Abstract
We study the null boundary problems of some classical evolution equations constrained by some special forms of Lax pairs. Furthermore, we present the constraint and nonlinearization of some supersymmetric (SUSY) equations with a special form of Lax pairs and solve the null boundary problems of these SUSY equations under the corresponding constraints.
1. Introduction
The method of nonlinearization of the Lax pair of the classical integrable equations has attracted many attentions in the soliton theory, including monononlinearization [1–3] and binary nonlinearization [4, 5]. The nonlinearization technique may be used to construct ordinary differential equations (ODEs) directly from soliton equations [6, 7] and get exact solutions of the integrable systems [8]. In particular, Wu et al. split the 1+1 dimensional soliton equation into two compatible Hamiltonian systems of ODEs in the finite-dimensional invariant set of the flow and illustrated that the Hamiltonian systems are well approximated by the so-called symplectic numerical methods [9]. Li et al. gave the constraint and nonlinear technique of the classical partial differential equations (PDEs) and solved their soliton-solution through the technique [10].
Fermionic extensions of integrable systems are interesting from both physical and mathematical point of views. In a series of papers, some important integrable models such as the super Ablowitz-Kaup-Newell-Segur (AKNS) system [11–13], the SUSY AKNS system [14–16], and the SUSY Kadomstev-Petviashvili (KP) hierarchy [17, 18] are studied. For the supersymmetric system, the nonlinearization technique may be more complicated. In Refs. [19, 20], Aratyn et al. gave the symmetry constraint of SUSY KP hierarchy, i.e., the constrained SUSY KP hierarchy. In Refs. [13, 21], He et al. established the binary nonlinearization approach of the spectral problem of the super AKNS system under an explicit constraint and proposed an implicit symmetry constraint between the potentials and the eigenfunctions of super AKNS system. Zhou and Kui [22] derived a fermionic extension of the Garnier system through the binary nonlinearization of the spectral problem to the super KdV equation of Kupershmidt [11]. To our best knowledge, there is no work on the constraint and nonlinear method of SUSY AKNS system, which will be considered in this paper. Furthermore, it is well-known that the boundary condition has many important applications [23, 24], so the null boundary problems of some classical evolution equations and SUSY equations constrained, respectively, by some special forms of Lax pairs will also be solved.
For convenience, the paper is organized as follows. In Section 2, we give an example to explain the nonlinear technique. In Section 3, we study the null boundary problems of some evolution equations constrained by the special forms of Lax pairs and obtain some soliton-solutions of the KdV equation. In Section 4, we present the constraint and nonlinear method of some SUSY equations constrained by a special form of Lax pairs. To illustrate the method, we solve the null boundary problem of Manin-Radul supersymmetric Korteweg-de Vries (MRSKdV) equation. In Section 5, we close with some conclusions and discussions.
2. Preliminaries
Numerical experiments show that the numerical integrations of most initial-boundary value problems (IBVPs) for PDEs are much more difficult than IBVPs for ODEs. But for some PDEs, the integrations may be done by only solving a series of ODEs. This celebrated property should be considered a kind of integration for the PDEs [10]. To explain it, let us study how the MRSKdV [18] is integrable in this sense, where is an anticommuting superfunction of the space parameter and one anticommuting Grassmannian variable , and the symbol is to denote the superderivative satisfying .
Expanding the superfunction with a bosonic field and a fermionic field , the equation (1) may be rewritten as the component form:
Let us consider the stationary flow , where with the recursion operator given by [25, 26]
Then, all the stationary equations can be written as the component form. Then, one may have two series of ODEs for the functions of and their derivatives with order no more than . Moreover, since the variables and with are mutually independent, so the above two systems are first-order ODEs for variables and, respectively. Moreover, from Equations (2) and (3), we can get the time-evolution for and as which are closed for the potentials and . Thus, we have four series of ODEs for the MRSKdV; two of them govern the space evolution for u, ξ, and the other govern the time-evolution of u, ξ. So, for any , the potentials u,; may be gotten from their initial values at by a space evolution and a time-evolution. In other words, for a special kind of initial value problems, the periodic MRSKdV is solvable by only integrating some ODEs.
3. Null Boundary Problems for the PDEs
Now, we try to solve soliton equations with the null boundary condition via the constraint and nonlinear techniques. For linear PDEs , let us separate as where the constraints for and are given by under some boundary conditions. Since and are both linear differential operators, we have
Following [1], the constraint and nonlinearization of the Lax pair for the KdV equation may be constructed as
Then, the nonlinearization of Lax pairs for the KdV equation is very similar to the classical separation of variables. Therefore, the nonlinearization of the Lax pair is also called separation of variables.
