42)

the KP-II equation (5) is reduced to the bilinear form (Hirota, 1976) 4 (DT DX + DX + 2 DY2 )f¯ · f¯ = 0,

where D denotes Hirota’s derivative defined, for example, by 
m
n  j j j j m n ¯    ¯ f (X, Y, T ) g(X ¯ , Y , T ) . − − DX DY f · g¯ =   jX jX  jY jY  X =X,Y  =Y The dependent variable f¯ is often called “-function”.

(43)

(44)

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

878

The multi-soliton solution of (43) may be obtained by the formal expansion f¯ =1+ f¯(1) + 2 f¯(2) +· · · (for example, Hirota, 1976). The one-soliton solution is given by f¯ = 1 + e ,

(45a)

 ≡ X + Y − T + 0 ,

(45b)

L(, , ) ≡ − + 4 + 2 2 = 0,

(45c)

u=

2



sech2 , 2 2

(45d)

where , ,  are constants satisfying relation (45c) which may be called the soliton dispersion relation and 0 is an arbitrary constant. The two-soliton solution is given by f¯ = 1 + e1 + e2 + A12 e1 +2 , j ≡ j X + j Y − j T + j 0

(46a) (j = 1, 2),

L(j , j , j ) ≡ −j j + 4j + 2 2j = 0 A12 = − = =

(46b)

(j = 1, 2),

(46c)

L(1 − 2 , 1 − 2 , 1 − 2 ) L(1 + 2 , 1 + 2 , 1 + 2 )

3(1 − 2 )2 − 2 (1 /1 − 2 /2 )2 3(1 + 2 )2 − 2 (1 /1 − 2 /2 )2 3(1 − 2 )2 − 2 (tan 1 − tan 2 )2 3(1 + 2 )

2

− 2 (tan

1

− tan 2 )

2

,

tan j ≡

j j

,

(46d)

where j (> 0), j , j (j = 1, 2) are constants satisfying relation (46c) and j 0 are arbitrary constants.

j (j = 1, 2) is the angle between the propagation direction of jth soliton and the positive X direction. For A12 > 0, that is, for

3(1 − 2 )2 > 2 (tan 1 − tan 2 )2

or

3(1 + 2 )2 < 2 (tan 1 − tan 2 )2 ,

(47)

u is regular. The pattern of this solution is stationary in an appropriate moving frame. The asymptotic form is ⎧ 2 1  ⎪ ⎪ sech2 1 for 1 = O(1), 2 → −∞, ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ 2  + 12 ⎪ ⎪ ⎨ 1 sech2 1 for 1 = O(1), 2 → +∞, 2 2 u∼ ⎪ 22  ⎪ ⎪ sech2 2 for 2 = O(1), 1 → −∞, ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ 2 ⎪  + 12 ⎩ 2 for 2 = O(1), 1 → +∞, sech2 2 2 2

10

0 -5 0 X

u

5

1.5 1 0.5 0

10 5 0

-10 -5

-5 5

879

Y

1.5 1 0.5 0 -10 0

Y

u

M. Oikawa, H. Tsuji / Fluid Dynamics Research 38 (2006) 868 – 898

-5 X

0

-10

5

10 -10

10

5 0

-5 X

1.5 1 0.5 0 -10

-5

0 5

10 5 0 -5

Y

10 u

1.5 1 0.5 0 -10

Y

u

Fig. 2. Two-soliton solutions. 1 =1.8, 1 =0.8, 2 =0.7, 2 =−0.3888 · · · , A12 =0.004 (left plot); 1 =1.2, 1 =−0.84, 2 =0.7, 2 = 0.85, A12 = 62.6629 · · · (right plot). 2 = 3, T = 0.

0 X

10 -10

-5 5

10

-10

Fig. 3. Resonant soliton solutions. 1 =1.8, 1 =0.8, 2 =0.7, 2 =−0.45888 · · · , A12 =0 (left plot); 1 =1.2, 1 =−0.84, 2 =0.7, 2 = 0.84, A12 = +∞ (right plot). 2 = 3, T = 0.

where 12 ≡ log A12 is the phase shift due to the interaction of two solitons and positive for A12 > 1 and negative for 0 < A12 < 1. Fig. 2 shows typical plots of the two-soliton solution with positive finite A12 . For A12 < 0, that is, for 3(1 − 2 )2 < 2 (tan 1 − tan 2 )2 < 3(1 + 2 )2 ,

(48)

u is singular. On the other hand, for A12 = 0 or A12 = +∞, that is, for 2 (tan 1 − tan 2 )2 = 3(1 − 2 )2

or

2 (tan 1 − tan 2 )2 = 3(1 + 2 )2 ,

(49)

the phase shift 12 becomes −∞ or +∞. The wave patterns are three-legged as shown in Fig. 3. The three legs are asymptotically three solitons satisfying the condition L(1 , 1 , 1 ) = 0, L(2 , 2 , 2 ) = 0, and L(1 − 2 , 1 − 2 , 1 − 2 ) = 0 or L(1 + 2 , 1 + 2 , 1 + 2 ) = 0 which are formally the same

