The Static Elliptic $N$-soliton Solutions of the KdV Equation

Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative B\"{a}cklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $\wp$-function type differential equation.

The name "soliton" has come through studies of the KdV equation. In nontrivial solutions of the KdV equation, there exists a solitary wave solution which can be regarded as an excitation of particle i.e. soliton. The KdV equation can also provide interacted configurations by its solutions. With proper time dependence, collision phenomena of solitons can be captured by such multi soliton solutions. A soliton solution can be visualized as a spatially localized object, and we in this paper refer this definition for soliton solutions. Having N localized excitation, we call this as an "N -soliton" solution.
Since the KdV equation is a nonlinear differential equation, it has been not obvious to find out N -soliton solutions due to lack of linear superposition. Nevertheless, it is now to be standard to construct N -soliton solutions from one soliton solutions by the Bäcklund transformation. In other wards, we could consider such a nontrivial nonlinear superposition in special cases.
In order to solve nonlinear differential equations, underlying symmetries which the systems possess may play a crucial role. In the AKNS formulation, the soliton equations such as the KdV, the mKdV, and the sine-Gordon equations are obtained as the integrability condition of real 2 × 2 matrix, which means the symmetry of the soliton systems lies on the Möbius (GL(2,R)) group symmetry.
In our previous paper [13], we have studied the algebraic construction of the N -soliton solutions. Using pieces of one-soliton solutions obtained by directly solving differential equations, we have algebraically constructed N -soliton solutions by using the commutative Bäcklund transformation for the KdV, the mKdV, and the sine-Gordon equations. In this algebraic construction, the commutative subgroup, i.e. commutative Bäcklund transformation of the Möbius group symmetry, has been essential. The N -soliton solutions which we had obtained were in the hyperbolic type (the exponential type). The addition formula of the hyperbolic function such as tanh(x + ξ) gives which is the global Möbius transformation with α = 1, β = tanh ξ , γ = tanh ξ, and δ = 1. The algebraic N -soliton construction in the previous paper [13] is the result from the local commutative Möbius transformation. This could be a realization of nontrivial superposition. So far we know only one-soliton solution of the elliptic type. Considering the Ising model, we observe that the SU(2) group symmetry and the elliptic function appear and they are mutually connected [14,15]. As the structures of the SU(2) and GL(2,R) is similar, we suppose it may be possible to access to elliptic N -soliton solutions through the commutative Bäcklund transformations.
The paper is organized as follows: In section 2, we briefly review the previous studies and make some preparations. Then explicit constructions of the static elliptic N -soliton solutions are presented in section 3. We devote the final section to the summary and the discussions. In order to find the one-soliton solution, we assume a linear dependence for x and t as ax + bt + δ =: X with constant parameters a, b, and δ. Setting u(x, t) = 2U (X) with the variable X, the KdV equation (2.1) becomes , we arrive at Now let us remind ourselves the Weierstrass's ℘-function which satisfies 2) with U (X) = a 2 ℘(X). Thus, in the original form, we have the elliptic onesoliton solution We discuss the time-dependent N -soliton solution in the summary and discussions, so that we first construct the static N -soliton solutions. Thus, we concentrate on the static case hereafter. The static elliptic one-soliton solution now has the form from Eq.(2.5), Before closing this subsection, it should be mentioned that the KdV equation can be rewritten as the ℘-function type differential equation. Integrating the static version of the KdV equation (2.1) twice, we directly obtain with integration constants C and D. Sending the constants to C = −2g 2 and D = −4g 3 , and redefining the function as u(x) = 2h(x), it is easy to see that Eq.(2.7) turns to be the same form as Eq.

Hyperbolic one-soliton solution by the Bäcklund transformation
Let us now introduce the Bäcklund transformation which can generate N -soliton solutions. Using the variable z x (x) = u(x), the Bäcklund transformation of the KdV equation [5] is given by with new arbitrary parameter λ. For the given soliton solution z(x), Eq.(2.15) provides a condition that the new soliton solution z ′ (x) must satisfy. It should be noted that this Bäcklund transformation is the only commutative one, as far as we know.
In our previous paper [13], we have constructed N -soliton solutions of the mKdV equation by using the KdV-type Bäcklund transformation [5] instead of the mKdV-type Bäcklund transformation [6] by making the connection between the mKdV equation and the KdV equation through the Miura transformation. The reason why we can construct N -soliton solutions by the KdV-type Bäcklund transformation is that it is the only commutative one. We had emphasized in our previous paper [13] that commutative Bäcklund transformations play an important role to construct N -soliton solutions algebraically.
Let us make use of the Bäcklund transformation to obtain soliton solution. As the trivial solution, we have z(x) = 0. In this case, the Bäcklund transformation Eq.(2.15) tells us that another soliton solution z ′ (x) satisfies the following "differential equation", One can solve the differential equation and get the hyperbolic type solution, with an arbitrary parameter δ. Thus, if we put z(x) = 0, we cannot obtain the elliptic Nsoliton solution via Bäcklund transformation. In the next section, we will show that both z(x) and z ′ (x) can be non-zero in the Bäcklund transformation Eq.(2.15). We can take elliptic type functions in such a way as both solutions are consistent with the KdV-type Bäcklund transformation Eq.(2.15). This fact is the key point for our construction of the elliptic Nsoliton solutions.

