# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 1, October 2019, Pages 57 - 94

# Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Authors
Tomas Nilsona, Cornelia Schiebolda, b
aDepartment of Mathematics and Science Education, Mid Sweden University, S-851 70 Sundsvall, Sweden
bInstytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Poland
Received 19 December 2018, Accepted 7 July 2019, Available Online 25 October 2019.
DOI
10.1080/14029251.2020.1683978How to use a DOI?
Abstract

The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting.

The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.

Open Access

## 1. Introduction

In a similar way as the Toda lattice is related to the KdV equation, the two-dimensional Toda (2d-Toda) lattice

2xylog(1+wn)=wn+12wn+wn1(1.1)
is the discretization of the Kadomtsev-Petviashvili equation
(4ut+6uux+uxxx)x+3σ2uyy=0,(1.2)
more precisely the KP-II where σ = 1 (in contrast to the KP-I where σ = i) [4]. Note that the unknown function wn = wn(x, y) in (1.1) depends on two continuous variables x, y ∈ ℝ and a discrete variable n ∈ 𝕑. Although (1.1) was already considered by Darboux [10] in 1915, its integrability was only established by Mikhailov [20] in 1979. For a review of recent research and extensive references, we refer to [37].

In the present article we develop an operator theoretic approach to the solution theory of the 2d-Toda lattice. Our model is the treatment of the KP equation given in [8, 32]. To be more precise, [8] establishes a solution formula for (1.2) depending on two parameters A, B which are bounded linear operators on some Banach space, satisfying the strong assumption [A, B] = 0. However it is shown in [32] how this assumption can be dropped, allowing for more flexibility in applications. For the 2d-Toda lattice (1.1), a solution formula with commuting parameters is obtained in [31], but the question whether the extra condition is necessary has remained open so far. Our main result in Theorem 3.6 fills in this gap (and introduces and additional operator parameter D which will become significant in the applications).

Inserting the dependent variable transformation wn = (1 + vn)/(1 + vn−1) − 1 in (1.1) yields

2xylog(1+vn)=1+vn+11+vn1+vn1+vn1,(1.3)
y((I+Vn)1xVn)=(I+Vn)1(I+Vn+1)(I+Vn1)1(I+Vn).(1.4)

Here we can view Vn = Vn(x, y) as an unknown function taking its values in (F), the space of bounded linear operators on a Banach space F, or more generally in a noncommutative Banach algebra. In Theorem 2.1 we generalize the soliton of (1.3) to the operator level. The proof of this (which makes extensive use of the tool box of operator identities provided in Appendix D) is the most involved step towards achieving our final goal, a solution formula for the scalar equation depending on operator parameters. Here it should be observed that discrete integrable systems have the tendency to require considerably more intricate computations than their continuous counterparts, as will also become apparent in our case. Once the operator soliton is known, we can follow more familiar roads to extract the desired solution formula for the scalar equation. This is elaborated in Section 3, building on related work in [2,27,31]. In Appendix C we extend our results to Hirota’s bilinear form of (1.1). We would like to mention a quite parallel recent approach, called Cauchy matrix approach in [22], see also [39, 40], where this approach is extended to a general method. We refer in particular to [39, Section 5.2] for explanations on the connection between the Cauchy matrix approach and the operator method employed in the present paper.

The solution formula we use in our applications, see Proposition 4.1, depends on four linear mappings A, B, C and D. In the finite-dimensional case all of these are induced by matrices (denoted by the same symbols), A and B are quadratic of size M × M and N × N, C is of size M × N and satisfies a 1-dimensionality condition depending on A and B, and D is of size N × M but otherwise arbitrary. Simplifying slightly for the sake of exposition, we may think of A, B and D as free parameters and of C as essentially determineda by A and B. The motivation of our applications part stems from the broader problem to classify the family of solutions obtained by all possible choices of these matrices in terms of their asymptotic behaviour.

The second author has treated the analogous problem for solution formulas for soliton equations in one space dimension. For the KdV, the classification is complete [9] and yields multiple pole solutions besides the familiar N-solitons. For the sine-Gordon (sG), the modified KdV (mKdV), and the Nonlinear Schrödinger (NLS) equations, the picture is reasonably complete: For the sG and mKdV besides solitons one obtains breathers (or pulsating solitons, constituting a bound state between a soliton and an antisoliton [23]) as solutions with particle character, to which the concept of multiple pole solutions extends naturally [28,36]; for the NLS one has solitons being complex in nature [35]. Note that whereas in the KdV case different solitons necessarily have different velocities, this is no longer the case for the NLS. General results on such degeneracies are obtained in [18, 35].

Most relevant for the present article are the results on the 1d-Toda lattice [29], which are roughly analogous to the KdV case. It should be stressed that in the latter cases wave packets of multiple pole solutions always consist of both regular solitons and singular antisolitons.

In the applications, we will provide evidence that the phenomena observable within the family of solutions coming from matrices are structurally much richer than what we have mentioned in one space dimension. We start from web structures (roughly speaking these are solutions built from line solitons with X- and Y-crossings), which exist for KP-II and the 2d-Toda lattice. Their asymptotic classification is a major contribution to the topic, see the articles [3, 4, 19] and the recent monograph [17]. We will show that the solutions studied in [4, 19] (including generalized cases with singularities, whose asymptotics are not studied yet) are included in . For the reader’s convenience, we summarise some background from [4] in Appendix B. The link between the Wronskian constructions used originally to obtain web structures and our formulas is of independent interest, see Appendix A.

In the remaining applications we study three solution classes in which differ from web-structures essentially. They are obtained by choosing A, B and D as 2 × 2-matrices with appropriate structure. To get a guideline to what may be interesting, we observe that one can assume A and B in Jordan normal form (Lemma 4.8).

Taking A, B as Jordan blocks (hence commuting) and D = I2, we obtain solutions whose intersection with a y-slice {y = c} looks similar to the familiar 2-pole solution of the KdV: A weakly bound wave packet moving with constant velocity as n → ±∞, which consists of a regular soliton and a singular antisoliton deviating logarithmically from the common center, switching sides under collision, see Figure 6.

For A, B as before, we continue with the case that D is arbitrary. Asymptotic analysis shows that the entry in the left lower corner of D plays a privileged role, leading to solutions which are essentially different from the case D = I2. In a y-slice, such a solution is a 2-pole, but moving on steeper logarithmic curves, and not changing sides any longer. Moreover, it can be arranged that the solution is asymptotically regular. It should be mentioned that singularities occur at the place where the solitons collide, see Figure 7.

Finally, taking A as Jordan block but B as diagonal matrix with two different entries leads to even more intriguing solutions: Here a y-slice shows two solitons moving on logarithmic paths, but these do not deviate from a common center, but each from a center of its own. Hence the solution looks as if it consists of the superposition of two 2-poles, where each of the 2-poles has lost one of its partners, see Figure 8. Splitting the asymptotic solitons into two pairs, regularity can be arranged for each of the pairs independently.

For the above mentioned three solution classes, the asymptotic analysis is given.

Similar phenomena are to be expected for the KP-II where comparable solution formulas in [32] can be exploited.

## 2. Solving the noncommutative two-dimensional Toda lattice

The aim of the first section is to find a general solution formula for its noncommutative (nc) version (1.4). Our result on the general operator level is the following.

### Theorem 2.1.

Let E, F be Banach spaces, and A(E), B(F) invertible. Assume that the families of operators Ln = Ln(x, y) ∈ (F, E), Mn = Mn(x, y) ∈ (E, F) satisfy the following set of base equations

Ln+1=ALn,xLn=ALn,yLn=A1Ln,Mn+1=BMn,xMn=B1Mn,yMn=BMn,(2.1)
and that (IF + MnLn) is invertible for all n ∈ 𝕑 and all (x, y) ∈ Ω, where Ω is an open subset of2. Then the ℒ(F)-valued function
Vn=B(IF+MnLn)1Mn(ALnLnB1).(2.2)
is a solution of the nc 2d-Toda lattice (1.4) on 𝕑 × Ω.

Theorem 2.1 generalizes [31, Theorem 2.1], where the result was obtained under the additional assumptions [A, B] = 0, which is very restrictive in applicationsb. One of our motivations was a similar generalization obtained for the noncommutative Kadomtsev-Petviashvili equations in [32].

The proof of Theorem 2.1 relies on a rather general tool box of operator identities which is presented in Appendix D.

### Proof.

Let us start with some preparations. To utilize the appendix, we introduce the following operator-functions

S1±(n)=(IE+LnMn)1(A±1LnLnB1),S2±(n)=(IF+MnLn)1(B1MnMnA±1),T1±(n)=(IE+LnMn)1(A±1+LnB1Mn),T2±(n)=(IF+MnLn)1(B1+MnA±1Ln),
and collect the identities needed for the proof.