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as the three-wave resonance condition for sinusoidal waves obeying a linear dispersion relation. So, the solutions in these cases are called the resonant soliton solutions. The solution for A12 = 0 is obtained from (42) and f¯ = 1 + e1 + e2 and the asymptotic form is ⎧ 2 1 1 ⎪ ⎪ 1 = O(1), 2 → −∞, ⎪ sech2 , ⎪ 2 2 ⎪ ⎪ ⎨ 2 2  u∼ 2 = O(1), 1 → −∞, sech2 2 , ⎪ 2 2 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ (1 − 2 ) sech2 1 − 2 ,  −  = O(1),  ,  → +∞. 1 2 1 2 2 2 The condition A12 = +∞ reduces to the condition A12 = 0 by the replacement (2 , 2 , 2 ) → (−2 , −2 , −2 ). The solution for A12 = +∞ may, therefore, be obtained from (42) and f¯ = 1 + e1 + e−2 and the asymptotic form is ⎧ 2 1  ⎪ ⎪ sech2 1 , 1 = O(1), 2 → +∞, ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ 2 2 2 2 , u∼ sech 2 = O(1), 1 → −∞, ⎪ 2 2 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ (1 + 2 ) sech2 1 + 2 ,  +  = O(1),  → ∞,  → −∞. 1 2 1 2 2 2 If we associate the KP-II equation (5) with (40) by use of the transformations (4) and (41) with S = 0 and  = 1 (consequently,  = 1, 2 = 3), then 1 /1 >1, 2 /2 >1, and 21 /3 = a1 , 22 /3 = a2 (a1 , a2 : the amplitudes of solitons normalized by h). In this case conditions (47)–(49) are written approximately as    

2 < ( 3a1 − 3a2 )2 or 2 > ( 3a1 + 3a2 )2 ⇔  : regular, (50)     ( 3a1 − 3a2 )2 < 2 < ( 3a1 + 3a2 )2 ⇔  : singular, (51)    

2 = ( 3a1 − 3a2 )2 or 2 = ( 3a1 + 3a2 )2 ⇔ resonance, (52) where ≡ 1 − 2 is the angle between the propagation directions of two solitons. Miles (1977b) found first such resonant solutions in shallow-water solitons and used them to explain the observed phenomena of the Mach reflection in shallow water. The reflection of an obliquely incident solitary wave at a rigid vertical wall is regular (equi-angular) if the angle of incidence is not small, but if the angle of incidence is small, the regular reflection does not occur. It seems that Russell recognized this fact (Freeman, 1980; Bullough, 1988). Experiment shows that for a small angle of incidence the Mach reflection takes place instead of the regular reflection (Perroud, 1957). Fig. 4 shows a sketch of the Mach reflection. The apex P of the incident wave and the reflected wave moves away from the wall at a constant angle ∗ . The reflected wave is smaller than the incident wave and diminishes with the angle of incidence. The third wave joining the apex P of the incident wave and the reflected wave to the wall is approximately perpendicular to the wall and is called “Mach stem”. Though the observed profile of the Mach stem might depart significantly from that of the shallow-water solitary wave, Miles (1977b) presumed the Mach stem also to be a solitary wave and constructed a resonant soliton solution which represents the asymptotic state of the Mach reflection. When ai , ar , aM designate the amplitudes of incident solitary wave, reflected

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B C

r

i

P θ*

O

A

Fig. 4. Mach reflection pattern. PB: incident solitary wave, PC: reflected solitary wave, PA: Mach stem, i : angle of incidence,

r : angle of reflection.

solitary wave, and Mach stem normalized by the undisturbed water depth h, respectively, the properties of the asymptotic solution are summarized as follows (Tanaka, 1993): (i) Critical angle:

√ Mach reflection,

i < 3ai , √ type of reflection = regular reflection, i > 3ai . 

(ii) Angle ∗ (see Fig. 4): √ √ ai /3(1 − i / 3ai )

∗ = 0

Mach reflection, regular reflection.

(iii) Amplitude of Mach stem aM (or the maximum run-up at the wall): √ ⎧ Mach reflection, (1 + i / 3ai )2 ⎪  aM ⎨ = 4/[1 + 1 − 3ai / 2i ] regular grazing reflection, ⎪ ai ⎩ 2 + [3/(2 sin2 i ) − 3 + 2 sin2 i ]ai regular non-grazing reflection.

(53)

(iv) Amplitude of the reflected wave ar :  2

/3ai Mach reflection, ar = i ai 1 regular reflection. (v) Angle of reflection r : √ 3ai Mach reflection,

r =

i regular reflection. For the Mach reflection, the angle of reflection r does not depend on i . In particular, it is pointed out that the maximum run-up at the wall aM attains the magnitude of four times the amplitude of the incident √ solitary wave at the critical angle of incidence i = 3ai . Funakoshi (1980) solved numerically initial-value problems of a Boussinesq system of equations for shallow water to study the evolutionary process of the oblique reflection of a shallow-water solitary wave due to a rigid wall. The numerical solution for a sufficiently long time confirmed all of the above properties of Miles’ model.

