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Integrable initial boundary value problems

The problem of searching boundary value problems for soliton equations consistent with the integrability property is discussed. A method of describing integrals of motion for the integrable initial boundary value problems for the Kadomtsev-Petviashvili equation is suggested via Green identity.


Introduction
Consider the IBV problem of the general form for the NLS equation q| t=0 = q 0 (x), q 0 (x)| x→+∞ → 0, which was studied by many authors. The usual scattering matrix s(λ, t) of the corresponding Dirac operator on the half-line x > 0 depends on t in a very implicit way. Namely, it satisfies the following matrix equation s t = 2iλ 2 [s, σ 3 ] + Z(q(0, t), q x (0, t), λ)s.
Equation contains unknowns s, q and q x . How to study such kind of equations? At the first glance it contains an extra unknown and it is underdetermined. But some implicit requirement should be valid: s(λ, t) preserves its analytical properties on the upper and lower half planes Imλ > 0 and Imλ < 0.
So really the equation is correctly defined.
Different approaches to study the equation (4) are discussed in the literature, for instance, recently were proposed the global relation method (Fokas) and the elimination by restriction method (Degasperis, Manakov, Santini). The following result allows to understand the essence of the problem (1), (2), (3).
Theorem, [1]. The entries α, β of the scattering matrix s(λ, t) satisfy the following system of equations Generally this system of equations with the variable coefficients is nonlinear the IBV problem is equivalent to the Cauchy problem for a pseudodifferential equation with two independent variables (generally nonintegrable). Integrability is lost when f (t) = 0. Only in the homogeneous case the DMS equation If f (t) is not identically zero then no hope to find exact solutions to the IBV problem. One of the ways here is to introduce a small parameter 0 < ǫ → 0, i.e. replace f (t) by ǫf (t) and to study the influence of the boundary by using the appropriately developed perturbation theory.
To extract integrable cases one can apply the integrability test to the system for the scattering matrix like (5)- (6). But we will testify the boundary condition using directly the Lax pair. Suppose the equation admits the Lax pair of the form Let a boundary condition of the form Example 1, see [2]. Consider the Korteweg-de Vries equation u t = u xxx − 6uu x . The coefficient matrices for the Lax pair are defined as Suppose the BC imposed at x = 0 is of the form To look for the discrete symmetry we must solve the equation The BC is of the form where a and b are arbitrary constants.
Example 2, [2]. The Harry Dym equation u t + u 3 u xxx = 0 admits two kinds of BC: where a, b are constants.
Example 3, [3]. The discrete Heisenberg model has the following Lax pair In this case the discrete involution and the cutting off condition are found from the equation The BC reads as where a, b, c are arbitrary constants and a 2 + b 2 = 0. 2 How to use the discrete symmetry in the ISM Let us discuss how to use the discrete symmetry when constructing solutions of the corresponding IBV problems. Take the IBV problem for the KdV equation with vanishing BC (see, [4]) In this case H is the unity matrix and h(λ) = λ −1+i √ 3 2 = λω. Actually, the discrete symmetry reflects only the fact that the evolution of the scattering matrix is invariant under the change λ → ωλ. This equation is really nonlinear but the discrete symmetry allows one to linearize it. Put z = λ 3 and define the matrices These two matrices satisfy the Riemann problem where p(z, t) = e −4izσ 3 t p(z, 0)e 4izσ 3 t . Now it is easy to see that the scattering matrix s(λ, t) is found from the linear equation (19). Using this fact one can prove Theorem. Let the initial value satisfy the conditions 1) u(x, 0) = u 0 (x) is smooth and vanishes; 2) the associated Sturm-Liouville operator has no discrete eigenvalues, 3) the scattering matrix is unbounded at λ = 0 then the problem (15), (16), (17) is uniquely solvable for all t > 0. The function u x (0, t) satisfies the following representation In this case two of three functions u(0, t), u x (0, t), u xx (0, t) are zero and the third one slowly decays. It is not ever in L 1 , only in L 2 .
Unfortunately, there is no regular soliton-like solutions of the KdV equation with the vanishing boundary conditions. If the parameters a and b are different from zero then regular exact soliton-like solutions (as well as finite-gap solutions) exist approaching C = a 2 − b/3 at x = ∞ and satisfying the BC at x = 0. They are described in [6]. In this case time evolution of the scattering matrix is reduced to a Riemann problem on a Riemann surface defined by the

Discrete symmetry and BC in multidimensional case
Consider the well known 2D-Toda chain with the following Lax pair Impose a cutting off constraint at n = 0 How to find all integrable cases only by using the equation (22)? To this end it is necessary to study the discrete symmetries of the equation (22), appearing under the BC. But now there is no λ and we are to find some generalization of the involution λ → h(λ). It is evident that the Toda chain is invariant under transform x ↔ t, so the following pair of equation gives also a Lax pair to the Toda chain.
Proposition, [5]. Suppose that there exists such an operator M = aD 2 x + bD x + c that for n = 0 for any solution ψ of the equation (25) the function φ = Mψ is a solution of (22). Then the BC (23) takes one of the forms below 1) e u(−1) = 0, The corresponding operator M is respectively of the form where a 0 , b 0 -arbitrary constant parameters, and u = u(0). Notice that the operator M 1 is a linear combination of the operators M 2 and M 3 .
All of the cutting off conditions above are known to be consistent with the integrability. Initiated by this example we formulate the discrete involution test for multidimensional equations. Two Lax pairs which are not connected by any conjugation transformation, should become conjugate after imposing the BC.
Apply now the test to look for boundary conditions to the KP equation admitting the Lax pair The equation (28) is invariant under the change y → −y, w → −w and by this reason the following system of equations is also a Lax pair for the KP Proposition, [5]. Suppose that there exists a differential operator M = aD 2 x + bD x + c such that for y = 0 for any solution ψ of the equation (30) the function defined as φ = Mψ is a solution to (32). Then one of the following equations holds The corresponding operator M is of the form 3) M = D 2 x + c.
If one replaces ψ ↔ φ one gets one more constraint