Next, we will explain how to solve the KdV equation with null boundary condition with the Lax pairs and
Example 1. The null boundary problem of the KdV equation admitting Lax pair (12) and (13) is the constrained system:
Since satisfies the null boundary conditions, at the first thought, should also have 0-boundary conditions. But this will lead to . From (16), we know that ; then, it follows so . Then, what is the right eigenvalue problem for (16)?
The right eigenvalue problem for (16) is that the boundary condition must be invariant under the evolution. Let us explain it in detail, from (16) and (17), one may have
The above equalities are correct for any ϕi. Without loss of generality, sometimes we omit the subscript. Hence, at the boundary (u =0), we have
The boundary is invariant under the evolution, which means
at the boundary.
Now let us turn to the following cases: (i)Case : we have and, thus will lead to (ii)Case : let and , . Solving by and substituting it into , we get
Here, or is equivalent to . The last term being 0 is the only nontrivial case. Differentiating it once and substituting the known equations, we get
Solving all the known equations, we finally get
Noting under the above solution, thus, will not provide further constraint. (iii)Case : by solving and , we get the following two sets of solutions: (i)Solution 1:(ii)Solution 2:
Then, the above results may be summarized as the following theorem:
Theorem 1. are invariant under (20) and (21).
Proof. The proof is straightforward. For example, we can calculate, under (20) and (21), Here, the boundary conditions for and seems solvable completely. But the Lax pair (12) and (13) seems not be able to solve all initial value problems for us. We even think that most of the static solutions of the KdV with could not be solved by separation of variables with Lax pair (12) and (13).
Example 2. The KdV equation with null boundary condition and the Lax pair (14) and (15), namely, the constrained system:
Setting , we arrive at and hence, one may have
From the boundary condition , the following equalities can be immediately got
Then, it is easy to verify the following first integrals
As and , the solutions are obtained:
For the case , there is somewhat difficulty to solve the algebraic system.
4. Constraint and Nonlinearization of the SUSY Equations
From the second part of this paper, we know that an effective method to nonlinearize the Lax pair is crucial for IBVP. However, nontrivial constraint for a Lax pair is not easy to obtain, but for Lax pairs in the following form (Form ), the constraint is at hand.
The Form : (i), where the subscript+ means the projection to the differential part of the super-pseudodifferential operator and is a function(ii) is a differential operator
Theorem 2. For a Lax pair in the Form , a possible constraint is , where s are eigenfunctions.
The proof is similar to that of the Theorem 1, and here, . In the following, we will solve the MRSKdV equation with null boundary condition through the above theory.
Example 3. The MRSKdV equation with null boundary and Lax pair and therefore, we have the constrained system
To calculate easily, setting and , then, the equations (44) can be written as follows:
Similar to the KdV equation, the right eigenvalue problem for (44) is that the boundary condition must be invariant under the evolution: at the boundary. From the equations (46), we obtain
where denote arbitrary , respectively. Then, at the boundary , we have
To get the right eigenvalue problem, let us consider the problem for the cases : (i)Case : the equalities and indicate that will lead to (ii)Case : we have and ,
Solving by and substituting them to , we get
Similar to the case of KdV equation, one may obtain
From the above equations, we know the null boundary value problem for the MRSKdV’s bosonic part is the same as that of the KdV equation and at the boundary, the Equation (49) is linear for .
5. Conclusions and Discussions
In this work, we investigate the null boundary problem of some evolution equations constrained by special forms of Lax pairs. Using the null boundary problems of the KdV equation with Lax pairs (12), (13), (14) and (15) and the corresponding constraints (17) and (35), we get some exact solutions of them. This nonlinearization theory is rare; maybe it is the special condition of the reality. Li et al. also imposed the constraints for general Lax pairs including nonlinear ones, so how to solve the null boundary problems of the evolution equations with general Lax pairs under the corresponding constraints is an interesting problem.
For some SUSY equations with special forms of Lax pairs, we also give the constraint and nonlinearization of them. By solving the null boundary problem of MRSKdV with a Lax pair (43) and the corresponding constraint (45), we obtain the corresponding solutions and find the relation between SUSY equations and corresponding classical integrable partial differential equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We also would like to thank Professor Y Q Li for useful discussions. This work is partially supported by the National Natural Science Foundation of China (11805071, 11747010, and 11871232), the Fujian Province Science Foundation for Youths (2019J05092), and the Fundamental Research Funds for the Central Universities (ZQN-803).