Grassmannian flows and applications to non-commutative non-local and local integrable systems

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Highlights

  • Linearisation of nonlocal and local integrable systems via Hankel operators.

  • Optimal approach with generalisation to the non-commutative case.

  • The nonlocal nonlinear integrable systems generated constitute Grassmannian flows.

  • Unified programme incorporating all matrix nonlocal nonlinear systems we consider.

  • Solution of initial value problem without numerically evolving the solution in time.

Abstract

We present a method for linearising classes of matrix-valued nonlinear partial differential equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation procedure originally developed by Pöppe based on solving the corresponding underlying linear partial differential equation to generate an evolutionary Hankel operator for the ‘scattering data’, and then solving a linear Fredholm equation akin to the Marchenko equation to generate the evolutionary solution to the nonlinear partial differential system. Our generalisation involves inflating the underlying linear partial differential system for the scattering data to incorporate corresponding adjoint, reverse time or reverse space–time data, and it also allows for Hankel operators with matrix-valued kernels. With this approach we show how to linearise the matrix nonlinear Schrödinger and modified Korteweg de Vries equations as well as nonlocal reverse time and/or reverse space–time versions of these systems. Further, we formulate a unified linearisation procedure that incorporates all these systems as special cases. Further still, we demonstrate all such systems are example Fredholm Grassmannian flows.

Introduction

The aim of this paper is to formulate a unified programme for the linearisation of certain types of nonlinear systems. We use the Grassmann–Pöppe method presented in Beck et al. [1], [2] and Doikou et al. [3]. This method combines the idea that linear flows on Fredholm Stiefel manifolds project onto nonlinear Riccati flows in a given coordinate patch of corresponding Fredholm Grassmann manifolds, together with the operator approach developed by Pöppe [4], [5] for nonlinear integrable systems. Indeed we streamline Pöppe’s approach, only requiring Pöppe’s kernel product rule, and also not requiring commutativity of the integral kernels involved. Further we generalise Pöppe’s approach by generalising the kernel product rule central to Pöppe’s method by not insisting on evaluating the trace in the product rule. The consequences of implementing these generalisations are that we provide a unified approach to the linearisation of the matrix nonlinear Schrödinger and modified Korteweg de Vries equations, as well as the matrix Korteweg de Vries equation itself and matrix nonlocal versions of these equations as presented in Ablowitz and Musslimani [6]. By nonlocal we mean the matrix nonlinearities involve factors with time or space reversal or both. Further, generalising the kernel product rule means we generate quite general matrix nonlinear partial differential equations for the underlying integral kernels in the method. These represent a wide class of integrable systems in themselves.

Let us now try to disentangle these statements as succinctly as possible. First let us address the Grassmannian and Riccati flow aspect mentioned. Central to the method presented in Beck et al. [1], [2] are a pair of time-dependent Hilbert–Schmidt operators Q=Q(t) and P=P(t). Suppose this pair of operators satisfy the linear evolutionary system, tQ=A(id+Q)+BPtP=C(id+Q)+DP, where A and C are known bounded operators, while B and D are known, possibly unbounded, operators. Assume a solution Q=Q(t) and P=P(t) exists to this system, at least up to some time T>0. We now introduce a third Hilbert–Schmidt operator G=G(t) via the relation, P=G(id+Q).We call this the Riccati relation. Formally by direct straightforward computation, G evolves according to the Riccati flow tG=C+DGG(A+BG).In Beck et al. [1], [2], we show there exists a unique, well-behaved solution G to the Riccati relation at least for some time T>0, which indeed evolves according to such a flow. As we explain in detail in Section 2.3, the Riccati flow for G represents a projected flow via the Riccati relation on the canonical coordinate patch of a Fredholm Grassmann manifold. Further, since the operators concerned are Hilbert–Schmidt integral operators, all of the above can be translated to corresponding equations for the kernels of P,Q and G. At the kernel level the Riccati relation has the form, p(x,y)=g(x,y)+g(x,z)q(z,y)dz.The interval of integration depends on the application. This is the Marchenko equation which plays a central role in the classical theory of integrable systems, for example in the Zakharov–Shabat scheme [7], [8], as well as the work of Ablowitz et al. [9]. Naturally we can interpret the Riccati equation above in terms of its integral kernel g=g(x,y;t). In Beck et al. [1], [2] and Doikou et al. [3] we show many classes of partial differential equations with nonlocal nonlinearities can have the representation given by the Riccati equation. Hence the solutions g=g(x,y;t) to those equations can be represented by the three linear equations for Q, P and G; the linear equation for G is the Riccati relation. In other words such equations can be linearised. We meet three notions of nonlocal nonlinearities herein. Example classes of nonlinear equations we can solve using the approach just mentioned are as follows. They demonstrate the first notion of nonlocal nonlinearities we refer to. One class of equations for g=g(x,y;t) we solve in Beck et al. [1], [2] has the nonlocal Korteweg de Vries form, tg(x,y;t)+x3g(x,y;t)+Rg(x,z;t)zg(z,y;t)dz=0.On the other hand in Doikou et al. [3] we use the method above to solve Smoluchowski-type coagulation equations for g=g(x;t) of the form, tg(x;t)=d(x)g(x;t)+0xg(xy;t)g(y;t)dyg(x;t)0g(y;t)dy. Here d=d(x) is a constant coefficient polynomial of =x. We associate this class of coagulation nonlocal nonlinear terms with the notion of nonlocal nonlinearity present in the nonlocal KdV equation just above.

