# Constraint and Nonlinearization of Supersymmetric Equations with Some Special Forms of Lax Pairs

Authors: Hongmin Li
Journal: Advances in Mathematical Physics
 (2020)

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## 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.


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## Body

## 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 Section2, 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]
(1)Φt=Φ3+3ΦDΦ1+3DΦΦ1is integrable in this sense, where Φi=∂i/∂xiΦ,Φ=Φx,t,θ is an anticommuting superfunction of the space parameter x and one anticommuting Grassmannian variable θ, and the symbol D is to denote the superderivative D=∂θ+θ∂x satisfying D2=∂x.Expanding the superfunctionΦ=θux,t+ξx,t with a bosonic field ux,t and a fermionic field ξx,t, the equation (1) may be rewritten as the component form:
(2)ut=u3+6uu1−3ξξ2,(3)ξt=ξ3+3uξ1.Let us consider the stationary flowfn=0, where fn=LnΦ1 with the recursion operator given by [25, 26]
(4)L=∂x2+2ΦD+2DΦ+ΦxD−1+DΦx∂x−1+DΦD−1Φ∂x−1+2Φx∂x−1Φ∂x−1.Then, all the stationary equationsfi=0,i=0,1,2,⋯,n can be written as the component form. Then, one may have two series of ODEs for the functions of u,ξ and their derivatives with order no more than 2n. Moreover, since the variables ui and ξi with 0≤i≤2n are mutually independent, so the above two systems are first-order ODEs for 4n+2 variables u,u1,u2,⋯,u2n andξ,ξ1,ξ2,⋯,ξ2n, respectively. Moreover, from Equations (2) and (3), we can get the time-evolution for u,u1,u2,⋯,u2n and ξ,ξ1,ξ2,⋯,ξ2n as
(5)ddtui=didxi6uu1+u3−3ξξ2,i=0,1,2,⋯,n,ddtξi=didxiξ3+3uξ1,i=0,1,2,⋯,n,which are closed for the potentials u,u1,u2,⋯,u2n and ξ,ξ1,ξ2,⋯,ξ2n. 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 x,t, the potentials u,u,u1,u2,⋯,u2n; ξ,ξ1,ξ2,⋯,ξ2n may be gotten from their initial values at x0,t0 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 PDEsvt=Av, let us separate v as
(6)v=∑i=0∞aipixqit,where the constraints for pix and qit are given by
(7)Apix=λipix,qi,tt=λiqit,under some boundary conditions. Since A and ∂t are both linear differential operators, we have
(8)Apixqit=λipixqit,pixqitt=λipixqit.Following [1], the constraint and nonlinearization of the Lax pair for the KdV equation ut=u3+6uu1 may be constructed as
(9)u=c0+∑i=1naiψi2,(10)∂x2+uψi=λiψi,(11)ψi,t=−∂x3+3u∂x+3∂xuψi.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 conditionu0,t=0 with the Lax pairs
(12)L1=∂x+u∂x−1,(13)P1=∂x3+3∂xu,and
(14)L2=∂x2+4u+2u1∂x−1,(15)P2=∂x∂x2+6u.Example 1.
The null boundary problem of the KdV equation admitting Lax pair (12) and (13) is the constrained system:
(16)∂2+uϕi=λiϕi,ϕi,t=∂2+3uϕi1,(17)u=∑i−1naiϕii.