The first three identities, which follow on applicationc of Lemma D.5, are the derivation rules

xT2+(n)=S2+(n)S1+(n),(2.3a)
yS1+(n)=T1(n)S1+(n),(2.3b)
yS2+(n)=T2(n)S2+(n).(2.3c)

The fourth identity, which followsd from Lemma D.2 a) and Corollary D.3, allows us to handle a certain product of the operator-functions,

S2+(n)T1(n)=T2+(n)S2(n).(2.3d)

We are now in the position to prove Theorem 2.1. Let us begin with the left-hand side of the nc 2d-Toda lattice (1.4). Observe first that

IF+Vn=IF+B(IF+MnLn)1Mn(ALnLnB1)=B(IF+MnLn)1((IF+MnLn)B1+Mn(ALnLnB1))=B(IF+MnLn)1(B1+MnALn)(2.4a)
=B(IF+MnLn)1B1(IF+Mn+1Ln+1).(2.4b)

Note that (2.4a) implies that

IF+Vn=BT2+(n).(2.4c)

Hence, we find

xVn=x(IF+Vn)=(2.4c)BxT2+(n)=(2.3a)BS2+(n)S1+(n).

Moreover,

(IF+Vn)1BS2±(n)=(2.4b)(IF+Mn+1Ln+1)1B(B1MnMnA±1)=S2±(n+1)(2.5)
so that
(IF+Vn)1xVn=S2+(n+1)S1+(n).

Starting from this identity, and using (2.3b), (2.3c), we get for the left-hand side of the nc 2d-Toda lattice (1.4)

y((IF+Vn)1xVn)=(yS2+(n+1))S1+(n)+S2+(n+1)(yS1+(n))=(T2(n+1)S2+(n+1)S2+(n+1)T1(n))S1+(n).(2.6)

We next turn to the right-hand side of the nc 2d-Toda lattice (1.4). To compute the second term, we use

yielding
(IF+Vn1)1(VnVn1)=(IF+Vn1)1BS2(n1)S1+(n)=(2.5)S2(n)S1+(n).(2.7)

Analogously,

Vn+1Vn=B((IF+Mn+1Ln+1)1Mn+1A(IF+MnLn)1Mn)(ALnLnB1)=B((IF+Mn+1Ln+1)1Mn+1AMn(IE+LnMn)1)(ALnLnB1)=B(IF+Mn+1Ln+1)1(Mn+1A(IE+LnMn)(IF+Mn+1Ln+1)Mn)(IE+LnMn)1(ALnLnB1)=B(IF+Mn+1Ln+1)1(Mn+1AMn)S1+(n)=B(IF+Mn+1Ln+1)1(Mn+1AB1Mn+1)S1+(n)=BS2+(n+1)S1+(n).

Note that (2.4b) implies that (IF + Vn)−1 = (IF + Mn+1Ln+1)−1B(IF + MnLn) B−1 = (IF + Mn+1Ln+1)−1(B + Mn+1A−1Ln+1) B1=T2(n+1)B1, showing

(IF+Vn)1(Vn+1Vn)=T2(n+1)S2+(n+1)S1+(n).(2.8)

Thus, for the right-hand side of the nc 2d-Toda lattice (1.4) we obtain

(IF+Vn)1(IF+Vn+1)(IF+Vn1)1(IF+Vn)=((IF+Vn)1(IF+Vn+1)+IF)((IF+Vn1)1(IF+Vn)+IF)=(IF+Vn)1((IF+Vn+1)(IF+Vn))(IF+Vn1)1((IF+Vn)(IF+Vn1))(2.9)
=(IF+Vn)1(Vn+1Vn)(IF+Vn1)1(VnVn1)=(2.7),(2.8)(T2(n+1)S2+(n+1)+S2(n))S1+(n).(2.10)

Comparing (2.6) with (2.10) it remains to show S2+(n+1)T1(n)=S2(n) to complete the proof. The obstacle here is that properties as in Lemma D.2 can only be applied if the operator-functions have the same arguments. Hence we need a little detour.

S2+(n+1)T1(n)=(2.5)(IF+Vn)1BS2+(n)T1(n)=(2.3d)(IF+Vn)1BT2+(n)S2(n)=(2.4c)S2(n),
which completes the proof.

## 3. A solution formula for the scalar 2d-Toda lattice depending on two independent operator parameters

In this section a scalarization process is introduced with which, starting from a solution of the non-commutative 2d-Toda lattice (1.4), a solution formula for the scalar lattice (1.3) can be constructed.

After introducing the necessary background from functional analysis in Subsection 3.1, we explain the main idea behind scalarization in Subsection 3.2. In Subsection 3.3, this strategy is carried out explicitely for the 2d-Toda lattice. Finally, in Subsection 3.4 we explain how the theory of elementary operators can be exploited to meet the requirements of scalarization and to simplify the solution formulas considerably.

## 3.1. Terminology

Before explaining our general strategy, we need some terminology. Let E, F be Banach spaces. A one-dimensional operator T(E, F) is an operator whose range is contained in a one-dimensional subspace of F. Every such operator T can be written as bc for a vector cF and a functional bE′, where

bc(x):=x,bcxE
(〈, 〉 is the usual dual pairing: for the functional bE′, 〈x, b〉 is the evaluation of b at the vector xE).

A finite-rank operator is an operator T(E, F) with finite-dimensional range, and the space of all finite-rank operators from E into F is denoted by (E, F). We set rank(T) = dim(ran(T)). Note that

(E,F)={j=1Nbjcj|bjE,cjF},

The class = ∪E,F(E, F) of finite-rank operators forms an operator ideal. Moreover, it is well-known that there is a unique trace tr : ∪F(F) → ℂ, which is given by

tr(T)=j=1Ncj,bj,
where T=j=1Nbjcj, bjF′, cjF, is an arbitrary (finite) representation of T(F) (see [24] for more detailed information on the concept of traces and determinants on quasi-Banach operator ideals).

## 3.2. Strategy

Let us briefly explain the idea of the scalarization process. Let us assume that Vn = Vn(x, y) ∈ (F) is an operator-valued solution of the nc 2d-Toda lattice (1.4). A natural ansatz to derive a solution vn for the scalar lattice (1.3) is to apply a continuous linear functional τ to Vn, i.e. to try vn = τ(Vn). Of course the functional τ has to be chosen in a way that the solution property is maintained under its application. Since the 2d-Toda lattice is nonlinear, τ needs to be multiplicative at least in a certain sense.

To meet this requirement we introduce, for a fixed functional bF′, the subalgebra

𝒮b(F)={bc|cF}
of (F) consisting of one-dimensional operators whose behavior is governed by the functional b. Observe that the restriction of the trace tr on the finite-rank operators (F) to 𝒮b(F) coincides with the evaluation of the functional b. Now the crucial observation is that tr is multiplicative on 𝒮b(F):

### Lemma 3.1.

For T1, T2𝒮b(F), it holds tr(T1T2) = tr (T1)tr (T2).

### Proof.

We verify

T1T2=(tr(T2))T1,
then the assertion follows from the linearity of the trace. Indeed, for T1 = bc1, T2 = bc2 with c1, c2F, we have
(T1T2)(x)=T1(T2(x))=T1((bc2)x)=T1(x,bc2)=x,bT1(c2)=x,b(bc1)c2=x,bc2,bc1=c2,bT1(x)
for xF.

This motivates the following choices:

1. (1)

We assume that the operator solution Vn belongs to 𝒮b(F) for a constant, fixed bF′. In other words, Vn is one-dimensional with fixed kernel.

2. (2)

For scalarization, we use the functional tr.

Then application of tr to Vn maintains the solution property.

### Proposition 3.2.

Let Vn = Vn(x, y) ∈ 𝒮b(F) be a solution of (1.4). Then vn = tr (Vn) solves (1.3).

### Remark 3.3.

For a more systematic explanation of the choices (1), (2) above, we refer to [2].

For the proof of Proposion 3.2, we need two more properties for operators in 𝒮b.

### Lemma 3.4.

Let T be an operator-valued function depending 𝒞1-smoothly on some variable ξ such that T(ξ) ∈ 𝒮b(F). Then also tr (T) depends 𝒞1-smoothly on ξ with

ddξ(tr(T))=tr(ddξT).

### Proof.

By assumption, we can write T = bc with a 𝒞1-smooth vector-function c : ξc(ξ) ∈ F. Hence ddξT=b(ddξc), in particular also ddξT takes its values in 𝒮b(F). Since evaluation of the functional b is continuous, we get

ddξ(c(ξ),b)=limh0c(ξ+h),bc(ξ),bh=limh0c(ξ+h)c(ξ)h,b=limh0c(ξ+h)c(ξ)h,b=ddξc(ξ),b,
showing
ddξ(tr(T))=ddξ(tr(bc))=ddξ(c,b)=ddξc,b=tr(b(ddξc))=tr(ddξT).

### Lemma 3.5.

For T1, T2𝒮b(F) with IF + T1 invertible, it holds

tr((IF+T1)1T2)=tr(T2)1+tr(T1).

### Proof.

The assumption follows from

(1+tr(T1))tr((IF+T1)1T2)==tr((IF+T1)1T2)+tr(T1)tr((IF+T1)1T2)=tr((IF+T1)1T2)+tr(T1(IF+T1)1T2)=tr((IF+T1)(IF+T1)1T2)=tr(T2)
and the fact that 1 + tr (T1) = det (IF + T1) ≠ 0 since IF + T1 is invertible. Note that we have used the multiplicity of tr in the boxed reformulation.

### Proof of Propostion 3.2.

Using the tools collected above, we can directly verify that vn = tr (Vn) satisfies (1.3).