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Melville (1980) made laboratory experiments to test Miles’ model of the Mach reflection. His measurements provided support for the predicted critical angle of incidence at which regular reflection is replaced by the Mach reflection but the measured run-up at the wall is considerably lower than the predicted value. Melville determined the length of the reflected wave in Miles’ model on the basis of mass and energy conservation and showed that this is not consistent with momentum conservation in the neighborhood of the end point of the reflected wave. So, he cast considerable doubt on the use of Miles’ model to describe Mach reflection at a rigid wall. On the contrary, Funakoshi (1981) supported Miles’ model on the basis of the results of numerical simulation. He asserted that the discrepancy between Miles’ model and the measurements in the run-up at the wall comes from insufficient interaction time in the measurements and viscous effect and inconsistency with respect to momentum conservation comes from defect of Melville’s model for the reflected wave. It is certain that Miles’ model is useful if the amplitudes are sufficiently small and viscous effects are negligible, though it may be difficult to confirm it in a laboratory experiment. However, when ai is not sufficiently small, Miles’ model may not give good approximation. In fact, numerical results of Tanaka (1993) based on fully nonlinear equations show that the effect of large ai tends to prevent the Mach reflection to occur. For ai = 0.1 Miles’ prediction gives good results, but for ai = 0.3 it gives considerable over-prediction for the critical angle, the stem length, the maximum run-up at the wall, and the angle of reflection. Tanaka’s numerical simulation shows also that though the amplitude of the Mach stem does not attain the magnitude of four times the amplitude of the incident solitary wave, it can become larger than that of the highest two-dimensional steady solitary wave for a given water depth. Johnson (1982) analyzed the oblique interactions between a large and a small solitary wave by means of a perturbation method. When i is sufficiently small and the Mach reflection occurs, Johnson’s result agrees much better with the Tanaka’s numerical result than Miles’ prediction does. There are many works also for the resonant interaction of ion-acoustic solitons. Yajima et al. (1978) investigated the interactions of plane ion-acoustic solitons of small amplitude in three-dimensional collisionless plasmas to give an analytical solution representing the interaction of two plane solitons and the conditions for the resonance quite similar to (50)–(52). Their solution is wider than two-soliton solution of the KP equation in the sense that the nearly unidirectional approximation is not used in their solution. Kako and Yajima (1980) solved numerically initial-value problems of two-dimensional ion-acoustic wave system to make clear the importance of soliton resonance in nonlinear development of the system. The initial condition is a superposition of two solitary waves. Their results show that even for an angle between the two solitons in which the steady solution should be singular, singular solution never appears but soliton resonance structure is created instead. Folkes et al. (1980) and Nishida and Nagasawa (1980) studied experimentally the two-dimensional interaction of two plane ion-acoustic solitons. Their observations agree qualitatively or quantitatively with the results of numerical simulation of Kako and Yajima (1980). The singular solution never appear but the resonant structure appears near the resonance condition. In particular, for = /2 ( : the angle between propagation directions of two solitons), Folkes et al. (1980) observed no phase-shift, which is one of the predictions of Yajima et al. (1978). Nishida and Nagasawa (1980) observed that the amplitude of the resonant soliton increases up to about four times as large as that of the colliding soliton near the resonance condition. Gabl and Lonngren (1984) performed the experiments on oblique interactions of unequal amplitude ion-acoustic solitons and interpreted the results in terms of the soliton resonance. The cylindrical and spherical solitons are known in ion-acoustic waves, though the term “soliton” here is used somewhat loosely. For the cylindrical and spherical solitons, refer also to chapter 9 of Infeld and Rowlands (1990). Maxon and Viecelli (1974a, b) showed that cylindrically or spherically symmetric

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inward-propagating ion-acoustic waves of small amplitude are described at large radius by the equation in the form: jU j

+

U jU 1 j3 U +U + = 0, 2 j 2 j3

(54)

where  = 1 for cylindrical case and  = 2 for spherical case. For outward-propagating waves, the form of the equation is the same though the scaled variables ,  are different from the case of inward propagation (Chen and Schott, 1976). Eq. (54) with  = 1 is integrable (Dryuma, 1976; Calogero and Degasperis, 1978; Johnson, 1979; Freeman, 1980) and called the cylindrical KdV (cKdV) equation. It also applies to shallow-water waves (Miles, 1978; Johnson, 1980). Maxon and Viecelli (1974a, b) investigated numerically inward-propagating solution of (54) for an initial condition corresponding to well-known sech2 profile which is an exact soliton solution of (54) without the term U/2. It was found that the inward-propagating solitary wave increases in amplitude and in propagation speed while decreasing in width, and a small-amplitude shelf develops behind the solitary wave. Cumberbatch (1978) showed that the amplitude of the solitary wave varies as ∝ ||−2/3 in the lowest order of the effect of geometry. For an inward-propagating wave,  is taken negative. Properly, an outward-propagating wave decreases in amplitude as time increases. Chen and Schott (1977) solved a similar initial-value problem numerically to find that the outward-propagating cylindrical soliton develops oscillatory tails. Ko and Kuehl (1979) considered the next order approximation to obtain the distortion of the solitary wave and the shelf height. The shelf was shown to be positive for an inward-propagating solitary wave and negative for an outward-propagating solitary wave (an oscillatory tail follows the negative shelf). They confirmed their analytical results by numerical integration of (54) and further found that for inward propagation a disturbance different from a one-dimensional soliton breaks up into two solitary waves with their respective shelf-like formations after a sufficiently long time. Hershkowitz and Romesser (1974) and Chen and Schott (1976) observed the ion-acoustic cylindrical solitons experimentally. Nishida et al. (1978) observed precisely formation and propagation of cylindrical ion-acoustic solitons to find that the experimental results agree well with theoretical predictions. Spherical ion-acoustic solitons were observed by Ze et al. (1979). Stepanyants (1981) performed an experimental investigation of cylindrical solitons in a two-dimensional electric LC-lattice to observe formation of solitons from non-soliton pulses and the soliton amplitude variation proportional to ||−2/3 . On the other hand, there exist two kinds of exact solution of the cKdV equation which are also called the cylindrical solitons. One is written in terms of the Airy function Ai (for example, Freeman, 1980). Its waveform is shown in Fig. 10 of Freeman (1980) and looks like the outward-propagating cylindrical solitons found in numerical simulations and in experiments. The other solution obtained by Nakamura and Chen (1981) is described in terms of the Airy function Bi (see also Nakamura, 1980) and looks like the inward-propagating cylindrical solitons found in numerical simulations and in experiments. The figures of the latter exact solution (in fact, two-soliton solution) can be found in the book by Infeld and Rowlands (1990). Johnson (1999) contended that the cylindrical solitons found in numerical simulations and in experiments are different from the exact soliton solutions of the cKdV equation and well described by an asymptotic solution of the cKdV equation for a given sech2 initial profile based on the small parameter  (≡ 1/large initial radius, in non-dimensional variables). The asymptotic solutions which he constructed agree very well with numerical solutions of cKdV equation both for inward and outward propagation. When two outward-propagating cylindrical (or spherical) solitons of large radius having different axes (or centers) interact, soliton resonance may take place as the intersecting angle of two solitons