Sinceu satisfies the null boundary conditions, at the first thought, ϕi1 should also have 0-boundary conditions. But this will lead to ux,t=0. From (16), we know that ϕi2x=0=0; then, it follows ϕinx=0=0,n≥2 so ϕi1x,t=0. 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
(18)ϕt=−λu+∑i=1naiλiϕi1−uϕiϕ+2u+λ2ϕ1(19)ϕt1=−λu−∑i=1naiλiϕi1−uϕiϕ1−λ∑i=1naiλiϕi1−uϕi+∑i=1naiλiλiϕi1−uϕi−aiuϕiϕ+ϕ∑i=1naiϕi∑j=1najλjϕj1−uϕj+2u+λ2λϕ1−uϕ.
The above equalities are correct for anyϕi. Without loss of generality, sometimes we omit the subscript. Hence, at the boundary (u =0), we have
(20)ϕt=λ2ϕ1−ϕ∑i=1naiλiϕi1,(21)ϕt1=λ3ϕ1+ϕ1−λϕ∑i=1naiλiϕi1−ϕ∑i=1naiλi2ϕi1+ϕ∑i=1naiϕi∑j=1najλjϕj1.
The boundary is invariant under the evolution, which means(22)dmdtmu=∑i=1naidmdtmϕi1=0,m=0,1,2⋯,at the boundary.Now let us turn to the following cases:(i)
Casen=1: we have u=a1ϕ11 and, a1≠0 thus ϕ11=0 will lead to u=0(ii)
Casen=2: let u=a1ϕ11+a2ϕ21 and a1≠0, a2≠0. Solving ϕ11 by u=0 and substituting it into d/dtu, we get(23)ut=a1λ1−λ2ϕ11λ12+λ1λ2−2a1λ1ϕ1−a2λ1ϕ2+λ22+a12ϕ12+a22ϕ22−a1λ2ϕ1−2a2λ2ϕ2+2a1a2ϕ1ϕ2Here,λ1=λ2 or ϕ11=0 is equivalent to u=0. The last term being 0 is the only nontrivial case. Differentiating it once and substituting the known equations, we get
(24)Z3=λ1λ2−a1λ2ϕ1−a2λ1ϕ2=0.Solving all the known equations, we finally get(25)ϕ1=λ12a1λ1−λ2,ϕ2=λ22/a2λ2−λ1,a1ϕ11+a2ϕ21=0.Notingd/dtZ3=0 under the above solution, thus, d/dtZ3=0 will not provide further constraint.
(iii)
Casen=3: by solving u=0,d/dtu=0,d2/dt2u=0 and d3/dt3u=0, we get the following two sets of solutions:
(i)
Solution 1:(26)a1ϕ11+a2ϕ21+a3ϕ31=0,(27)a1ϕ1+a2ϕ2+a3ϕ3=λ1+λ2+λ3,(28)λ1a1ϕ1+λ2a2ϕ2+λ3a3ϕ3=λ1λ2+λ1λ3+λ2λ3,(29)a1ϕ11λ1+a2ϕ21λ2+a3ϕ31λ3=0.(ii)
Solution 2:(30)a1ϕ11+a2ϕ21+a3ϕ31=0,(31)a1ϕ1λ1+a2ϕ2λ2+a3ϕ3λ3=1,(32)λ12ϕ11ϕ1=λ22ϕ21ϕ2=λ32ϕ31ϕ3.Then, the above results may be summarized as the following theorem:Theorem 1.
∑i=1maiϕi1=0,∑i=1maiϕi/λi=1,λi2ϕi1/ϕi=λj2ϕj1/ϕj are invariant under (20) and (21).Proof.
The proof is straightforward. For example, we can calculate, under (20) and (21), d/dtλi2ϕi1/ϕi=λi2ϕi1/ϕi2. Here, the boundary conditions for ϕ and ϕ1 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 ux=0=0 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:
(33)∂x2+4u+2u1∂x−1ψ=λψ,(34)ψt=∂x2+6uψ1,(35)u1=∑i=1naiψi.Settingψ=ϕ1, we arrive at
(36)∂x3+4u∂x+2u1ϕ=λϕ1,(37)ϕt=∂x2+6uϕ1,(38)u=c0+∑i=1naiϕi,and hence, one may have
(39)ϕt=λϕ1+2uϕ1−2u1ϕ,ϕt1=λϕ2+2uϕ2−2u2ϕ,ϕt2=λλϕ1−4uϕ1−2u1ϕ+2u1ϕ2+2uϕ3−2u2ϕ1−2∑i=1nαiλiϕi1−4uϕi1−2u1ϕiϕFrom the boundary conditionux=0=0, the following equalities can be immediately got
(40)ϕt=λϕ1−2ϕ∑i=1naiϕi1,ϕt1=λϕ2−2ϕ∑i=1naiϕi2,ϕt2=λ2ϕ1+2ϕ2−2λϕ−4c0ϕ∑i=1naiϕi1−2ϕ∑i=1naiλiϕi1−2ϕ1∑i=1naiϕi2=λ2ϕ1+2ϕ2−2λϕ∑i=1naiϕi1−2ϕ1∑i=1naiϕi2.Then, it is easy to verify the following first integrals(41)12λiϕi2+12ϕi12−ϕiϕi2=ki.Asn=1 and n=2, the solutions are obtained:
