A Darboux Transformation for Ito Equation

1 Introduction

In his investigation of Hirota’s bilinear equations possessing 3-soliton solutions, Ito [1] discovered the following equation:

Dt(Dt + Dx3)f ⋅ f = 0,

where f=f(x, t), and D_t and D_x are the well-known Hirota derivatives. By means of the dependent variable transformation u=2(ln f)_xx, Ito’s equation is brought into its nonlinear version

utt + uxxxt + 3(2uxut + uuxt) + 3uxx∫−∞xutdx = 0.

If we let

u → 13u, t → −t,

then the aforementioned equation becomes

Introducing a new variable. v=v(x, t) by u_t=3v_x, (1) may be rewritten as

vt = vxxx + (uv)x.

Thus, the non-evolutionary and nonlocal (1) is equivalent to the following coupled system:

In the subsequent discussion, (2) will be referred to Ito’s equation. Ito showed that his equation has N-soliton solution, a bilinear Bäcklund transformation, and a Lax representation. It is interesting to note that Ito’s equation appeared in other contents and has been studied extensively. Jimbo and Miwa [2] listed it as a system related with the Kac-Moody algebra D3(2). Drinfeld and Sokolov presented it as an example of their work on the Lie algebras and equations of KdV type [3, 4] (see also [5]). Konopelchenko and Oevel [6] found Ito’s equation (2) as the Kupershmidt reduction of a nonstandard hierarchy. Matsuno [7] constructed its conservation laws out of the Bäcklund transformations. Hu and Li [8] succeeded in presenting a nonlinear superposition formula of the equation within the framework of Hirota’s bilinear method. Liu [9] obtained the bi-Hamiltonian formalisms for (2). Hirota et al. generalised Ito’s equation and its vector version was obtained [10]. Also, a supersymmetric extension was given in [11].

It is well known that Darboux transformations constitute an important part of soliton theory and play an vital role in the study of integrable systems [12]. In particular, a Darboux transformation may provides a convenient tool to construct particular solutions of an integrable, nonlinear differential equation. To our knowledge, Darboux transformation of (2) has not been constructed.

The aim of this article is to construct a proper Darboux transformation for (2). We show that the Bäcklund transformation follows from the constructed Darboux transformation directly. Also, a nonlinear superposition formula, which is of differential-algebraic, is worked out. As applications, the results are used to calculate a 2-soliton solution of Ito’s equation.

2 Darboux Transformation

For a given a nonlinear system, to construct its Darboux transformation, we must first have a Lax or zero curvature representation. It is interesting to notice that (2) possesses two Lax operators. One of them is due to Drinfeld and Sokolov [3, 4] (see (B8) of [5] also) and reads as

ℒt = [𝒫, ℒ],

where

ℒ = (∂x5 + u∂x3 + ∂x3u + (12u2 − 2v)∂x + ∂x(12u2 − 2v))∂x, 𝒫= − 12(∂x3 + u∂x).

The other one, found by Ito [1] (see also [6]), takes the following form:

where

L = ∂x3 + u∂x + ∂x−1v,   P = ∂x3 + u∂x.

Lax equation (3) is the compatibility condition of the following linear problems:

In principal we may use either ℒ or L to build the Darboux transformation for Ito’s equation. However, because the latter is of a lower order, it is easier to handle. Thus, next we work with (4). To find a Darboux transformation, we first reformulate the linear spectral problem as the matrix form. Introducing σ_x=vϕ, and Φ=(ϕ, ϕ_x, σ, ϕ_xx)^T, then we may rewrite (4) in matrix form, that is,

Now we suppose that there exists a gauge transformation, 𝒯:

such that Φ_[1] solves

where 𝒰_[1], 𝒱_[1] are the matrices 𝒰, 𝒱 but with u, v replaced by the new field variables u_[1], v_[1]. To be qualified as a Darboux transformation, the matrix 𝒯 has to satisfy

Suppose 𝒯=λ𝒜+ℬ, 𝒜=(a_ij)_4×4, ℬ=(b_ij)_4×4, and after certain analysis we find that the matrices 𝒜, ℬ take the following forms:

A = (1000a10000−1012a2 + axa01),ℬ = (b + η12a2 − ax−2abx − 2v−12aax + 14a3 + η−a12a2−12(b + η)2 + 12η2(bx − 2v)x − a(bx − v)b + η2v − bxbxx − 2vx − av(b − 12aax + 14a3)x − 2v − 12a2u− 12a2 − ax12aax + 14a3 + η),

where

b = axx − 32aax + au + 14a3.