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decreases with time (it is noted that we exclude here the cylindrical solitons as the exact solutions of the cKdV equation from the object of consideration). In fact, Ze et al. (1979) performed experiments on the collision of two identical spherical outward-propagating ion-acoustic solitons having different centers to observe the creation of a new, large-amplitude two-dimensional nonlinear object. Ze et al. (1980) observed not only the collision of two spherical ion-acoustic solitons but also the collision of two identical cylindrical outward-propagating solitons having different axes to find that many characteristics of the interactions are similar to those of the solutions to the KP equation. Tsukabayashi and Nakamura (1981) also investigated experimentally the collision of outward-propagating cylindrical ion-acoustic solitons to observe a resonant interaction and trailing structure at a critical intersecting angle. Maxworthy (1980) observed the formation of a new wave in the collision of two cylindrical outward-propagating internal gravity solitons. Kaup (1981) investigated theoretically the experimental results of Ze et al. (1979) to find that they can be explained at least qualitatively in terms of nonlinear resonance of solitons. Kako and Yajima (1982) studied numerically the collision process between two identical cylindrical or spherical outward-propagating ion-acoustic solitons having different axes or centers and gave theoretical analysis for the simulation results in terms of soliton resonance. They used, as the initial condition, the superposition of two asymptotic solutions of their equation for large radius. 4. Oblique interactions of solitary waves in non-KP systems In Section 3, the oblique interaction of solitary waves in the KP systems has been described. Essentially interesting ingredients of them are included in the KP equation. We have derived, in Section 2.2, the 2DBO equation describing the strong interaction of the BO solitons. Similar weakly two-dimensional extension is possible for other one-dimensional soliton equations. It is very interesting to investigate the oblique interaction of solitary waves in such two-dimensional equations. Do the soliton resonance, the Mach-type reflection, and the creation of large-amplitude new wave take place? So far, Tsuji and Oikawa (2001) have investigated the 2DBO equation in the form

2  j ju ju j u j2 u +u +H + = 0, (55) jX jT jX jX 2 jY 2 the MKP equation (Tsuji and Oikawa, 2004)
 j ju j3 u j2 u 2 ju + 6u + + = 0, jX jT jX jX 3 jY 2 and the EKP equation (Tsuji and Oikawa, 1993, 2006)
 j ju ju ju j3 u 1 j2 u + 6pu − 6qu2 + + = 0, jX jT jX jX jX 3 2 jY 2

(56)

q > 0.

(57)

The MKP equation is a weakly two-dimensional extension of the MKdV equation ju jT

+ 6u2

ju jX

+

j3 u jX 3

= 0,

(58)

and describes weakly two-dimensional ion-acoustic waves in a plasma composed of electrons, positive, and negative ions (Watanabe, 1984; Tsuji and Oikawa, 2004) and also describes weakly two-dimensional

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electromagnetic waves in a cold collisionless plasma (Bindu and Kuriakose, 1998). The MKdV equation is one of typical soliton equations and exactly solvable by the inverse scattering method (Wadati, 1972, 1973). The EKP equation is a weakly two-dimensional extension of the EKdV equation ju jT

+ 6pu

ju jX

− 6qu2

ju jX

+

j3 u jX 3

= 0,

q > 0,

(59)

and describes weakly two-dimensional long interfacial waves in a two-layer fluid having a thickness ratio of the two layers near the critical one which is the square root of the density ratio and for which the quadratic nonlinear term in (57) vanishes, that is, p = 0. The EKdV (59) is also the soliton equation and exactly solvable by the inverse scattering method (Perelman et al., 1974 a, b). Eqs. (55)–(57) are (probably) not integrable, so exact analytical solutions to them are not available. Tsuji and Oikawa, therefore, investigated numerically the oblique interaction of solitary waves in those equations. Two types of computations were made. The initial condition in Type 1 computation is a simple superposition of two identical solitary waves (or two identical periodic waves of a very long wavelength) with different propagation directions and the periodic boundary condition. This type of computations were made in Tsuji and Oikawa (1993, 2001). In these computations, the disturbances which are generated by the interaction of the solitary waves and propagate downward appear upward due to the periodic boundary condition and contaminate the main part of the wave field. Consequently, this type of computation is inappropriate for a detailed quantitative discussion. The initial condition in Type 2 computation is shown in Fig. 5. In this initial condition, AB and AC are crest lines of the two identical solitary waves and the pattern is symmetric with respect to the line PQ. This condition simulates the oblique reflection of a solitary wave due to a rigid wall PQ. i is the angle of incidence. So, we use the terms “incident waves”, “reflected waves”, and so on in the following. The boundary condition for the computational domain LX × LY is basically periodic. In addition to the periodic boundary condition the wave field is reset only near the boundaries Y = 0 and LY (or, in certain cases, Y = ±LY /2) at every time step as if the solitary waves propagate steadily far from the wall at Y = LY /2 (or Y = 0). Tanaka (1993) adopted successfully a similar boundary condition. Without this procedure serious deformation of the incident solitary waves near the boundaries Y = 0, LY (or Y = ±LY /2) takes place. Far from the solitary wave near the boundaries, u is actually zero. Hence, these boundaries impose virtually the