(42)n=1:c0=−λ12,ϕ1=λ12a1n=2:c0=−12λ1+λ2,a1ϕ1=λ122λ1−λ2,a2ϕ2=λ222λ2−λ1,a1ϕ11λ1+a2ϕ21λ2=0.For the casen=3, 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 (FormI), the constraint is at hand.The FormI:
(i)
L=L++LFD−1, where the subscript+ means the projection to the differential part of the super-pseudodifferential operator and LF is a function(ii)
P is a differential operatorTheorem 2.
For a Lax pair in the FormI, a possible constraint is LF=∑i=1naiψi, where Lψi=λiψi,i.e.ψis are eigenfunctions.The proof is similar to that of the Theorem 1, and here,ψ=Dϕ. In the following, we will solve the MRSKdV equation with null boundary condition u0,t=0,ξ0,t=0 through the above theory.Example 3.
The MRSKdV equation with null boundary and Lax pair(43)L=∂x+ΦD−1,P=L3+=∂x3+3ΦD∂x+3Φ1D,and therefore, we have the constrained system
(44)∂x+ΦD−1Dψi=λiDψi,Dψi,t=∂x3+3ΦD∂+3Φ1DDψi,(45)Φ=∑i=1naiDψi.To calculate easily, settingΦ=θux,t+ξx,t and Ψ=θηx,t+gx,t, then, the equations (44) can be written as follows:
(46)ηi1+ξgi=λiηi,gi2+ugi−ξηi=λigi1,ηi,t=ηi3+3ξgi2+3ξ1gi1,gi,t1=gi4+3ugi2−3ξηi2+3u1gi1−3ξ1ni1,u=∑i=1naigi1,ξ=∑i=1naiηi.Similar to the KdV equation, the right eigenvalue problem for (44) is that the boundary condition must be invariant under the evolution:
(47)dmdtmΦ=∑i=1naidmdtmDψi,m=0,1,2,⋯,at the boundary. From the equations (46), we obtain
(48)ηt=λ3η+g1−λg∑i=1naiλiηi+g∑i=1naigi∑i=1naiλiηi−g∑i=1naiλi2ηi+−λ2g+λg∑i=1naigi+λg1−ug+g∑i=1naiλigi−g∑i=1naigi2−g1∑i=1naigiξ,gt=λ2g1−η∑i=1naiλiηi−g∑i=1naiλigi1+λn+n∑i=1naigiξ+g∑i=1naigi+2g1−λgu,gt1=λ3g1−λg∑i=1naiλigi1−η∑i=1naiλi2ni−uηξ+n∑i=1naigi∑i=1naiλiηi−g∑i=1naiλi2gi1+g∑i=1naigi∑i=1naiλigi1+g1∑i=1naiλigi1−u2g+λg1−λ2g+λg∑i=1naigi+g∑i=1naiλigi−g∑i=1naigi2−g1∑i=1naigiu+n∑i=1naiλigi−n∑i=1naigi2ξ,whereη,g denote arbitrary ηi,gi, respectively. Then, at the boundary u=0,ξ=0, we have
(49)ηt=λ3η+g1−λg∑i=1naiλiηi+g∑i=1naigi∑i=1naiλiηi−g∑i=1naiλi2ηi,gt=λ2g1−η∑i=1naiλiηi−g∑i=1naiλigi1,gt1=λ3g1+g1−λg∑i=1naiλigi1+η∑i=1naigi∑i=1naiλiηi−η∑i=1naiλi2ηi−g∑i=1naiλi2gi1+g∑i=1naigi∑i=1naiλigi1.To get the right eigenvalue problem, let us consider the problem for the casesn=1,n=2:
(i)
Casen=1: the equalities u=a1g11,ξ=a1η1 and α1≠0 indicate that g11=0,η1=0 will lead to u=0,ξ=0(ii)
Casen=2: we have u=a1g11+a2g21,ξ=a1η1+a2η2 and a1≠0, a2≠0Solvingg11,η1 by u=0,ξ=0 and substituting them to d/dtu, we get
(50)ut=a2λ2−λ1g21λ12+λ1λ2−2a1λ1g1−a2λ1g2+λ12+a12g12+a22g22−a1λ2g1−2a2λ2g2+2a1a2g1g2.Similar to the case of KdV equation, one may obtain(51)g1=λ12a1λ1−λ2,g2=λ22a2λ2−λ1,a1g11+a2g21=0,a1η1+a2η2=0.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 η.


## 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.



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