Also in Doikou et al. [3] we generalised the approach above to consider linear equations for the operators P and Q, and a Riccati relation for G, for example as follows: itP=x2P,Q=PP,P=G(id+Q), where P is the adjoint of P. In addition we assume P is a Hankel operator with an integral kernel of the form p=p(y+z+x;t). We say this system is linear. This is because, to solve for G, first, we need to solve the linear partial differential equation for P. Then Q is explicitly given as a quadratic term in P so we do not need to solve another equation for Q. Then second, we need to solve the linear Riccati relation for G. As shown in Doikou et al. [3], and we also see in Section 3, in practice as a result of this procedure we can linearise partial differential equations for g=g(y,z;x,t) of the form, itg(y,z;x,t)=x2g(y,z;x,t)+2g(y,0;x,t)g(0,0;x,t)g(0,z;x,t). We call this type of equation a kernel equation, though note the special form of nonlocal nonlinearity present. This is the second notion of nonlocal nonlinearity to which we refer. Note if we specialise this last equation so y=z=0, then g=g(0,0;x,t) satisfies the cubic nonlinear Schrödinger equation. This approach for the kernel form of the nonlinear Schrödinger equation shown just above, is based on the approach Pöppe [4], [5] developed. Though the connection with Grassmannian flows can be glimpsed via the formulation of the linear operator equations above, to maintain conciseness for now we refer the reader to Section 2.3 for the full details. The procedure above lends itself naturally to the non-commutative setting and, as we see in Section 3, the integral kernels for P, Q and G can be matrix-valued. Further we have thusfar glossed over an important component of Pöppe’s method, which is the major insight underlying the method. This is the aforementioned kernel product rule, in which the Hankel property of P plays a crucial role. Suppose F is a linear operator with kernel f(y,z;x). Note we assume f depends on a parameter x. Recall from above if F is a Hankel operator, then f=f(y+z+x). Let us denote by [F] the kernel of F, i.e. [F]=f. Now suppose F, F, H and H are all Hilbert–Schmidt operators with continuous kernels, and in addition assume H and H are Hankel operators. Then the fundamental theorem of calculus implies [Fx(HH)F](y,z;x)=[FH](y,0;z)[HF](0,z;x).This is the crucial kernel product rule to which we refer. It generalises the product rule used by Pöppe who used the ‘trace’ form of this rule in the sense that the rule was applied with y=z=0. The kernel product rule above is the only property we assume in Doikou et al. [3] and herein. Further the reader can now begin to fathom how the nonlinear term in the kernel equation for g=g(y,z;x,t) was formed—by applying this product rule twice to the appropriate product of operators. Indeed one application of the kernel product rule generates [](y,0;x)[](0,z;x), while after two applications we get [](y,0;x)[](0,0;x,t)[](0,z;x).

Herein we use the generalisations just mentioned to show how matrix-valued kernel equations analogous to that for g=g(y,z;x,t) above can be linearised. Hence the corresponding standard matrix-valued local nonlinearity partial differential systems can be linearised. The matrix-valued systems we establish this for include the nonlinear Schrödinger equation, the modified Korteweg de Vries equation and the Korteweg de Vries equation. Further we can also linearise in this manner corresponding matrix-valued nonlocally nonlinear versions of these equations, including the reverse space–time and reverse time nonlocal nonlinear Schrödinger equation as well as the reverse space–time nonlocal modified Korteweg de Vries equation. For example the latter equation has the following form for g=g(x,t): tg+x3g=3gg̃(xg)+3(xg)g̃g,where g̃(x,t)=gT(x,t), i.e. the transpose of g with space and time reversal. This is the third notion of nonlocal nonlinearity to which we refer. Such equations can be found in Ablowitz and Musslimani [6]. Herein we only focus on the latter two notions of nonlocal nonlinearities. To distinguish them, we refer to the third notion just above as the nonlocally nonlinear equations, while the second notion above involving the kernels g=g(y,z;x,t) we refer to as nonlocal/kernel or simply ‘kernel equations’.

