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Elliptic Solitons with Free Constants and their Isospectral Deformations

New exact solvable elliptic potentials with free constants for the spectral problems of the third order are found. A time dependence of such potentials gives their isospectral deformations and solutions of nonlinear integrable equations.

There are the following main problems in the theory of elliptic solitons: the search for the potentials u i (x), formula for the Ψ-function, corresponding algebraic curve and its realization in the form of covering of torus, isospectral deformations of u i with respect to integrable nonlinear equations.
Originally (1872), Hermite developed an approach to the integration of the Lamé potentials for (1) using an ansatz for the Ψ-function as a meromorphic two-periodic function of the second kind by Picard [3,26] (Hermite used Jacobi's H, Θ, Z-functions): where the poles Ω j of the potential u i (x) independent of λ. In such ansatz, zeroes of the Ψ-function α i depend on the spectral parameter λ and are determined from the several essentially transcendental equations. By this reason the ansatz (2) is not convenient and not effective. Later, Hermite and Halphen considered the following ansatz where the Φ-function (élément simple by Halphen) is defined by the expression Thus, instead of unknown parameters α i (λ), k(λ) in (2) we have only two ones k, α in the framework of ansatz (3). Indeed, after the substitution (3) into the spectral problem and expanding the principal part of poles at x = Ω j , we will have equations being polynomial in k and transcendental in α. In fact, α is a parameter of the covered torus Evidently, that the obtained equations are linear ones with respect to a nj and therefore elementary solvable. The classical potentials of Lamé, Halphen, Treibich-Verdier and others have a fixed location of poles. So, considering the potential for the spectral problem (1), we get the restriction: b = −a, ℘ ′′ (a) = 0. In this paper, we find new potentials containing free parameters and poles (see also the most recent information in [25]). Assuming their dependence on time t we will obtain the isospectral deformation of initial condition u(x) as a solution of the corresponding nonlinear integrable equation.

Algorithm for the solution of the spectral problems
The general algorithm for integrating the spectral problem (4) with elliptic coefficients u i of x and arbitrary of λ was proposed by the author in [27]. It has a pure algebraic characterization and is reduced to a system of polynomial equations solvable by the polynomial algebra methods (Gröbner bases, characteristic sets and others [16,17]). Indeed, the equations on poles of the expressionLΨ, where the Ψ-function is given by the formula (3), contain the following unknown quantities: Using the addition theorems for elliptic functions and two important equalities where we introduce the short notation ℘ α ≡ ℘(α) etc., the above mentioned equations on poles are "algebraized": they are transformed into the polynomial ones with respect to the variables k, [27] for the details). Thus, we have a polynomial ideal in a ring Q(λ, and the final solution of the spectral problem (4) is defined by the structure of this ideal [15,17,27].
All the associated curves are the trigonal ones. λ is a meromorphic function with only 3-rd order pole. The 2-pole elliptic potentials in (5) have the form Using the Θ-functional approach, one can show that the finite-gap potential for (5) is a second logarithmic derivative of Θ and thus, must not has residues. Therefore C = 0. The simplest nontrivial example is The corresponding solution is as follows [27]: The generalization of Halphen's equation was considered by Unterkofler [20] in the framework of the ansatz (2). The algebraic curves were written, but important formulas for the covering of torus R(℘(α), ℘ ′ (α), λ) = 0 (i.e. the transcendental equation for the determining of α by λ) have not been obtained. Let us consider the spectral problem The most general 1-pole elliptic potential for the equation (10) has the form: The case n = 0 in the ansatz (7) yields the solution: Under n = 1 in (7) we arrive at the nontrivial solution. Indeed, the first of three polynomials in the original base has the form: where without loss of generality we set b = 12 − a. In general case, an attempt to solve the corresponding system of algebraic equations failed. For example, in some particular cases, all the solutions lead to the dependence of parameters a, c, d on spectral one λ.
The next nontrivial example is a simplest 2-pole potential in (10).
This equation is integrable with arbitrary Ω and A (see [27,23] for the solution) and the restriction B 2 = 4 ℘ Ω − 4 ℘ A .

Isospectral deformations
If Ψ-function depends on a parameter t, then the arbitrariness of location of poles Ω i in potential means its isospectral deformation with Ω i (t). Let us consider examples.
The spectral problem (5) corresponds to the Sawada-Kotera equation [19] u A deformation of the potential (8) is stationary. Chazy [21, p. 380] found the more general stationary solution of the equation (19) u(x, t) = 6 ℘ 1 + 6 ℘ 2 , where we introduced the designations: Now we use a connection between the equation (19) and the Kaup-Kupershmidt, Fordy-Gibbons [28] and Tzitzeica equations respectively by means of the Miura maps After the substitution v = − ln ′ ψ, the first Miura's map in (23) is linearized and we arrive at a problem of integration of the Schrödinger equation without the spectral parameter: Using the solution (9) under λ = 0 (the passage to the limit α → ω i ) we will have After the substitution Ψ ′ = ψ, we get the result (under the coinciding invariants g 3 =g 3 ): The straightforward verification shows that it will be also valid under g 3 =g 3 . Thus we obtain the 5-parametric stationary solutions of the equations (20,21): The same technique leads to the solution of the Tzitzeica equation (22), but the time dynamics is not stationary: Let us consider the time deformation of 2-pole potential (18) u with respect to the Boussinesq equation Substituting the ansatz (24) in (25) and equating the poles to zero we will obtain the equations on the poles Ω 1,2 (t) -the Calogero-Moser system of two particles with a repulsion potential and immovable center of mass: Ruling out its trivial solution Ω 1 − Ω 2 = C, we get Ω 2 = C − Ω 1 and an equation on Ω 1 : The substitution X = ℘(2 Ω 1 − C) transforms the equation (27) into the algebraic form: (28) is the algebraic curve F (X,Ẋ) = 0 of genus 1. After the birational change X →X we obtain an equation onX in the canonical Weierstrass's form dX dt 2 = 4X 3 −g 2X −g 3 ,X(t) = ℘(t − t 0 ;g 2 ,g 3 ) ≡℘(t − t 0 ) with new invariantsg 2 = 2 8 g 2 ℘ 2 A + 768 g 3 ℘ A + 64 3 g 2 2 , g 3 = −2 12 g 3 ℘ 3 A − 2 11 3 g 2 2 ℘ 2 A − 2 10 g 2 g 3 ℘ A + 2 9 27 g 3 2 − 2 10 g 2 3 . Thus, we get the complete solution of the equations (26): t 0 , C are arbitrary constants and the elliptic integral ℘ −1 corresponds to the invariants g 2 , g 3 . See also [24] for the solution in the context of the solutions of the Kadomtsev-Petviashvili equation.