Note on the 2-component Analogue of 2-dimensional Long Wave-Short Wave Resonance Interaction System

y ) -S (1) xx + LS (1) = 0 , i(S

xx + LS (2) = 0 , L t = 2(|S (1) | 2 + |S (2) | 2 ) x .

(1

This system is an extension of the two-dimensional long wave-short wave resonance interaction system [7,8] and describes the two-dimensional resonant interaction between an interfacial gravity wave and two surface gravity packets propagating in directions symmetric about the propagation direction of the interfacial wave in a two-layer fluid.

In this paper, we will study this system and its integrable modification, i(S

t + S

(1) 2) .

(

where * means complex conjugate. In our recent paper [9], we studied i(S

t + S

(1)

xx + LS (1) = 0 , i(S

(2)

Note that this system is different from the system (1) only in the sign of y-derivative term S

(2) y .

Consider a two-component analogue of two-dimensional long wave-short wave resonance interaction (2c-2d-LSRI) system (2). Using the dependent variable transformation L = -(2 log F) xx , S (1) = G/F, S (2) = H/F, Q = -K * /F , we obtain

These bilinear forms have the three-component Wronskian solution [10,11,12]. Consider the following three-component Wronskian:

where ϕ, ψ and χ are

1≤ j≤M 1≤i≤N+M+L and χ = (∂ j-1

x 1 χ i ) 1≤ j≤L 1≤i≤N+M+L , and ϕ i is an arbitrary function of x 1 and x 2 satisfying ∂ x 2 ϕ i = ∂ 2

x 1 ϕ i , and ψ i and χ i are arbitrary functions of y 1 and z 1 , respectively. The above Wronskian satisfies

we have the following bilinear forms:

By the change of independent variables

Thus we obtain

Consider solutions satisfying the following condition

where G is a gauge factor. Then, for F = f G, G = gG, H = hG, K = kG, we will obtain the bilinear equations of the 2c-2d-LSRI system (4). Thus the 2c-2d-LSRI system has a three-component Wronskian solution.

To satisfy the condition (5), we consider the following constrained case:

, where p i , s i , q i , r i are wave numbers and η i0 , ζ i0 are phase constants. The parameters a i and b i must be determined from the condition of complex conjugacy. By using the standard technique [13], a i and b i are determined as

Figure 1: Single line soliton of eqs.(2), which is obtained by tau-functions of ( 6).

(a) -L , (b) |S (1) |, (c) |S (2) |, (d) Re [S (1) ], (e) Re [S (2) ]. The parameters are p = 1 + i, q = -1 + 2i, r = -2 + i.

and the condition ( 5) is satisfied for the gauge factor,

This solution represents the (M + L)-soliton, i.e., M solitons propagate on the first component of short wave S (1) whose complex wave numbers are given by p i , q i and complex phase constants are η i0 , and L solitons propagate on the second one S (2) whose complex wave numbers and phase constants are s i , r i and ζ i0 . For instance by taking M = L = 1, (1+1)-soliton solution is given as

where c = -|(ps)(p + s * )| 2 and we dropped the index 1 for simplicity. In order to satisfy the regularity condition f = 0, we can take Re p > 0, Re s > 0, Re q < 0 and Re r < 0. After removing the gauge and constant factors, by choosing the same wave number in x direction for the above two solitons, i.e., s = p, we obtain the single soliton solution,

where ξ = pxip 2 y, η = q(yt) + η 0 and ζ = -r(y + t) + ζ 0 . Figure 1 shows the plots of this single soliton solution. L shows V-shape soliton, |S (1) | and |S (2) shows solitoff behaviour [14].

We consider the 2c-2d-LSRI system (1) without the fourth field Q in (2). This system (1) describes waves in the two-layer fluid. Setting L = -(2 log F) xx , S (1) = G/F, S (2) = H/F , we have

Here we consider the case of c = 0.

Using the procedure of the Hirota bilinear method, we obtain the single soliton solution

,

Here q j is a real number. We can rewrite A 11 as

.

Thus we have

, S (2) = b 1 exp(ξ 1 )

∂x 2 log F do not include y, all solitons propagate in the x direction.

There is an exact solution depending on y-variable, S (1) = A 1 exp(px + qy + rt)

and p, q, k 1 , l 1 , l 2 are arbitrary parameters. In figure 2, we see that waves in S (1) and S (2) have different modulation property, i.e., carrier waves in S (1) and S (2) has different directions of propagation. Note that the solutions of equations (2) also have this property.

It seems that eqs.(1) are nonintegrable and do not admit general N-soliton solution. Similar system (2) has an N-soliton solution, but its physical derivation has not been done yet.

We have studied solutions of a new integrable two-component two-dimensional long wave-short wave resonant interaction (2c-2d LSRI) system (2). We presented a Wronskian formula for 2c-2d LSRI system (2) with complex conjugacy condition. We have also presented solutions of the system (1) in the case of two-layer fluid, i.e. the 2c-2d LSRI system without Q. In this case, the