2xylog(1+vn)=yxvn1+vn=yxtr(Vn)1+tr(Vn)=L3.4ytr(xVn)1+tr(Vn)=L3.5ytr((IF+Vn)1xVn)=L3.4tr(y((IF+Vn)1xVn))=(1.4)tr((IF+Vn)1(IF+Vn+1)(IF+Vn1)1(IF+Vn))=tr((IF+Vn)1(Vn+1Vn)(IF+Vn1)1(VnVn1))=()tr((IF+Vn)1(Vn+1Vn))tr((IF+Vn1)1(VnVn1))=L3.5tr(Vn+1)tr(Vn)1+tr(Vn)tr(Vn)tr(Vn1)1+tr(Vn1)=vn+1vn1+vnvnvn11+vn1=1+vn+11+vn1+vn1+vn1.

Note that for Step (⋆) above, linearity of tr, more precisely tr (T1 + T2) = tr (T1) + tr (T2) for T1, T2𝒮b(F), is used, to which end we observe that (IF +Vn)−1(Vn+1Vn) ∈ 𝒮b(F) for all n.

## 3.3. Solution formulas

Application of the scalarization technique from last subsection to the operator solution in Theorem 2.1 provides us with a first solution formula for (1.3).

### Theorem 3.6.

Let E, F be Banach spaces, and A(E), B(F) invertible. Let Ln = Ln(x, y) ∈ (F, E), Mn = Mn(x, y) ∈ (E, F) be operator functions which satisfy the base equations (2.1) and the one-dimensionality condition

Mn(ALnLnB1)𝒮b(F)(3.1)
for some bF′.

If (IF + MnLn) is invertible for all n ∈ 𝕑 and all (x, y) ∈ Ω, where Ω is an open subset of2, then

vn=tr(B(IF+MnLn)1Mn(ALnLnB1))(3.2)
is a solution of the scalar 2d-Toda lattice (1.3) on 𝕑 × Ω.

### Proof.

Since Ln(x, y), Mn(x, y) satisfy the base equations (2.1), Theorem 2.1 implies that Vn = B(IF + MnLn)−1Mn(ALnLnB−1) solves of the nc 2d-Toda lattice (1.4) on 𝕑 × Ω. Now the one-dimensionality condition (3.1) tells Mn(ALnLnB−1) = bcn with some vector function cn = cn(x, y) ∈ F, and hence

Vn=B(IF+MnLn)1Mn(ALnLnB1)=B(IF+MnLn)1(bcn)=b((IF+MnLn)1cn).

This shows Vn𝒮b(F), and the assertion follows from Proposition 3.2.

If the operator function Ln = Ln(x, y) even takes its values in a quasi-Banach operator ideal 𝒜 which is equipped with a nice, generalized determinant, the solution formula in Theorem 3.6 can be improved considerably.

### Theorem 3.7.

Let the assumptions of Theorem 3.6 be satisfied. Let, in addition, Ln = Ln(x, y) ∈ 𝒜(F, E), where 𝒜 is a quasi-Banach operator ideal admitting a continuous determinant δ. Then the solution (3.2) can be expressed as

vn=δ(IF+Mn+1Ln+1)δ(IF+MnLn)1.

### Proof.

Let τ be the trace associated to the determinant δ by the trace-determinant theorem [24], and note this trace coincides with tr on (F), by uniqueness of tr. Hence we get

vn=tr(B(IF+MnLn)1Mn(ALnLnB1))=τ(B(IF+MnLn)1Mn(ALnLnB1)).

Next, using the fact that 1 + τ(T) = δ(I + T) holds for one-dimensional operators T, we obtain

vn=δ(IF+B(IF+MnLn)1Mn(ALnLnB1))1=δ(B(IF+MnLn)1(B1+MnALn))1=δ((IF+MnLn)1(B1+MnALn)B)1=δ((IF+MnLn)1(IF+MnALnB))1=δ(IF+MnALnB)δ(IF+MnLn)1=δ(IF+BMnALn)δ(IF+MnLn)1=δ(IF+Mn+1Ln+1)δ(IF+MnLn)1,
which was to be shown.

## 3.4. Elementary operators

A natural way to satisfy the base equations (2.1) is by choosing

Ln(x,y)=Anexp(AxA1y)C,Mn(x,y)=Bnexp(B1x+By)D
with arbitrary constant operators C(F, E), D(E, F). Then the one-dimensionality condition (3.1) is met provided that ACCB−1 is one-dimensional or, equivalently, provided that C is a solution of the Sylvester equation
AXBX=bc(3.3)
for some bF′ and cE. In the present subsection we review some relevant results on the Sylvester equation and their impact on our solution formulas.

To this end, let 𝒜 be a quasi-Banach operator ideal, and ΦA,B : 𝒜(F, E) → 𝒜(F, E) the operator defined by

ΦA,B(X)=AXBX.(3.4)

The operator ΦA,B belongs to the larger class of so-called elementary operators the structural properties of which have been extensively studied in the literature (see the survey [26] and references therein).

For the spectrum of ΦA,B the following striking formula

spec(ΦA,B)=spec(A)spec(B)1:={αβ1|αspec(A),βspec(B)}(3.5)
holds (see [11] for fundamental work on the spectra of elementary operators on Banach operator ideals, and [1] for the extension to quasi-Banach operator ideals). It shows that invertibility of ΦA,B can be read off from the spectra of A and B, independently of the underlying quasi-Banach operator ideal 𝒜.

In particular, since we are interested in solving the Sylvester equation (3.3) with right-hand side bc(F, E), and the finite-dimensional operators are contained in any quasi-Banach operator ideal, we are then free to choose 𝒜 arbitrarily.

Provided that 1 ∉ spec(A) · spec(B), the Sylvester equation (3.3) has the (unique) solution ΦA,B1(bc). Moreover, ΦA,B1(bc)𝒜(F,E) for any quasi-Banach operator ideal 𝒜.

A particularly convenient choice for 𝒜 is the quasi-Banach operator ideal 𝒩2/3 of 2/3-nuclear operators in the sense of Grothendieck. Recall that a bounded operator T: FE belongs to 𝒩2/3(F, E) iff there is a representation

T=j=1bjcjwithj=1(bjcj)2/3<,
where bjF′, cjE. Note that 𝒩2/3 becomes a quasi-Banach operator ideal with respect to the quasi-norm
T|𝒩2/3=inf{(j=1(bjcj)2/3)3/2},
where the infimum is taken over all possible representations.

It was already shown by Grothendieck [12] that 𝒩2/3 is of eigenvalue-type 1, i.e. operators in 𝒩2/3 possess absolutely summing eigenvalues. For quasi-Banach operator ideals of eigenvalue-type 1, a deep result of White [38] states that the spectral sum trλ (i.e. the sum of the eigenvalues) is a continuous trace. By the trace extension theorem [24], this trace is unique. Using the relationship between traces and determinants on quasi-Banach operator ideals [24, Chapter 4.6], there also is a unique continuous determinant detλ on 𝒩2/3, which is spectral.

We sum up:

### Proposition 3.8.

Let E, F be Banach spaces and A(E), B(F) invertible with 1 ∉ spec(A)· spec(B). Let bF′, cE, and D(E, F), and set

Ln(x,y)=Anexp(AxA1y)ΦA,B1(bc),Mn(x,y)=Bnexp(B1x+By)D,
such that MnLn𝒩2/3(F).

If detλ (IF + MnLn) ≠ 0 for all n ∈ 𝕑 and all (x, y) ∈ Ω, where Ω is an open subset of2, then

vn=detλ(IF+Mn+1Ln+1)detλ(IF+MnLn)1
is a solution of the scalar 2d-Toda lattice (1.3) on 𝕑 × Ω (and detλ denotes the unique continuous and spectral determinant on 𝒩2/3).

## 4. Applications

In the concluding section we discuss first applications of the solution formula in Proposition 3.8. First we re-derive line solitons [14, 16] in Subsection 4.1, then we study how resonance and web structure phenomena [4, 19] fit into the picture in Subsection 4.2. Finally, Subsection 4.3 contains some examples beyond the soliton solutions.

Usually one visualizes solutions for (1.1). Recall from the introduction that the connection between (1.3) and (1.1) is given by the dependent variable transformation wn = (1 + vn)/(1 + vn−1) − 1. One then plots the solutions for fixed values of y, the time variable, as functions of (x, n). To facilitate comparison with the literature, we start with summing up the contents of Propositions 3.8 and C.1 for (1.1).

### Proposition 4.1.

Let A be an M × M-matrix, B an N × N-matrix, both invertible with 1 ∉ spec(A) · spec(B), and D an N × M-matrix. Let b ∈ ℂN, c ∈ ℂM, and let the M × N-matrix C satisfy

ACBC=cbt
(where bt denotes the transposed of b)e. Set
Ln(x,y)=Anexp(AxA1y)C,Mn(x,y)=Bnexp(B1x+By)D,
and assume that det(IN + MnLn) ≠ 0 for all n ∈ 𝕑 and all (x, y) ∈ Ω, where Ω is an open subset of2. Then
wn(x,y)=2xylogp(n,x,y)
with
p(n,x,y)=det(IN+Mn(x,y)Ln(x,y)),(4.1)
is a solution of (1.1) on 𝕑 × Ω.