B

i

Q

P

Y A

C X

Fig. 5. Sketch of the initial condition in Type 2 computation. AB and AC are crest lines of the solitary waves. PQ is the axis of symmetry.

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boundary condition u = 0 far from the solitary waves. Sufficiently large LX was also taken. Here, the results of Type 2 computation are mainly described. 4.1. 2DBO equation The initial condition for (55) is ⎧ A ⎪ ⎨ 1 − B cos[(2 /LX )(X − 7LX /8 + (Y − LY /2))] u= A ⎪ ⎩ 1 − B cos[(2 /LX )(X − 7LX /8 − (Y − LY /2))]

for 0
Y
LY /2, (60) for LY /2
Y
LY .

Here,  = tan i determines the directions of the crest line and of propagation. This is constructed from the periodic solution of (55): A , 1 − B cos[(2 /LX )(X + Y − (a/4 + 2 )T )]  
2 32 8 2 A≡ , B ≡ 1− aLX aL2X

u=

(61)

with the period LX in the X direction, and the parameter a is the amplitude of the soliton solution obtained in the limit aLX → ∞: u=

a . 1 + a 2 [X + Y − (a/4 + 2 )T ]2 /16

(62)

Since aLX ?1 in their computation, (61) cannot be distinguished graphically from (62) in the computational domain. The parameter a is taken as 2. Figs. 6(a)–(c) show the wave patterns for  = 2, T = 10.56, for  = 1, T = 25.29, and for  = 0.5, T = 43.49, respectively. It is noted that these times are those at which the location of the peak of wave amplitude attains X = 765. It is also noted that the scale of the Y-axis varies with . For  = 2 (Fig. 6(a)), the reflected waves newly generated behind the incident waves have the same amplitude, the same propagation directions, and the same profile as the incident solitary waves. Consequently, the reflection in this case is evidently regular (equi-angular). It seems that there exist very small phase shifts. For  = 1 (Fig. 6(b)) and  = 0.5 (Fig. 6(c)), a new wave parallel to the Y-axis and propagating in the X direction is generated. We call this the stem wave. The stem has the profile of the BO soliton. However, the reflected waves have not the profiles of the solitary wave solutions. So, we cannot say that the soliton resonance takes place. The behavior of the stem is, nevertheless, similar to that of the Mach reflection in the KP systems. Fig. 7 shows the dependence of the asymptotic value of the stem amplitude on various . Actually each asymptotic value of stem amplitude is measured at the time when the stem attains the location X = 765. For  = 1, 1.33, 1.43 the stem amplitude is still slowly increasing at the time. This result is qualitatively very similar to that of the KP equation. In the KP equation the possible largest amplitude of the stem

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887

50 40

4 2 0

30 50 20

40 700

30 20

740 760

X

0 700 720 740 760 780 800 X

10 780

(a)

10

Y

720

0

100 80 100 80

40

60 720

40

740 X

(b)

760

20

Y

700

0 700 720 740 760 780 800 X

20 780

60 Y

8 6 4 2 0

0

200 4 2 0

200 150

700 720

50

100 X

Y

740

(c)

100

Y

150

50

760 780

0

0 700 720 740 760 780 800 X

Fig. 6. Interaction patterns of solitary waves in the 2DBO equation (55) for (a)  = 2 and T = 10.56; (b)  = 1 and T = 25.29; (c)  = 0.5 and T = 43.49. The bird’s eye view (left side) and the contour map (right side).

wave is four times as much as the amplitude of the incident solitary wave. On the other hand, the possible maximum amplitude of the stem in (55) exceeds evidently the value 4a (=8) for the KP equation. Fig. 8 shows the evolution of the stem length for various . While the stem length reaches the steady state for relatively large , it grows almost linearly for relatively small . In the present case of a = 2, the stem length grows almost linearly at least for 1.43.

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10 9

u max

8 7 6 5 4

0

0.5

1

1.5

2

Ω

Fig. 7. Dependence of the stem amplitude on . The stem amplitude umax is measured just when the stem (or the peak of the wave field) attains X = 765.  = 0.5, 0.67, 1, 1.33, 1.43, 1.67, 2.

25 20

Ls

15 10 5 0 710

720

730

740

750

760

770

X

Fig. 8. Evolution of the stem length LS for various . +:  = 0.5, ×:  = 1, :  = 1.43, :  = 1.67, :  = 2. Fig. 9 in Tsuji and Oikawa (2001) should be replaced with this figure, because it includes a mistake.