Thus, all the entries of the Darboux matrix 𝒯 are represented in terms of a. If this function satisfies the following equations:

then we have

To obtain an explicit Darboux transformation, we now seek a solution of (10)–(12). Indeed, impose on a the following equation

then it is easy to check that (10), (11) and (12) hold identically. Thus, the Darboux matrix 𝒯 may be formulated entirely in terms of a and the field variables, and (14) serves as the (spatial part) of Bäcklund transformation for (2). The temporal part of Bäcklund transformation may be obtained as

Now, we reformulate the Darboux matrix 𝒯 in terms of the solutions of the linear system (5) and (6) so that it takes a more explicit form. This may be achieved by exploring the kernel of the Darboux matrix 𝒯. Indeed, suppose that Y=(y₁, y₂, y₃, y₄)^T is a particular solution of (5, 6) at λ=η, and then 𝒯Y|_λ=η=0 leads to

a = − 2y2y1.

Finally, we relate our findings to the known results. According to the Ito equation [1], there is the following bilinear Bäcklund transformation:

Set λ=0 and let

f = exp(12∫−∞xwdx), f′ = exp(12∫−∞xw[1]dx), w[1] − w = −a,

we obtain from (16) and (17)

axxx = 18a4 + 32a2u − 32a2ax + 32ax2 + 2aaxx − 3(ua)x − 2wt − μa,at = 34a2ax − 12aaxx − 18a4 − 32ua2 + 2wt + μa,

which is nothing but the Bäcklund transformation (14) and (15) up to some scaling.

3 Nonlinear Superposition Formula

For a given Bäcklund transformation, it is desirable to find the corresponding nonlinear superposition formula.because on the one hand, such formula often provides a convenient way to construct solutions to the associated nonlinear system, it may also supply a (semi) discrete system on the other hand.

To find a nonlinear superposition formula of the Darboux/Bäcklund transformation, we obtained in last section, we supposed that (u, v) is an arbitrary solution of (2), then with the help of η=η_k (k=1, 2), we may get the new solutions u_[k], v_[k], and Φ[k] = T|η = ηkΦ. That is, we consider a pair of Darboux transformations:

where u_[k]=u−a_k,_x. Then, by means of the compatibility of last two equations or the Bianchi permutability depicted by the diagram

we have

ℳ[2]N = N[1]ℳ,

where

ℳ[2] = T|η = η1, a =a21,   N[1] = T|η = η2, a = a12,

u[21] = u[2] − 3a21x,   u[12] = u[1] − 3a12x.

From u_[12]=u_[21], v_[12]=v_[21], Φ_[12]=Φ_[21], we obtain

u[12] = u + 3[4(η1 + η2)hh(a12 + a22) − 6(a1a1x − a2a2x) − 2(a1a2x − a1xa2) + 4(uh + hxx) + 4(η1 − η2)     ]x,v[12] = v + [4(η1 + η2)hh(a12 + a22) − 6(a1a1x − a2a2x) − 2(a1a2x − a1xa2) + 4(uh + hxx) + 4(η1 − η2)   ]t,

where h=a₁−a₂. This is the nonlinear superposition formula for Ito’s equation (2). As stated in the introduction, a nonlinear superposition formula in bilinear form had been constructed by Hu and Li [8] and reads as Dxf0 ⋅ f12 = kDxf1 ⋅ f2,k is a constant. We observed that this formula does take a very simple form, but in general to apply it, we need to evaluate the integral, which may not be easy. Our formula is of differential-algebraic form and can be used to find solutions directly.

We conclude by illustrating the applications of our results. The 1-soliton solution can be easily found by taking the vacuum seeds u=v=0 and solving the corresponding linear problem

y1x = y2, y2x = y4, y3x = 0, y4x = k13y1 − y3,y1t = k13y1 − y3, y2t = k13y2, y3t = 0, y4t = k13y4,

which leads to

y1 = 1 + exp(k1x + k13t), y2 = k1exp(k1x + k13t), y3 = k13,y4 = k12exp(k1x + k13t).

Therefore, we have a = −2k1exp(k1x + k13t)1 + exp(k1x + k13t),

and u = 32k12sech212(k1x + k13t),   v = 12k14sech212(k1x + k13t),

which is nothing but the 1-soliton solution of Ito’s equation.

Now, we may easily construct a 2-soliton solution by choosing

a1 = −2k1exp(k1x + k13t)1 + exp(k1x + k13t),   a2 = 2k2exp(k2x + k23t)1 − exp(k2x + k23t),

and substituting them into the aforementioned nonlinear superposition formula, so the final solution reads as

u[12] = 6[ln(k13 − k23k13 + k23 − exp(k1x + k13t) − exp(k2x + k23t) + k1 − k2k1 + k2exp((k1 + k2)x + (k13 + k23)t)) ]xx,

v[12] = 2[ln(k13 − k23k13 + k23 − exp(k1x + k13t) − exp(k2x + k23t) + k1 − k2k1 + k2exp((k1 + k2)x + (k13 + k23)t))  ]xt.

To avoid the singularity, we impose the condition −|k₂|<|k₁|<|k₂|. The examples for u component are shown in Figure 1.

Figure 1:

u component of the 2-soliton solution with k₁=0.8, k₂=1.2.