Note that we formulated Proposition 4.1 in the finite-dimensional setting, which is sufficient for the applications we have in mind, but it should be mentioned that there are numerous applications relying on analysis in infinite-dimensional spaces [5, 7, 8, 33].

## 4.1. Line solitons

Let us start with explicitly evaluating the solution formula in Proposition 4.1 for the following choices: Let M = N and let A, B are diagonal matrices, say

A=(p100pN),B=(1/q1001/qN)
with pj, qj ≠ 0 and pi/qj ≠ 1 for all i, j. Note that this implies that A, B are invertible, and the spectral condition 1 ∉ spec(A) · spec(B) is satisfied.

Furthermore let D = IN.

Given b, c ∈ ℂN, one verifies directly that the unique solution of the one-dimensionality condition ACBC = cbt is given by

C=(bjcipi/qj1)i,j=1N.

Finally, An exp(AxA−1y) = diag{[pj]|j}, Bn exp(−B−1x + By) = diag{1/[qj]|j}, where we have set

[p]=[p](n,x,y)=pnexp(pxy/p).(4.2)

Then we get for the determinant (4.1) in Proposition 4.1 that

p=det(δij+bjcipi/qj1[pi]/[qi])i,j=1N=1+n=1Ni1<<indet(bijcijpij/qij1[pij]/[qij])j,j=1n=1+n=1Ni1<<indet(1pij/qij1)j,j=1nj=1nbijcij[pij][qij]=1+n=1Ni1<<inj=1nbijcijpij/qij1[pij][qij]j,j=1j<jn(pijpij)(1/qij1/qij)(pij/qij1)(pij/qij1).(4.3)

The calculation of det(1pi/qj1)i,j=1n in the last step is a slight modification of the calculation of det(1pi+qj)i,j=1n, which can be traced back to a classical remark of Cauchy [6], see also [25]. For details and generalizations with impact on integrable systems see [34].

In particular, for N = 1 we get

p=1+b1c1p1/q11[p1][q1].

Taking p1 > q1 > 0 and b1c1 > 0, to avoid singularities, we recover the 1-soliton solution (line soliton)

w=14(p1q1)(1p11q1)sech2(12(θ[p1]θ[q1]+ϕ1)),
where we have set
θ[p]=log(p)n+pxy/pandϕ1=logb1c1p1/q11.

For fixed y, the line soliton is situated on a line in xn-space with slope d. (For visualization it is convenient to regard n as continuous variable). It is characterized by the latter and its amplitude a, where we have

a=(p1q1)24p1q1,d=p1q1log(p1)log(q1)<0,
see Fig. 1 for illustration.

For N = 2, inspection of (4.3) shows that the solution is regular if pj > qj > 0, bjcj > 0 for j = 1, 2, and

A12=(p1p2)(1/q11/q2)(p1/q21)(p2/q11)>0.(4.4)

Note that A12 encodes the position shift of the asymptotic paths of the line-solitons, which is caused by their collision.

We may assume p1 > p2 without loss of generality. Then (4.4) can be satisfied by either one of the following conditions

• (O) 0 < q2 < p2 < q1 < p1,

• (A) 0 < q1 < q2 < p2 < p1.

The resulting solution describes the elastic collision of two line-solitons. For condition (O), the involved solitons belong to the parameter constellations (p1, q1) and (p2, q2), constituting a collision of so-called ordinary type. For condition (A), they belong to (p1, q2) and (p2, q1), a collision of so-called asymmetric type, see [4] for details.

### Example 4.2.

In Fig. 2 we provide an illustration for

{p1,p2,q1,q2}={15,2,0.5,0.1},(4.5)
compare [4, Figure 3], and φ1 = φ2 = 0.

The plot on the right corresponds to condition (O), which implies that the parameter constellation is q2 = 0.1, p2 = 0.5, q1 = 2, p1 = 15. The resulting solution shows the collision of two solitons with parameters (15, 2), (0.5, 0.1). It is plotted for y = 1.

In contrast, the plot on the left corresponds to condition (A) with the parameter constellation q1 = 0.1, q2 = 0.5, p2 = 2, p1 = 15. In this case the solution shows the collision of two solitons with parameters (15, 0.1), (2, 0.5). It is plotted for y = 10.

In [31] it is shown that the line solitons are comprised in the solution class discussed above by establishing the link to their representation in terms of Casorati determinants [14, 16].

## 4.2. Resonance and web structures

In [4] a solution class was studied, which not only includes elastic collision of line-solitons [15], but also soliton resonances and web structures [19]. In this subsection we first briefly recall the construction in [4]. Then we show how this class can be realized in terms of Proposition 4.1. Finally we give some examples.

The following solution class for the 2d Toda lattice (1.1) was studied in [4]:

wn=2xylogdet(CeΘK),(4.6)
where C is a (real) N × M-matrix with NM, and K the M × N-matrix
K=(pji1)j=1,,Mi=1,,N=(1p1p1N11pMpMN1)
with 0 < p1 < ··· < pM. Finally, Θ is the M × M-diagonal matrix with θ[pj] on the diagonal, where
θ[pj](n,x,t)=log(pj)n+pjxy/pj+θj.

In fact, this class contains line solitons [15] for C = C1, and fully resonant solutions [19] for C = C2, where

C1=(11000011),C2=(11p1pMp1N1pMN1).

For this class, the following assumptions can be made:

1. (1)

N < M.

2. (2)

C is in reduced row echelon form (RREF) with rank(C) = N.

3. (3)

The matrix C in addition satisfies that (i) each column of C contains at least one nonzero element, and (ii) each row of C contains at least one nonzero element in addition to the pivot.

Assumption (1) is no restriction since for N = M one obtains the trivial solution. Assumption (2) can be made without loss of generality. The reason is that the transformation CC′ = GC with G ∈ GLN(ℝ) leaves the solution invariant. Note also that unless C has full rank, the determinant in (4.6) vanishes identically, and the solution is undefined. Assumption (3) serves to avoid redundancies.

### Remark 4.3.

(4) all N × N-minors of C are non-negative implies regularity of the solutions (which can be easily seen by expanding the determinant).The corresponding solution class is characterised asymptotically in [4]. For the reader’s convenience, we describe in Appendix B how the asymptotics of the solution depends on the RREF of C.

The main aim of the subsection is to explain how this solution class can be realized in terms of Proposition 4.1. Note that we will not need Assumption (4). We start with the following simplification of the coefficient matrix C.

### Lemma 4.4.

Without loss of generality, C = (IN D) with some N × (MN)-matrix D. Here IN denotes the N × N-unit matrix.

### Proof.

Since C is in RREF with rank(C) = N, there is a matrix Π, of size M × M, such that CΠ = (IN D) with some N × (MN)-matrix D. Now

det(CeΘK)=det((CΠ)(Π1eΘΠ)(Π1K))=det((IND)eΠ1ΘΠ(Π1K)).

Inspection shows that Π−1ΘΠ corresponds to a permutation of p1,..., pM, and that Π−1K changes the Vandermonde matrix K accordingly.

Similar to the decomposition of the coefficient matrix C in an N × N-matrix and an N ×(MN)-matrix in the above lemma, we write

K=(KNKMN)andΘ=diag{ΘN,ΘMN}
with N × N-matrices KN, ΘN, an (MN) × N-matrix KMN, and an (MN) × (MN)-matrix ΘMN. Then
det(CeΘK)=det((IND)(eΘN00eΘMN)(KNKMN))=det(eΘNKN+DeΘMNKMN)=det(eΘN(IN+eΘNDeΘMNKMNKN1)KN),
which generates same solution wn as
det(IN+eΘNDeΘMNKMNKN1).

To see how the corresponding solution is realized using Proposition 4.1, we set E = ℝMN, F = ℝN, and define

A=diag{pN+1,,pM},B=diag{1p1,,1pN}.

By Proposition A.4, see Appendix A, the (MN) × N-matrix KMNKN1 is a solution of the Sylvester equation AXBX = cbt for appropriately chosen b ∈ ℝN, c ∈ ℝMN. Hence,

eΘMNKMNKN1=Anexp(AxA1y)C0,eΘND=Bnexp(B1x+By)D0,
where
C0=diag{eθN+1,,eθM}KMNKN1andD0=diag{eθ1,,eθN}D.

As a consequence, the solution class (4.6) is included in Proposition 4.1 with the two generating matrices A, B being diagonal, but not necessarily of the same size.

Let us illustrate the above by having a closer look at collisions of two line-solitons. Recall 0 < p1 < p2 < p3 < p4, and let θ1 = θ2 = θ3 = θ4 = 0. Elastic collision of two line-solitons correponds to the following choices for C [4, Lemma 4.1]:

• (O) ordinary type: C=(11000011),

• (A) asymmetric type: C=(10010110),

• (R) resonant type: C=(101101cd) with c > d > 0.

Note that in the case (O), the second and third column of C have to be interchanged in order to bring C into the form of Lemma 4.4. Hence the following settings are needed in our formalism:

• (O) A=(p200p4), B=(1/p1001/p3), and D=(1001),

• (A) A=(p300p4), B=(1/p1001/p2), and D=(0110),

• (R) A=(p300p4), B=(1/p1001/p2), and D=(11cd) with c > d > 0.