Eq. (55) is invariant under the transformation u u∗ = , a

X∗ = aX,

Y ∗ = a 3/2 Y,

T ∗ = a2T ,

(63)

where a is an arbitrary positive constant. In terms of these new variables, Eq. (62) is rewritten as u∗ =

1 1 + [X ∗

+ (/a 1/2 )Y ∗

− ( 41 + 2 /a)T ∗ ]2 /16

.

(64)

Consequently, for aLX >1, the initial condition in the new variables depends on a and  through the combination /a 1/2 . If the boundary condition used in the computations simulates well the theoretical

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boundary condition, the solution must satisfy the above scaling property and the interaction properties must depend only on the combination /a 1/2 . In fact, the computational results for a = 1 are consistent with this scaling property. 4.2. MKP equation The MKP equation (56) has positive and negative solitary wave solutions u = ±A sech[A(X − X0 + Y − (A2 + 2 )T )],

(65)

where the positive parameter A is the amplitude and X0 is a constant. Tsuji and Oikawa (2004) investigated two kinds of interaction—positive–positive and positive–negative solitary waves. The initial condition for the positive–positive interaction is  A sech[A(X − X0 + Y )] for − LY /2 < Y < 0, (66) u= A sech[A(X − X0 − Y )] for 0 < Y < LY /2. The initial condition for the positive–negative interaction is  −A sech[A(X − X0 + Y )] for − LY /2 < Y < 0, u= A sech[A(X − X0 − Y )] for 0 < Y < LY /2.

(67)

A is certain function of Y introduced in order to smooth out the discontinuity at Y = 0 and to suppress a non-essential initial transient behavior. Eq. (56) is invariant under the transformation u (68) u∗ = , X∗ = aX, Y ∗ = a 2 Y, T ∗ = a 3 T , a where a is an arbitrary constant. The initial conditions written in terms of these new variables with a = A contain only the parameter /A. So, the wave field depends on the parameter /A. The behavior of the interaction of two positive solitary waves is qualitatively the same as that of 2DBO equation. For large /A, the regular reflection takes place. On the other hand, for small /A, the stem wave is newly generated and has the MKdV soliton profile. However, reflected waves have not soliton profiles. The stem length increases linearly with the propagation distance for /A3 and initially increases but soon settles down to a constant for /A4.5 (Fig. 5(a) in Tsuji and Oikawa, 2004). So, large and small /A may mean /A4.5 and /A3, respectively. Fig. 9 shows the plots of umax /A versus /A, where umax is the asymptotic value of the maximum of u (the maximum of u is the stem amplitude or the peak amplitude of u at the intersection of the solitary waves). The overall features of this plot as well as those of Fig. 7 are very similar to those of the analytical result for the KP equation due to Miles (Fig. 5 in Miles, 1977b). umax /A attains its maximum value at some value near /A = 4, and the maximum value may be near 4. Nakamura et al. (1999) investigated experimentally the oblique interaction of the MKdV solitons. Though they observed a newly generated stem wave, at the same time they found that the stem amplitude attains at most 2.2 times that of the incident solitary wave. The reason for this discrepancy is to be considered in the future. The initial condition (67) for the interaction between positive and negative solitary waves is antisymmetric with respect to Y = 0 and (56) is invariant under the transformation Y → −Y . The solution to (56)

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4

u max /A

3

2

1 0

2

4

6

8

10

Ω /A

Fig. 9. Dependence of the asymptotic value umax /A of the normalized stem amplitude on /A for three values of A. +: A = 0.5, ⱓ: A = 0.75, : A = 1.

is, therefore, also antisymmetric. So, this problem may be considered as the oblique reflection problem of a solitary wave due to a hypothetical wall at Y = 0 where the condition u = 0 is imposed. Some examples of wave field are shown in Fig. 6 of Tsuji and Oikawa (2004). For large /A the reflection is regular, though the reflected waves change their signs. For small /A, while the incident waves decrease in amplitude near Y = 0 and curve backward, the reflected waves develop wavy structures. It is noted that even for  = 0 a similar wave pattern develops.

4.3. EKP equation The EKdV equation (59) has the soliton solution u=

a sech2  , 1 − (a/(2p/q − a))tanh2 

 =
(X − 4
2 T ),
=

1 a(2p − qa), 2

(69)

where p > 0 can be assumed without loss of generality. Unlike the BO soliton and the MKdV soliton, the amplitude of this soliton solution is restricted to values lower than the critical height ac ≡ p/q. The interaction properties in (57) may be different from those in the 2DBO and the MKP equations. Tsuji and Oikawa (1993) investigated (57) by means of Type 1 computation to find that for aq/p = 0.1501 · · · the solutions of (57) are very close to those of the corresponding KP equation (57) with q = 0 and the soliton resonance condition is approximately satisfied, but for aq/p = 0.3250 · · · and 0.5425 · · · the solutions of (57) are significantly different from those of the corresponding KP equation. Recently, Tsuji and Oikawa (2006) have investigated more quantitatively the EKP equation of the form j jX




ju jT

+ 6u

ju jX

− Qu2

ju jX

+

j3 u jX 3

 +

1 j2 u = 0, 2 jY 2

Q>0

(70)

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by means of Type 2 computation. The initial condition is ⎧ a sech2 + ⎪ for 0 < Y < LY /2, ⎪ ⎨ 1 − (a/(12/Q − a))tanh2  + u= ⎪ a sech2 − ⎪ ⎩ for LY /2 < Y < LY , 1 − (a/(12/Q − a))tanh2 − where ± =
(X − X0 ± Y ),


= 21



a(2 − aQ/6).