The Static Elliptic N -soliton Solutions
We work with the Bäcklund transformation of the KdV equation given by Eq.(2.15). We prepare two elliptic one-soliton solutions which have the forms of Eq.(2.6), where we have introduced z x (x) and z ′ x (x) for the sake of using Bäcklund transformation. Using the relation between the ℘-and ζ-functions, we have We now look at the relation, and adjust the parameters in (3.4) and (3.5) so as to get consistency between Eqs.(3.6) and (3.7). We first take a 1 = a 2 and put η 1 = 0 without loss of generality by the constant shift of x. Thus, choosing the parameters as we can accommodate Eq.(3.6) to the following form which suits the relation Eq.(3.7). As the result, we can obtain the pair of elliptic one-soliton solutions z(x) and z ′ (x) in the Bäcklund transformation Eq.(2.15), which are consistently coexist, in the form z(x) = −2ζ(x) and z ′ (x) = −2 ζ(x + δ) − ζ(δ) . By changing the parameter δ, we obtain infinitely many one-soliton solutions: In the next section, using these one-soliton solutions z 0 (x) and z i (x), we can algebraically construct N -soliton solutions by the commutative Bäcklund transformation. In terms of z x (x) = u(x) = 2h(x), we fix our "KdV equation" to be solved as Eq.(2.7) with C = −2g 2 and D = −4g 3 , i.e.
which can be related with the ℘-function type differential equation (2.8).
We sketch the graphs of z 0 (x) and z 12 (x) in Figure 1 and Figure 2, respectively. We can observe that the pole at x = 0 in z 0 (x) disappears in z 12 (x), which can be seen by expanding Eq.(3.13) around x = 0. We can also see that z 12 (x) becomes narrower than z 0 (x) in width.
Because of the commutativity of the Bäcklund transformation, the expression in Eq.(3.17) becomes in the cyclic symmetric form for z 1 (x), z 2 (x) and z 3 (x), which confirms that 3!independent construction of z 123 (x) gives the same result as above. Then we call this solution as the static elliptic 3-soliton solution.
We can recursively show the commutativity of the Bäcklund transformation by identifying in the proof of 2 + 1-soliton solution.
We sketch the graph of z 123 (x) in Figure 3. We can see three localized clusters in this solution.
The static elliptic KdV 5-soliton solution for Eq.(3.11) is given by The expression of z 12345 (x) with z 0 (x), z 1 (x), z 2 (x), z 3 (x), z 4 (x), and z 5 (x) is given in the form We have numerically confirmed that both the static elliptic (4+1)-soliton and the 5-soliton solutions really satisfy the static KdV equation (3.11).
In the same manner, we could recursively construct (1+(even number))-soliton solutions and (odd number)-soliton solutions. In the (odd number)-soliton solutions, z 0 cancels out and does not appear in the final soliton solutions. General structures of static elliptic solutions could be discussed elsewhere.

Summary and Discussions
Regarding soliton solutions for the elliptic type, only the one-soliton solution has been available so far. We have obtained the KdV static elliptic N -soliton solutions by using the commutative Bäcklund transformations. We understand that the key point of the algebraic construction of the KdV static elliptic N -soliton solution is the existence of the Möbius (GL(2,R)) group symmetry and the one-soliton solutions of the algebraic functions such as the trigonometric, the hyperbolic or the elliptic types for the KdV equation. The local algebraic addition formula of the algebraic functions, which comes from the commutative Bäcklund transformation, seems to be essential.
For the time-dependent solution, we can construct a certain time-dependent solution by the static solution, which can be constructed in our paper, by just the following replacement. We denote the static solution u (static) (x), which can be written in the form Then we replace x → x + bt in this static solution and we have u (static) (x + bt) = F f 1 (x + bt + δ 1 ), f 2 (x + bt + δ 2 ), · · · .
This time-dependent solution u(x, t) is the special generalization of the time-dependent elliptic solution Eq.(2.5).
Our N -solitons and well-known one-soliton as elliptic type solutions of the KdV equation are both singular. Originally the KdV equation is derived as the wave equation of the shallow water by taking the special limit. Then the KdV equation is an idealistic equation, so that our singular solutions will correspond to the much milder solitary waves in the real shallow water. However, what we prefer here is to emphasize a deep relationship between mathematics and underlying physics. If we consider the ℘-function type differential equation as the static KdV equation, we have infinitely many elliptic soliton solutions for the ℘-function type differential equation. In other wards, we find a family of the ℘-function via the physical integrable KdV system. This might be quite interesting not only for physics but also for mathematics.