Note however, that the constants θj and the φj from our formalism not necessarily coincide.

### Example 4.5.

For p4 = 15, p3 = 2, p2 = 0.5, p1 = 0.1, the elastic collision of two line-solitons of resonant type is visualized in Fig. 3 with c = 2, d = 1, compare [4, Figure 4].

A similar transition can be made for the inelastic collision of two line-solitons. The conditions given in [4, Lemma 4.1] are:

typeI:C=(110r0011),typeII:C=(10rr0111),typeIII:C=(100r0111),typeIV:C=(10r10110)
with r > 0. The corresponding settings for the realization in our formalism are the following:
typeI:D=(1r01),typeII:D=(rr11),typeIII:D=(0r11),typeIV:D=(r110),
with
A=(p200p4),B=(1/p1001/p3)fortypeI,A=(p300p4),B=(1/p1001/p2)fortypeIIIV.

### Example 4.6.

For p4 = 15, p3 = 2, p2 = 0.5, p1 = 0.1, inelastic collision of two line-solitons is visualized in Fig. 4 with r = 1, compare [4, Figure 5].

We conclude the present subsection with the realization of Miles structures [21]. Choosing

A=diag{p1,,pM},B=(1/q),
and the 1 × M-matrix D = (1, 1,...,1) in Proposition 4.1, we get for the solution of the one-dimensionality condition ACBC = c bt with b ∈ ℝ, c ∈ ℝM the M × 1-matrix
(bcjpj/q1)j=1M,
p=1+j=1Nbcjpj/q1[pj]/[q],(4.7)
where [p] is defined as in (4.2). Regularity of the corresponding solution can for example be guaranteed by pj > q > 0 and bcj > 0 for all j.

Note that (4.7) can be considered as the degenerated case q1 = ··· = qM = q of (4.3). In fact, starting from A = diag{p1,..., pM}, B = diag{1/q1,...,1/qM}, one arrives at the same solution class. This class has also been discussed in [31]. For fixed values of y the solutions represent tree-like structures with M line solitons for n → ∞ and 1 line soliton for n → −∞.

A similar discussion can be made for solutions generated from

A=(p),B=diag{1/q1,,1/qN}.

### Example 4.7.

The solutions in Fig. 5 correspond to the case M = 3 and the parameter constellation p1 = 0.5, p2 = 2, p3 = 15 and q = 0.1. The initial phase shifts are determined by (bcj)/(pj/q − 1) = 1.

## 4.3. Further examples

In this subsection we give some selected applications of Proposition 4.1 beyond the case of diagonal matrices A, B.

First we will discuss the case that A, B are 2 × 2 Jordan blocks and D = I2. Note that the corresponding solutions can be easily realized in the framework of [31], where a solution formula was established for commuting A and B, without the additional matrix parameter D. In the next case we drop the assumption D = I2. Finally we consider the case that A is a 2 × 2 Jordan block and B a 2 × 2-diagonal matrix with two different eigenvalues.

We start with the observation that the assumption that A, B are in Jordan form can be made without loss of generality.

### Lemma 4.8

Let U, V be the transformation matrices of the M × M-matrix A, the N × N-matrix B into Jordan canonical form JA, JB, respectively (i.e., we have A = U−1JAU and B = V −1JBV).

Then the solution in Proposition 4.1 is not altered if we replace simultaneously A, B by JA, JB, the matrix D by VDU−1 and the vectors b, c by (V −1)tb, Uc.

### Proof.

Let D˜=VDU1, b˜=(V1)tb, c˜=Uc, and C˜=UCV1.

First we show that replacing A, B by JA, JB and b, c by b˜, c˜ amounts to replacing C by C˜. Since C solves the Sylvester equation AXBX = cbt,

c˜b˜t=(Uc)((V1)tb)t=U(cbt)V1=U(ACBC)V1=(UAU1)(UCV1)(VBV1)UCV1=JAC˜JBC˜,
implying that C˜ is the (unique) solution of JAXJBX=c˜b˜t.

Furthermore, by definition of the exponential function as power series, we have Anexp(AxA1y)=U1JAnexp(JAxJA1y)U, and Bnexp(B1xBy)=V1JBnexp(JB1x+JBy)V. This yields

ULn(x,y)V1=JAnexp(JAxJA1y)C˜,VMn(x,y)U1=JBnexp(JB1x+JBy)D˜.

By multiplicity of the determinant, p(n, x, y) = det(IN + Mn(x, y)Ln(x, y)) hence coincides with the determinant of

IN+JBnexp(JB1x+JBy)D˜JAnexp(JAxJA1y)C˜,
generated using the data JA, JB, D˜, and b˜, c˜ in Proposition 4.1.

## 4.3.1. A, B (commuting) 2 × 2-Jordan blocks and D = I2

Let us now first consider the case that

A=(p10p),B=(1/q101/q),
with p, q > 0. It is not too hard to compute the determinant in Proposition 4.1 explicitly. The result is
p(n,x,y)=1+f(n,x,y)(n,x,y)(n,x,y)2,
where
(n,x,y)=εexp(θ[p]θ[q]+ϕ),
f(n,x,y)=(pq)(P+Q+γ)/q.

Here we have introduced the ploynomials

P=P(n,x,y)=(χ[p]p/(pq))/p,Q=Q(n,x,y)=q(χ[q]p/(pq)),
with
θ[p]=log(p)n+pxy/p,χ[p]=n+px+y/p,
andf the parameters φ, γ and the sign ε are given by
εexp(ϕ)=b1c2(p/q1)2,γ=b1c1+b2c2b1c21+pqpq.

In order to do an asymptotic investigation, we fix y and look at the solution for n → ±∞. Following the kind of arguments laid down in [29] for the one-dimensional Toda lattice, we arrive atg

wn(x,y)w1±(n,x,y)+w2±(n,x,y)forn+,
with solitons
wj±(n,x,y)=(pq)2pqj±(n,x,y)(1+j±(n,x,y))2,
where
1±(n,x,y)=εC±1exp(θ[p]θ[q]±log|n|+ϕ),2±(n,x,y)=εC1exp(θ[p]θ[q]log|n|+ϕ),
and
C=pqq((1+p/d)/p+q(1+q/d)),d=pqlogplogq.

This shows that, for fixed y, the solution constitutes a wave packet consisting of two partners. As a whole, the wave packet has slope d in xn-space, with the partners deviating logarithmically as n → ±∞.

Note that the asymptotics also shows that one of the partners is a (regular) soliton and the other a (singular) antisoliton. Identifying them by their regularity, one also observes that, from n → −∞ to n → ∞, the solitons cross sides. In fact, for the one-dimensional Toda lattice, which can be obtained from (1.1) by reduction [4], this kind of appearance of negaton solutions is well reflected [29].

### Example 4.9.

For p = 1, q = 2, the solution discussed above is visualized in Figure 6, with b1 = 1/10, b2 = −1, c1 = 2, c2 = 1. Note that the solution is plotted in the frame (n, x − log(2)n, 0), corresponding to the fact that a line-soliton with parameters p = 1, q = 2 has slope −1/log(2) in xn-space.

## 4.3.2. The general case of (commuting) 2 × 2-Jordan blocks A, B

In the next case we once more consider

A=(p10p),B=(1/q101/q)
with p, q > 0, but now we take advantage of the additional matrix parameter D. To this end, let
D=(d1d2d3d4).

Inspection of the determinant in Proposition 4.1 in this case shows that in order to capture the leading terms in the asymptotic analysis it suffices to choose

b=(b1,0)t,c=(0,c2)t
with b1c2 ≠ 0. For the determinant we then get
p(n,x,y)=1+f(n,x,y)(n,x,y)det(D)(n,x,y)2,
where all ingredients are defined as in the case before except
f(n,x,y)=(pq)(d2+d1P+d4Q+d3(PQ+q2/(pq)2))/q.

Assume d3 ≠ 0 and det(D) ≠ 0. Asymptotic analysis (for fixed y) confirms that, for n → ±∞, the solution constitutes a wave packet consisting of two solitons wj±(n,x,y) as before, but now build with the exponentials

(n,x,y)=εCexp(θ[p]θ[q]+2log|n|+ϕ+logδ),
2(n,x,y)=σεC1exp(θ[p]θ[q]2log|n|+ϕ+logδ),
where
C=d3δpqp(1+p/d)(1+q/d)
and we have set det(D) = σδ2 with a sign σ and δ > 0.

Observe the additional factor 2 in front of the logarithm, which is due to the fact that the polynomial appearing in the determinant p(n, x, y) is of degree 2. The even degree of the polynomial is also responsible for the fact that the solitons no longer cross sides. It is remarkable that choosing d3 and det(D) appropriately, we now can arrange that both solitons are regular.

It should be stressed that the first case (D = I2) actually is degeneration of the case above.

### Example 4.10.

For p = 2 and q = 1, the solution discussed above is visualized in Figure 7, with b1 = c2 = 1 and d1 = −10, d2 = −2, d3 = −1, d4 = 1. It is plotted in the frame (n, x − log(2)n, 0).

## 4.3.3. Noncommuting A, B (of same size)

Finally we look at the case

A=(p10p),B=(1/q1101/q2),
(i.e. with non-commuting matrix parameters A and B) for p, q1, q2 > 0 and, without loss of generality, q1 > q2.