891

(71)

(72)

Keeping the initial amplitude a = 1, the variation of the wave field with variation of the parameter Q is examined. For Q = 0.5, ac = 6/Q = 12 is much larger than the initial amplitude a = 1. In this case, the solutions are very close to those of the corresponding KP equation (Q = 0). The equi-angular reflection for large  (=4) and the Mach reflection for small  (=1) take place. It is noted that for  = 4, the amplitude of the hump at the intersection of the solitary waves is about 2.6 and a wavy disturbance develops in the region between the reflected waves. For  = 1, the reflected waves are also solitons and the soliton resonance condition is almost satisfied. For Q = 5.99, ac = 1.0016 · · · is very close to the initial amplitude. For  = 4, the reflection pattern is equi-angular and the amplitude of the reflected waves is the same as that of the incident waves. However, the reflected waves have not the solitary wave profile but a fatter profile. Further the peak amplitude of the intersection of the solitary waves is about 1.7, being lower than the value 2 in the simple superposition. A wavy disturbance also develops in the region between the reflected waves. Even for  = 1, the stem wave is not formed but a hump is formed at the intersection of the incident waves. The amplitude of the hump is about 1.3 which is lower than that of the case  = 4. Unlike the case  = 4, the reflected waves hardly develop and curve backward. In this case, a wavy disturbance develops again in the region between the reflected waves. Fig. 10 shows the dependence of the asymptotic amplitude of the stem or the hump on  for various values of Q. For Q=0, umax increases linearly as  increases, attains the maximum at a value of  (we call such a value of  ‘c ’, though it is difficult to determine c precisely) and then decreases monotonically as  increases. While for Q3, umax () behaves as in the case Q = 0, c decreases with increasing Q and the corresponding maximum of umax also decreases. For Q > 3, c does not exist and umax increases monotonically with increasing . It is interesting to note that for Q > 3, the amplitude of the stem or the hump exceeds the critical height (ac = 6/Q) for a wide range of . 5. Solutions of the KP equation The KP equation is very important both from theoretical standpoint and from standpoint of application and besides it is integrable. So, there are innumerable studies about solutions and various properties of the KP equation. The inverse scattering method is applicable to the KP equation. A detailed description of it is given in the book by Ablowitz and Clarkson (1991). Konopelchenko (1993) and Boiti et al. (1995) may be also useful. One of the advantages of the inverse scattering method is that it gives the procedure for solving an initial-value problem. However, in general, it is extremely difficult to carry

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u max

3 2.5 2 1.5 1

0

1

2

3

4

5

Ω

Fig. 10. Dependence of the asymptotic value of the amplitude umax of stem or hump on  for various values of Q. : Q = 0 (KP), : Q = 0.1, ×: Q = 1, : Q = 1.5, +: Q = 2, 䊉: Q = 3, : Q = 4, : Q = 5, : Q = 5.99.

out the procedure. Solutions can also be constructed by use of the inverse scattering method without referring to an initial-value problem. The bilinear method (for example, Hirota, 1976) is another powerful method for obtaining various solutions including soliton solutions and is much easier to approach than the inverse scattering method. Sato (1981) revealed, by means of the method of algebraic analysis, the algebraic structures which the soliton equations represented by the KP equation possess in common. For an elementary introduction to the Sato theory, refer to Ohta et al. (1988). Here, we briefly survey relatively recent development of solutions of the KP equation. 5.1. KP-I equation As has been mentioned, the KP-I equation has the rational soliton solutions which decay algebraically in all directions in the (X, Y ) plane. Zaitsev (1983) first obtained two kinds of stationary “periodic soliton” solutions to the KP-I equation: both of them propagate in X direction and are constructed by superposition of rational solitons of the same amplitude. One is periodic in the Y direction and decays exponentially as X → ±∞. The other is periodic in the X direction and decays exponentially as Y → ±∞. Abramyan and Stepanyants (1985) rediscovered the same solutions using another method. Tajiri and Murakami (1989) obtained a “periodic soliton” solution which represents an inclined sequence of rational solitons and includes the above two solutions as special cases. At the same time they found that a solution representing the interaction between different periodic solitons (or between different kinds of solitons) may be constructed. Tajiri and Murakami (1990) investigated the interaction between two Y-periodic solitons—periodic solitons which are periodic in the Y direction and propagate in the X direction—to find that there is a parameter region where the interaction generates a phase shift of solitons not only in the direction of propagation but also in the transverse direction and that a resonant interaction can occur, which is associated with parametric points on the boundary between the region for transverse phase shift and that for no transverse phase shift of solitons. In this connection, it is noted that there is no phase shift in the interaction between rational solitons. The details on the interaction between two Y-periodic solitons were reported in Murakami and Tajiri (1991). Murakami and Tajiri (1992) also investigated the interaction