Let us now discuss the ‘unified programme’ mentioned at the very beginning. Herein we extend the system in Doikou et al. [3], which is based on the second system of equations for P, Q and G above. Indeed we generalise and inflate the system as follows, this is the application linear system in Definition 3.3: tP=μ1x2P+μ2x3PtP̃=μ̃1x2P̃+μ̃2x3P̃Q=P̃PP=G(id+Q), where μ1,μ2,μ̃1,μ̃2 are constant parameters. Note we have separate linear equations for the linear operators P and P̃, and we also include the linear operators Q̃=PP̃ and G̃, where G̃ satisfies P̃=G̃(id+Q̃). All the matrix nonlocal/kernel and nonlocal nonlinear systems we consider herein linearise to the system above for appropriate choices of the parameters and P̃. For example for the complex matrix modified Korteweg de Vries equation we set μ1=μ̃1=0, μ2=μ̃2=1 and P̃=P; see Remark 3.9. The inflated system above still naturally generates a Grassmannian flow; see Section 2.3. Thus all the nonlinear systems we consider herein are examples of such flows, adding to the large class of nonlocal nonlinear systems (in the sense of the first notion on nonlocal nonlinearity we mentioned above) we have identified as such, for example the Smoluchowski coagulation flows considered in Doikou et al. [3] and all the nonlocal nonlinear systems considered in Beck et al. [1], [2].

Our work herein, in Beck et al. [1], [2] and Doikou et al. [3], was motivated by the work of Ablowitz et al. [9], Dyson [10], McKean [11] and by a series of papers by Pöppe [4], [5], [12], Pöppe and Sattinger [13] and Bauhardt and Pöppe [14]. Of particular importance for us was the realisation by Pöppe that the solution to a soliton equation is given by some function of the Fredholm determinant of the solution to the linearised soliton equation. That the scattering operators are Hankel operators is another key ingredient in Pöppe’s method. Hankel operators have received a lot of recent attention, see Grudsky and Rybkin [15], [16], Grellier and Gerard [17] and Blower and Newsham [18]. Non-commutative integrable systems, see Fordy and Kulisch [19], Nijhoff et al. [20], Nijhoff et al. [21], Fokas and Ablowitz [22], Ablowitz, Prinari and Trubatch [23], and latterly nonlocal integrable systems have also recently received a lot of attention, see Ablowitz and Musslimani [6], Fokas [24] and Grahovski, Mohammed and Susanto [25]. With regard to the non-commutative NLS in particular, see Manakov [26] for the first instance of the multi-component NLS, and for its discretisation, see the work by Degasperis and Lombardo [27], [28], Ablowitz et al. [23] and, more recently, Doikou et al. [29] and Doikou and Sklaveniti [30].

Other work in such directions includes Fokas’ unified transform method, see Fokas and Pelloni [31] and the references therein, and the scheme by Zakharov and Shabat [7], [8]. Mumford [32, p. 3.239] also took a similar viewpoint providing solutions to the Sine–Gordon, KdV and KP equations using θ-functions. Classical integrability involves the existence of a Lax pair (L̃,D̃) which satisfies the auxiliary linear problem L̃Ψ=λΨ and tΨ=D̃Ψ for an auxiliary function Ψ and spectral parameter λ. Requiring these two equations be compatible, one arrives at the so-called zero curvature condition tL̃=[D̃,L̃], which in turn yields the nonlinear integrable equation. In this context, the existence of the Lax pair provides extra symmetries and, hence, integrability. In our formulation, which is based on Pöppe’s method, we impose the linear evolutionary condition tΨ=D̃Ψ and the Hankel property for Ψ only; see Beck et al. [2, pp. 5–6]. The connection between classical integrable systems and Grassmannians was first explored by Sato [33], [34] and developed further by Segal and Wilson [35].