Computing the determinant in Proposition 4.1 for this parameter choice, one sees that to keep the leading terms it is sufficient to choose

b=(b1,b2)t,c=(0,c2)t
with b1c2 and b2c2 ≠ 0. For the determinant we then get
p(n,x,y)=1+d1P(1)(n,x,y)(1)(n,x,y)+d3P(2)(n,x,y)(2)(n,x,y)det(D)A2(1)(n,x,y)(2)(n,x,y),
where
(j)(n,x,y)=εjexp(θ[p]θ[qj]+ϕj),P(j)(n,x,y)=(pqj)(χ[p]p/(pqj))/p,
with the parameters φj and the signs εj given by εj exp(φj) = bjc2qj/(pqj), and where
A12=q1q2(pq1)(pq2).

Assume d1, d3 ≠ 0 and det(D) ≠ 0. Recall that without loss of generality q1 > q2. For the asymptotic analysis we also fix p > q1 (note that q1 > p > q2 and q2 > p can be treated similarly). As a result, for n → ±∞ the solution consists of two solitons w(j),± (n, x, y), build with

(1),±(n,x,y)=ε1σ1±(C(1))1exp(θ[p]θ[q1]log|n|+ϕ1+logδ1+ϕ1±),(2),±(n,x,y)=ε2σ2±(C(2))±1exp(θ[p]θ[q2]±log|n|+ϕ2+logδ2+ϕ2±),
where
C(j)=|d1d3det(D)|(1p+1d(j)),d(j)=pqjlogplogqj,σ1+=sgn(d3det(D)),σ1=sgn(d1),σ2+=sgn(d3),σ2=sgn(d1det(D)),δ1=|d1det(D)/d3|,δ2=|d3det(D)/d1|.
and
ϕ1+=ϕ2=logA12,ϕ1=ϕ2+=0.

Hence the solution consists of two different solitons, corresponding to the parameter constellations (p, q1) and (p, q2), respectively, which deviate logarithmically from their respective slopes d(1) and d(2).

Note that the collision ot the two solitons causes a phase-shift determined by A12 = (1/q1 − 1/q2)/((p/q2 − 1)(p/q1 − 1)), compare with (4.4).

Observe also that regularity of the solitons can be manipulated by the signs of d1, d3 and det(D) in the following way: One can pair the solitons arbitrarily and assign to each pair separately whether both solitons of the pair are regular or singular.

### Example 4.11.

For p = 2, q1 = 1, q2 = 1/2, the corresponding solution is depicted in Figure 8 with b1 = c2 = 1 and d1 = 1, d2 = −1, d3 = −10, d4 = −1.

The two involved solitons correspond to the parameters (2, 1) and (2, 1/2). Observe that a line-soliton with parameters (2, 1) has slope −1/log2 in xn-space and one with parameters (2, 1/2) has slope −3/(4log2). The plots in the lower row of Figure 8 show the solution, plotted for y = 0, in the corresponding frames (n, x − log(2)n, 0) (left) and (n, x − 4log(2)n/3, 0) (right), the logarithmic paths of the solitons being clearly visible.

The plots in the upper row are plotted in (n, x − 7log(2)n/6, 0), where −7log(2)/6 is the arithmetic middle of −log(2) and −4log(2)/3.

## A. Factorizing solutions of the Sylvester equation AX + XB = C by Vandermonde matrices

Let A = diag{α1,...,αm}, B = diag{β1,...,βn} be diagonal matrices, not necessarily of the same size, with 0 ∉ spec(A) + spec(B). Note that this spectral condition guarantees the invertibility of the elementary operator

ΨA,B(X)=AX+XB(A.1)
on the m × n-matrices over ℂ.

In what follows we give a factorization of ΨA,B1(C) for arbitrary one-dimensional matrices C in terms of Vandermonde-type matrices. The results below generalize work in [30] for A, B of the same size.

To the data given above, we assign the m × n-Vandermonde matrix VA,n and the n × n-matrix WB as follows

VA,n=(αij1)i=1,,mj=1,,n=(1α1α1n11αmαmn1)
and
WB=(κ1<<κi1j{κ1,,κi1}βκ1βκi1)i,j=1n=(11j1βjjnβjj<jj,j1βjβjj<jj,jnβjβjj1βjjnβj).

Furthermore, we introduce the involution Jn=(0110) of size n × n.

### Lemma A.1.

The following identity holds:

VA,nJnWB=(κ=1κjn(αi+βκ))i=1,,mj=1,,n.

### Proof.

By direct verification, VA,nJn=(αinj)i=1,,mj=1,,n.

Hence, denoting the ij-th entry of a matrix T as usual by Tij, matrix calculation shows

(VA,nJnWB)ij=αin1+αin2κjβκ+αin3κ,κjβκβκ++κjβκ=κj(αi+βκ).

As a consequence, VA,nJnWB is a solution of the Sylvester equation AX + XB = C for an appropriately chosen one-dimensional right-hand side C.

### Corollary A.2.

It holds that A(VA,nJnWB)+(VA,nJnWB)B=D(fmfnt), where fk ∈ ℂk is the vector with all entries equal to 1 and

D=diag{κ=1n(αi+βκ)|i=1,,m}.(A.2)

In summary, we have shown the following factorization result.

### Theorem A.3.

Let A = diag{α1,...,αm} and B = diag{β1,...,βn} such that 0 ∉ spec(A)+spec(B). Then, for any b ∈ ℂn, c ∈ ℂm, the following factorization holds:

ΨA,B1(cbt)=DcD1(VA,nJnWB)Db,
where, for f=(fκ)κ=1kk, Df denotes the k × k-diagonal matrix with the entries f1,..., fk on the diagonal and D is given in (A.2).

### Proof.

From Corollary A.2 we have VA,nJnWB=ΨA,B1(D(fmfnt)). Since D commutes with A, it immediately follows that D1(VA,nJnWB)=ΨA,B1(fmfnt), and, similarly,

DcD1(VA,nJnWB)Db=ΨA,B1(Dc(fmfnt)Db)=ΨA,B1(cbt).

The following variant of the Theorem A.3 for the elementary operator ΦA,B defined in (3.4) is used in Subsection 4.2.

### Proposition A.4.

Let 0 < p1 < ··· < pM, and let A = diag{pN+1,..., pM} and B = diag{1/p1,..., 1/pN} with N < M. Furthermore, set

KN=(pij1)i,j=1N,KMN=(pN+i1j1)i=1,,MNj=1,,N.

Then

KMNKN1=ΦA,B1(C)
for an appropriately chosen one-dimensional matrix C.

For the proof it is useful to express the inverse of a Vandermonde matrix in the notation at hand. Note that this is a direct consequence of Lemma A.1 for n = m and B = −A.

### Corollary A.5

Let the αj be pairwise different. Then

where DA=diag{κi(αi+ακ)|i=1,,m}.

### Proof ofProposition A.4.

Note first that, in the notation of this section, we have m = MN, n = N, and KMN = VA and KN = VB−1.By Corollary A.5,

(VB1)1=JNWBD11,whereD1=diag{κ=1κiN(pipκ)|i=1,,N}.

Hence KMNKN1D1=VAJNWB1, and by Corollary A.2 the latter is a solution of the Sylvester equation

AX+X(B1)=D2(fMNfNt),whereD2=diag{κ=1N(pN+i+1/pκ)|i=1,,MN}.

Since B and D1 commute, it follows that KMNKN1 solves

AXBX=D2(fMNfNt)(D1B)1=(D2fMN)((D1B)1fNt)
with the right-hand side being one-dimensional.

## B. Classification of line solitons

For the reader’s convenience, we rehearse results from [4], where a classification of the solutions (4.6) under Assumptions (1)–(4) is given. More precisely, their asymptotic behaviour for n → ±∞ at a fixed time y is obtained in dependence of the coefficient matrix C. Note that the asymptotics is essentially independent of the particular value of y.

Recall that a single line soliton is determined by two of the parameters 0 < p1 < ··· < pM (up to a possible initial shift), and we use the abbreviation [i, j] for the soliton determined by pi < pj. Note that the pi are assumed to be generic in an appropriate sense, see [4]. Line solitons appearing asymptotically for n → −∞ are called incoming, for n → ∞ outgoing line solitons.

1. 1.

There are precisely N incoming line solitons, which are characterised by [i1,j1],,[iN,jN] (with ik<jk), where i1,,iN are the numbers of the pivot columns of C.

2. 2.

There are precisely MN outgoing line solitons, which are characterised by [i1+,j1+],,[iM+N+,jNM+] (with ik+<jk+), where j1+,,jNM+ are the numbers of the non-pivot columns of C.

To identify the incoming and outgoing line solitons completely, we introduce the matrices Cij± build from the following columns of C:

• columns no. 1,..., i − 1 and j + 1,..., M for Cij,

• columns no. i + 1,..., j − 1 for Cij+.

Observe that we admit empty matrices with rank 0. The following statements hold:

1. 1.

If [i, j] is an incoming line soliton, then

1. a)

rank(Cij)<N,

2. b)

augmenting Cij by (i) column no. i, (ii) column no. j or (iii) columns no. i and j from C, increases the rank by 1.

2. 2.