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between a line soliton and a Y-periodic soliton to classify the interaction into several types according to the phase shift and to point out that resonant interaction can occur. Tajiri and Murakami (1993) and Chow (1994) obtained the “lattice soliton” solution which is a doubly periodic array of the localized structures. Ablowitz et al. (2000) constructed “multi-pole lump” solutions through binary Darboux transformations and presented the properties of these solutions, related potential and eigenfunctions of the non-stationary Schrödinger equation which is the linear eigenvalue equation associated with the KP-I equation in the inverse scattering method. The “multi-pole lump” solutions are real and non-singular and they decay algebraically. The name is due to the fact that they are related to the solutions of the non-stationary Schrödinger equation which in general possess multiple poles in the spectral parameter. In the case of simple poles the solutions reduce to lump solutions (rational solitons). 5.2. KP-II equation Krichever (1976, 1977) first found that quasi-periodic solutions of N phases are written in terms of the Riemann theta function of genus N (see also Krichever, 1989). For low genera, Dubrovin (1981) and Segur and Finkel (1985) investigated these solutions in detail and gave them concrete expressions which permit us numerical valuation. In order to use the KP solution of genus 2 (that is, periodic in two spatial directions) as models of two-dimensional periodic gravity waves in shallow water, the latter presented an algorithm to infer the parameters in these solutions from measurements of wavelengths and wave velocity. Hammack et al. (1989) generated experimentally genus 2 gravity waves with symmetric pattern in a wave basin and compared an experimental result with the best-fit KP solution for it. It was found that the measured wave is described well by the corresponding KP solution over the entire parameter range of the experiments even for considerably large amplitude. It was also found that such waves are easy to generate experimentally and propagate stably. Further, Hammack et al. (1995) obtained similar conclusion for genus 2 waves with asymmetric pattern. In this new experiments the depth variation in the laboratory basin was reduced and the agreement between theoretical and experimental results was improved further. Recently, physical mechanisms of the rogue wave phenomena have attracted much attention of researchers and many mechanisms have been proposed (Kharif and Pelinovsky, 2003). According to Tomita et al. (2006), rogue or freak wave is a sort of abnormal wave in the ocean, which appears as a majestic feature among the surrounding waves. However, it is noted that there exists no decisive definition of rogue or freak wave yet. Some authors took up the two-soliton interaction of the KP-II equation as a possible mechanism of generation of large-amplitude wave in shallow water to analyze in detail the two-soliton solution (Peterson et al., 2003; Soomere, 2004; Soomere and Engelbrecht, 2005). Peterson and van Groesen (2000) found an interesting decomposition of the two-soliton solution into the asymptotic parts of solitons and the interacting part (they call this part “interaction soliton”) in an attempt to reconstruct the amplitudes of two-dimensional surface waves from observation of wave patterns. The three-soliton solutions of the KP-II equation were also considered by Anker and Freeman (1978) and Ohkuma and Wadati (1983). The interaction in N-soliton solution is very complicated when the soliton resonances take place for N > 2. Recently, Biondini and Kodama (2003) investigated a class of solutions of the KP-II equation in which the -function is given by Hankel determinants to show that all the solutions (except the one-soliton solution) are of resonant type (that is, the interaction of any pair of line solitons is in resonance) and if arbitrary N− incoming solitons for Y → −∞ interact to form arbitrary N+ outgoing solitons for Y → ∞, the resonant interaction creates a web-like structure

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having (N− − 1)(N+ − 1) holes. This web-like structure is the same as that found by Isojima et al. (2002) for a coupled KP equation. Pashaev and Francisco (2005) constituted similar solutions by means of quite different method. Kodama (2004) gave a general formulation of the classification of interaction patterns of the N-soliton solution of the KP-II equation in terms of Young diagrams (see also Biondini and Chakravarty, 2006). Such a sequence of soliton resonance may generate a large-amplitude wave temporarily in shallow water (Maruno et al., 2006).

6. Concluding remarks We have summarized studies on the oblique interaction of solitary waves in KP and non-KP systems. It seems that the experiments support the theory and numerical solutions related to the soliton resonance for the interaction of line solitons (or planar solitons) in KP systems if the restrictions in the theory and numerical solutions, i.e. weak nonlinearity, no dissipation, are taken into account. As is expected, the generation of a resonant soliton-like wave due to the interaction of two cylindrical (or spherical) solitons having different axes (or centers) of large radius takes place in experiments and in numerical simulation for ion-acoustic waves. Numerical studies on the oblique interaction of solitary waves, that is, the oblique reflection of a solitary wave due to a rigid wall, in the non-KP systems were surveyed. In 2DBO and MKP equations, for relatively large angles of incidence the reflection is regular and for relatively small angles of incidence Mach-like reflection takes place. Since the reflected wave is not a solitary wave solution, we cannot say that soliton resonance takes place. However, the maximum amplitude of the stem attains to about four times as much as the amplitude of the incident solitary wave for certain angle of incidence as in the case of the KP equation. It is interesting to investigate the reflection problem for fully nonlinear equations. For MKP equation there is a discrepancy between numerical result (Tsuji and Oikawa, 2004) and experimental result (Nakamura et al., 1999). The reason for this should be considered. In EKP equations, for sufficiently small Q the solution is very close to that of the KP equation (i.e. Q = 0). So, the soliton resonance occurs approximately. On the other hand, for large Q, because of the restriction in the amplitude of one-dimensional soliton, the stem does not develop and the growth of the possible maximum amplitude is suppressed. In the KP-I equation, because of instability of the soliton for transverse perturbation, there are various solutions including the rational soliton which decays in all directions in the (X, Y ) plane. On the other hand, as the soliton solution is stable to transverse perturbations in the KP-II equation, the multi-soliton solutions are realistic ones. Classification of pattern of the multi-soliton solutions to the KP-II equation including soliton resonance is a very recent outcome.

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