To summarise, what is new in this paper is that, for a collection of classical integrable nonlinear systems, we:

  • (i)

    Optimise and simplify the method of Pöppe. We show how a linear equation for a Hankel operator, based on the linearised version of the system, together with a linear integral relation, generates the solution to a corresponding nonlocal/kernel version of the integrable nonlinear systems. The integrable nonlinear system itself is generated by a further projection. The approach is optimal/minimal as it only uses the product rule of the Pöppe approach, and none further;

  • (ii)

    Generalise this optimal approach to the non-commutative case. We do not require the operator kernels involved to commute and thus the entire linearisation procedure for the integrable systems involved applies to matrix-valued systems;

  • (iii)

    Demonstrate via the non-commutative Pöppe procedure, the nonlocal/kernel nonlinear integrable systems generated are examples of evolutionary Grassmannian flows. We show how the evolutionary linear system for the Hankel operator generates an infinite dimensional Stiefel manifold flow. The projection of that flow onto the underlying Fredholm Grassmannian in a given coordinate chart is the nonlocal/kernel nonlinear integrable system under consideration;

  • (iv)

    Show how some matrix-valued nonlocal nonlinear partial differential systems, including nonlocal reverse time and reverse space–time versions of classical integrable systems, can also be linearised using the non-commutative Pöppe procedure we advocate;

  • (v)

    Reveal how the Miura transformation is the result of a trivial quadratic operator decomposition into linear operator factors at the level of the two separate non-commutative linear systems for the modified Korteweg de Vries and Korteweg de Vries equations;

  • (vi)

    Present a unified programme that incorporates all the matrix kernel, nonlocal nonlinear and local nonlinear integrable systems we consider. We show the different systems result from different choices of the parameters and underlying Hankel operators P and P̃;

  • (vii)

    Discuss how, for given initial data, this linearisation approach can be used to find time-evolutionary solutions to such non-commutative integrable nonlinear systems analytically, or efficiently numerically. In other words we can generate the evolutionary solution at any time by evaluating, analytically, the solution to the underlying linear partial differential equation for P and if required P̃ at any given time t, determining Q and if required Q̃ at that time, and then solving, usually numerically, the linear Fredholm equations defining G and if required G̃ at that time t. We do not have to numerically evolve the solution to the nonlinear system in time. This is one of the practical ‘gains’ we have made.

Our paper is outlined as follows. In Section 2 we introduce some preliminary notions and results we use throughout this paper, in particular the kernel bracket, observation functional and product rule. We then describe the unification scheme in its most general form. In Section 3 we present the main results of this paper. We start by establishing existence and uniqueness properties and proceed to prove Theorem 3.4, Theorem 3.8, where the linearisation of both nonlocal/kernel and local versions of different integrable systems is achieved without assuming commutativity. Finally, in Section 4 we discuss possible extensions to the work herein.

Section snippets

Preliminaries

Let us first describe the general framework by introducing the types of operators we use and providing some necessary definitions. As in Doikou et al. [3], we consider Hilbert–Schmidt integral operators which depend on both a spatial parameter xR and a time parameter t[0,). The class of Hilbert–Schmidt operators, J2, are representable in terms of square-integrable kernels, and so given an operator F=F(x,t) with FJ2, there exists a square-integrable kernel f=f(y,z;x,t) such that (Fϕ)(y;x,t)=

Existence and uniqueness results

Before we can proceed to the derivation of the nonlinear PDEs from the linear system, we need to establish some existence and uniqueness results. To begin we introduce some notation we require. For w:RR+, we denote by Lw2(R;n×m) the space of functions f:Rn×m whose L2 norm weighted by w is finite, that is, fLw2trRf(x)f(x)w(x)dx<,where f denotes the complex-conjugate transpose of f and ‘tr’ is the trace operator giving the sum of the diagonal elements of a matrix. We set H(R;n×m)l{0

Discussion

In this paper we have presented a unified approach to linearise and thereby solve many matrix-valued integrable systems with local and nonlocal nonlinearities. We have also shown that all the evolutionary nonlinear flows we present are evolutionary Grassmannian flows. There are however, many interesting issues still requiring further resolution. We discuss these here.

First, there are many further systems with both local and nonlocal nonlinearities in particular mentioned in Ablowitz and

CRediT authorship contribution statement

Anastasia Doikou: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. Simon J.A. Malham: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. Ioannis Stylianidis: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We thank the referees for their very useful comments which helped to significantly improve the original manuscript. We are also very grateful to one of the referees for bringing the article by Ercolani and McKean [40] to our attention. I.S. was supported by the EPSRC via a DTA Scholarship.

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      Recently Doikou et al. [6] streamlined Pöppe’s approach, demonstrating that only Pöppe’s celebrated kernel product rule is required to establish the linearisation of the classical Korteweg–de Vries and nonlinear Schrödinger equations. Subsequently Doikou et al. [7] demonstrated the approach, as considered by Bauhardt and Pöppe [5], naturally extends to the non-commutative nonlinear Schrödinger and Korteweg–de Vries equations. Malham [8] then used this approach to establish the linearisation of the non-commutative fourth order quintic nonlinear Schrödinger equation.

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