If [i, j] is an outgoing line soliton, then

1. a)

rank(Cij+)<N,

2. b)

augmenting Cij+ by (i) column no. i, (ii) column no. j or (iii) columns no. i and j from C, increases the rank by 1.

For the justification of the above statements we refer to [4]. Here we confine ourselves to mentioning that the above conditions yield the asymptotic behaviour of collisions of two solitons (i.e. solutions with 2 incoming and 2 outgoing solitons) in Table 1.

type asymptotic line solitons
n → −∞ n → ∞
ordinary (O) [1, 2], [3, 4] [1, 2], [3, 4]
asymmetric (A) [1, 4], [2, 3] [2, 3], [1, 4]
resonant (R) [1, 3], [2, 4] [1, 3], [2, 4]
I [1, 3], [3, 4] [1, 2], [2, 4]
II [1, 2], [2, 4] [1, 3], [3, 4]
III [1, 3], [2, 4] [2, 3], [1, 4]
IV [1, 4], [2, 3] [1, 3], [2, 4]
Table 1.

Summary of the asymptotics of (elastic and inelastic) 2-soliton solutions.

## C. The two-dimensional Toda lattice in bilinear form

The aim of this appendix is to present a solution formula for the bilinear form of the 2d-Toda lattice

τn2τnyxτnyτnx=τn+1τn1τn2.(C.1)
(see [13] and references therein) in terms of a generalized determinant. Note that the dependent variable transformation
vn=τn+1τn1
maps solutions τn(x, y) of (C.1) to solutions of the 2d Toda lattice (1.3).

Our main result is

### Proposition C.1.

Let E, F be a Banach spaces, A(E), B(F) invertible, and 𝒜 a quasi-Banach operator ideal admitting a continuous determinant δ. Let Ln = Ln(x, y) ∈ 𝒜(F, E), Mn = Mn(x, y) ∈ 𝒜(E, F) be operator-functions with values in 𝒜, which satisfy the base equations (2.1) and the one-dimensionality condition (3.1) with some bF′. Then

τn=δ(IF+MnLn)
is a solution of (C.1) on Ω = {(n, x, y) ∈ 𝕑 × ℝ2 | τn(x, y) ≠ 0, τn−1(x, y) ≠ 0}.

### Proof.

Let τ be the trace associated to the determinant δ according to the trace-determinant theorem [24]. Recall that for a smooth operator-function ξT(ξ), with values in the endomorphisms on some Banach space, the following derivation rule holds [24]

(ddξδ(I+T))/δ(I+T)=τ((I+T)1ddξT).(C.2)

We will verify that τn solves

y((τnx)/τn)=(τn+1τn)/(τnτn1)1,(C.3)
which implies (C.1). Let us start with the left-hand side of (C.3). Application of (C.2), the base equations (2.1), and linearity of the trace yield
(τnx)/τn=τ((IF+MnLn)1x(MnLn))=τ((IF+MnLn)1(B1Mn+MnA)Ln)=τ((IF+MnLn)1MnALn)τ((IF+MnLn)1B1MnLn).

Using the trace property τ(ST) = τ(TS) we get τ((IF + MnLn)−1B−1MnLn) = τ(MnLn(IF + MnLn)−1B−1) = τ((IF + MnLn)−1MnLnB−1), which gives

(τnx)/τn=τ((IF+MnLn)1Mn(ALnLnB1))=τ(B1Vn),
where Vn is the operator-soliton defined in (2.2). By continuity of τ, we therefore find
y((τnx)/τn)=τ(B1yVn),(C.4)

Next we turn to the right-hand side of (C.3). Using again the base equations (2.1), we find

τn+1τn=δ(IF+Mn+1Ln+1)δ(IF+MnLn)=δ(IF+BMnALn)δ(IF+MnLn)=δ(IF+MnALnB)δ(IF+MnLn)=δ((IF+MnLn)1(IF+MnALnB))=δ((IF+MnLn)1((IF+MnLn)+(MnALnBMnLn)))=δ(IF+(IF+MnLn)1Mn(ALnBLn))=δ(IF+(IF+MnLn)1Mn(ALnLnB1)B)δ(IF+B(IF+MnLn)1Mn(ALnLnB1))=δ(IF+Vn).

Hence, the right-hand side of (C.3) becomes

(τn+1τn)/(τnτn1)1=δ(IF+Vn)δ(IF+Vn1)1=δ((IF+Vn1)1(IF+Vn))1=δ((IF+Vn1)1((IF+Vn1)+(VnVn1)))1=δ(IF+(I+Vn1)1(VnVn1))1.

By (3.1), we know that Vn = B(IF + MnLn)−1(bdn) with some vector function dn, i.e. Vn = bvn for vn = B(IF +MnLn)−1dn. In particular, also (IF + Vn)−1(VnVn−1) = b ⊗ ((IF + Vn)−1(vnvn−1)) is one-dimensional. Using that δ(I + T) = 1 + τ(T) holds for one-dimensional operators T, we conclude

(τn+1τn)/(τnτn1)1=τ((IF+Vn1)1(VnVn1)).(C.5)

Comparing (C.5) with (C.4), we see that it is sufficient to show

B1yVn=(IF+Vn1)1(VnVn1).

To this end we fall back on some identities from the proof of Theorem 2.1. together with the following identity derivedh from Appendix D,

yT2+(n)=S2(n)S1+(n).(C.6)

Namely, we get

B1yVn=B1y(IF+Vn)=(2.4c)yT2+(n)=(C.6)S2(n)S1+(n)=(2.7)(IF+Vn1)1(VnVn1).

This completes the proof.

### Remark C.2.

If Ω is dense in 𝕑 × ℝ2, Proposition C.1 yields that τn satisfies (C.1) everywhere by continuity. This will be our way to verify (C.1) in applications.

## D. A toolbox of operator identities

The operator identities collected in the subsequent lemmata origin from and extend results in [30, Chapter 3]. It is worth mentioning that they also are crucial in the operator treatment of continuous integrable systems like the KP equation.

### Lemma D.1.

Let G1, G2 be Banach spaces, and T1(G2, G1), T2(G1, G2) bounded linear operators such that IG1 +T1T2 (and hence also IG2 +T2T1) is invertible. Then the following identities hold

1. (a)

T2(IG1 + T1T2)−1 = (IG2 + T2T1)−1T2,

2. (b)

T2(IG1 + T1T2)−1T1 = IG2 − (IG2 + T2T1)−1.

### Proof.

Assume that IG1 + T1T2 is invertible. Then

(IG2T2(IG1+T1T2)1T1)(IG2+T2T1)=IG2(IG2+T2T1)T2(IG1+T1T2)1T1(IG2+T2T1)=IG2+T2T1T2(IG1+T1T2)1(IG1+T1T2)T1=IG2+T2T1T2T1=IG2.

The same holds if one computes the product in reversed order. Hence also IG2 + T2T1 is invertible and

(IG2+T2T1)1=IG2T2(IG1+T1T2)1T1,
which gives (b).

Note that the key in the above argument was the identity

(IG1+T1T2)T1=T1(IG2+T2T1).

To verify (a), we start from the analogous identity (IG2 + T2T1)T2 = T2(IG1 + T1T2) and multiply by (IG1 + T1T2)−1 from the right, and by (IG2 + T2T1)−1 from the left.

The next lemma treats products of operators of a particular form which play a key role in establishing general solution formulas for non-commutative integrable systems such as (1.4) or the nc KP equation.

### Lemma D.2.

Let G1, G2 be Banach spaces, and L1(G2, G1), L2(G1, G2) such that IG1 + L1L2 (and hence also IG2 + L2L1) is invertible.

Let A1(G1), A2(G2) be invertible operators and f1[α](z), f2[α](z), α ∈ {a, b}, be complex functions such that the spectrum of A1 (resp. A2) is contained in the domain where f1[α] (resp. f2[α]) are holomorphic, and set

f1[c]:=f1[a]f1[b],f2[c]:=f2[a]f2[b].

Define

S1[α]=(IG1+L1L2)1(f1[α](A1)L1+L1f2[α](A2)),T1[α]=(IG1+L1L2)1(f1[α](A1)L1f2[α](A2)L2),
for α ∈ {a, b, c}, and analogously S2[α], T2[α] built in the same way but with the role of the lower indices interchanged. Then the following identities between products of these operator-functions hold
1. (a)

T1[a]S1[b]S1[a]T2[b]=S1[c],

2. (b)

T1[a]T1[b]+S1[a]S2[b]=T1[c].

### Proof.

To improve readability, we write fj[α] instead of fj[α](Aj). On application of Lemma D.1 on all boxed terms, we obtain

(f1[a]L1f2[a]L2)S1[b]=(f1[a]L1f2[a]L2)(IG1+L1L2)1(f1[b]L1+L1f2[b])=f1[a](IG1+L1L2)1f1[b]L1+f1[a](IG1+L1L2)1L1f2[b]L1f2[a]L2(IG1+L1L2)1f1[b]L1L1f2[a]L2(IG1+L1L2)1L1f2[b]=f1[a](IG1L1(IG2+L2L1)1L2)f1[b]L1+f1[a]L1(IG2+L2L1)1f2[b]L1f2[a](IG2+L2L12)1L2f1[b]L1L1f2[a](IG2(IG2+L2L1)1)f2[b]=(f1[a]f1[b]L1L1f2[a]f2[b])+(f1[a]L1+L1f2[a])(IG2+L2L1)1(f2[b]L2f1[b]L1)=(f1[c]L1+L1f2[c])+(f1[a]L1+L1f2[a])T2[b],
yielding the identity in (a) after multiplying with (IG1 + L1L2)−1 from the left, and, similarly,
(f1[a]L1f2[a]L2)T1[b]=(f1[a]L1f2[a]L2)(IG1+L1L2)1(f1[b]L1f2[b]L2)=f1[a](IG1+L1L2)1f1[b]f1[a](IG1+L1L2)1L1f2[b]L2L1f2[a]L2(IG1+L1L2)1f1[b]+L1f2[a]L2(IG1+L1L2)1L1f2[b]L2=f1[a](IG1L1(IG2+L2L1)1L2)f1[b]f1[a]L1(IG2+L2L1)1f2[b]L2L1f2[a](IG2+L2L1)1L2f1[b]+L1f2[a](IG2(IG2+L2L1)1)f2[b]L2=(f1[a]f1[b]+L1f2[a]f2[b]L2)(f1[a]L1+L1f2[a])(IG2+L2L1)1(f2[b]L2+L2f1[b])=(f1[c]L1f2[c]L2)(f1[a]L1+L1f2[a])S2[b]
applying (b), again after multiplying with (IG1 + L1L2)−1 from the left.

### Corollary D.3.

If fj[b]=σ/fj[a] with σ ∈ {±1}, we have S1[c]=0, and Lemma D.2 in particular yields

T1[a]S1[b]=S1[a]T2[b].

Next we turn to identities involving operator functions depending on a (scalar) variable ξ. Let us first recall the derivation rule for inverses.

### Lemma D.4.

Let G be a Banach space and T be an ℒ(G)-valued function depending 𝒞1-smoothly on some variable ξ. If T(ξ) is invertible for all ξ, then T −1(ξ) is 𝒞1-smooth and

ddξT1(ξ)=T1(ξ)T(ξ)T1(ξ).

Now we look at derivation rules for operator functions with a similar structure as in Lemma D.2.

### Lemma D.5.

Let G1, G2 be Banach spaces, and L1, L2 are operator-valued functions depending 𝒞1-smoothly on some variable ξ with L1(ξ) ∈ (G2, G1), L2(ξ) ∈ (G1, G2) such that IG1 +L1L2 (and hence also IG2 + L2L1) is invertible.

Let fj[a](z), fj[b](z), j = 1, 2, be complex functions, and let Aj(Gj) be invertible operators such that the spectrum of Aj is contained in the domain where fj[a], fj[b] are holomorphic.

Assume that the following base equations are satisfied

ddξLj=fj[a](Aj)Lj,
and define
S1[α]=(IG1+L1L2)1(f1[α](A1)L1+L1f2[α](A2)),T1[α]=(IG1+L1L2)1(f1[α](A1)L1f2[α](A2)L2),
for α ∈ {a, b}, and analogously S2[α], T2[α] build in the same way but with the role of the lower indices interchanged.

Then the following derivation rules hold:

1. (a)

ddξS1[b]=T1[a]S1[b],

2. (b)

ddξT1[b]=S1[a]S2[b].

### Remark D.6.

Note that if fj[a]=fj[b], we have Sj[a]=Sj[b], and Tj[a]=Tj[b] for the operator functions in the above lemma. Hence we also obtain derivation rules for Sj[a], Tj[a] as a particular case.

### Proof.

Observe first that, by the non-commutative product rule, and on use of the base equations, we have

ddξ(L1L2)=(ddξL1)L2+L1(ddξL2)=f1[a]L1L2+L1f2[a]L2=(f1[a]L1+L1f2[a])L2.

Hence, by Lemma D.4,

ddξS1[b]=(IG1+L1L2)1(ddξ(L1L2))(IG1+L1L2)1(f1[b]L1+L1f2[b])+(IG1+L1L2)1ddξ(f1[b]L1+L1f2[b])=(IG1+L1L2)1(f1[a]L1+L1f2[b])L2(IG1+L1L2)1(f1[b]L1+L1f2[b])+(IG1+L1L2)1(f1[b]f1[a]L1+f1[a]L1f2[b])=(IG1+L1L2)1((f1[a]L1+L1f2[a])L2+f1[a](IG1+L1L2))S1[b]=(IG1+L1L2)1(f1[a]L1f2[a]L2)S1[b]=T1[a]S1[b].

Second, using Lemma D.1(a) for the boxed identity, we compute

ddξT1[b]=(IG1+L1L2)1(ddξ(L1L2))(IG1+L1L2)1(f1[b]L1f2[b]L2)+(IG1+L1L2)1ddξ(f1[b]L1f2[b]L2)=(IG1+L1L2)1(f1[a]L1+L1f2[b])L2(IG1+L1L2)1(f1[b]L1f2[b]L2)+(IG1+L1L2)1(f1[a]L1f1[b]L1f2[b]f2[a]L2)=S1[a](L2(IG1+L1L2)1(f1[b]L1f2[b]L2)+f2[b]L2)=S1[a]((IG2+L2L1)1L2(f1[b]L1f2[b]L2)+f2[b]L2)=S1[a](IG2+L2L1)1(L2(f1[b]L1f2[b]L2)+(IG2+L2L1)f2[b]L2)=S1[a](IG2+L2L1)1(f2[b]L2+L2f1[b])=S1[a]S2[b],
which concludes the proof.

## Acknowledgement

The authors would like to thank the referees for careful reading and valuable remarks.

## Footnotes

a

The 1-dimensionality condition involves two further parameters, which are less significant for the dynamic behaviour and can be neglected in this discussion.

b

The lattice (1.4) differs from the one studied in [31] only by the coordinate change y ↦ −y.

c

For both (2.3a) and (2.3c), use L1(n, x, y) = Mn(x, y), L2(n, x, y) = Ln(x, y) and A1 = B, A2 = A. Then, for (2.3a) one sets

f1[a](z)=f1[b](z)=z1,f2[a](z)=f2[b](z)=z,
with which one has T2+(n)=T1[b]. By Lemma D.5 b), we get xT1[b]=S1[a]S2[b] and one identifies S1[a]=S2+(n), S2[b]=S1+(n).

For (2.3c) on the other hand, with

f1[a](z)=f2[b](z)=z,f1[b](z)=f2[a](z)=z1
we find S2+(n)=S1[b]. Hence Lemma D.5 a) gives yS1[b]=T1[a]S1[b], where T1[a]=T2(n).

Turning to (2.3b), we now use L1(n, x, y) = Ln(x, y), L2(n, x, y) = Mn(x, y), A1 = A, A2 = B, and

f1[a](z)=f2[b](z)=z1,f1[b](z)=f2[a](z)=z.

Then S1+(n)=S1[b], and again yS1[b]=T1[a]S1[b] by Lemma D.5 a), where now T1[a]=T1(n).

d

Use again L1(n, x, y) = Mn(x, y), L2(n, x, y) = Ln(x, y) and A1 = B, A2 = A. Furthermore one sets

f1[a](z)=f2[b](z)=z1,f1[b](z)=f2[a](z)=z,
such that S1[a]=S2+(n), T2[b]=T1(n). Note that f1[a]f1[b]=f2[a]f2[b]=1, hence Corollary D.3 applies, yielding S1[a]T2[b]=T1[a]S1[b]. Finally, we identify T1[a]=T2+(n), S1[b]=S2(n) .

e

Recall that all functionals β on ℂN are of the form β : xbtx for some b ∈ ℂN, and βc(x) = 〈x, βc = (btx)c = (cbt)x.

f

assuming b1c2 ≠ 0

g

Here we use the following notion of convergence, see [29]: For fixed x, fx(n) := f(n, x) is viewed as a mapping to the Riemann number sphere ^={}. Equipping ^ with the metric d(w, z) = |π−1(w) − π−1(z)|, where π denotes the stereographic projection π:S2^, we say that two functions f(n, x) and g(n, x) have the same asymptotic behaviour for n → ±∞, briefly f(n,x) ≈ g(n, x) for n±∞, if the family (fg)n:^ converges to zero as n → ±∞ uniformly with respect to the metric d.

h

Use L1(n, x, y) = Mn(x, y), L2(n, x, y) = Ln(x, y) and A1 = B, A2 = A, and

f1[a](z)=f2[b](z)=z,f2[a](z)=f1[b](z)=z1.

Then T2(+)(n)=T1[b], and Lemma D.5 b) implies yT1[b]=S1[a]S2[b], with the identifications S1[a]=S2(n) and S2[b]=S1+(n).

## References

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[30]C. Schiebold, Integrable Systems and Operator Equations, Jena, 2004. Habilitation Thesis
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 1
Pages
57 - 94
Publication Date
2019/10/25
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1683978How to use a DOI?
Open Access

TY  - JOUR
AU  - Tomas Nilson
AU  - Cornelia Schiebold
PY  - 2019
DA  - 2019/10/25
TI  - Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour
JO  - Journal of Nonlinear Mathematical Physics
SP  - 57
EP  - 94
VL  - 27
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1683978
DO  - 10.1080/14029251.2020.1683978
ID  - Nilson2019
ER  -