1 Introduction

The box–ball system (BBS) is an integrable nonlinear dynamical system that has connections to both classical and quantum integrable systems. The Korteweg–De Vries (KdV) equation is one such classical system, which describes shallow water waves in a one-dimensional channel [11]. It was shown by Kruskal and Zabusky that solutions to the KdV equation separate into solitonic waves that move with speed proportional to their amplitude and maintain their shape under collision with other solitons [14]. It was later discovered that the ultradiscretisation of these soliton solutions produces the BBS [15]. In addition to its connection to classical systems, the BBS also emerges from quantum integrable systems such as the six-vertex lattice model from statistical mechanics [1]. The symmetries of this particular system are governed by the quantum group \(U_q'(\mathfrak {sl}_2)\). Under crystallisation (the \(q \rightarrow 0\) limit), the system is frozen to the ground state and produces the BBS [7]. The position of BBS within the realm of classical and quantum integrable systems opens its analysis to a variety of methods. Moreover, the discrete nature of the system provides an important connection to combinatorics.

An important development in the analysis of the BBS comes from the construction of crystal bases by Kashiwara [9, 10]. This allowed for the reformulation of crystallisation in terms of the crystal theory of quantum affine algebras [8], leading to a crystal theoretic formulation of the BBS. This formulation utilises the ‘classical’ crystal \(B^{1,s}\), which is the crystal base of an s-fold symmetric tensor representation of \(U_q(\mathfrak {sl}_n)\) promoted to the Kirillov–Reshetikhin (KR) crystal of \(U_q'({\widehat{\mathfrak {sl}}}_n)\) [12] by adding additional crystal operators. States of the system are then defined as elements of \( (B^{1,1})^{\otimes \infty }\). The time evolution of these states utilises the existence of the combinatorial R-matrix, a unique isomorphism between the tensor product of KR crystals, \(R:B \otimes B' \rightarrow B' \otimes B\) [12]. The time evolution is given by repeated applications of this R-matrix together with a carrier. Below is an example of a BBS constructed from the KR crystal \(B^{1,1}\) in \(A_1^{(1)}\).

Example 1.1

For \(B^{1,1}\) in \(A_1^{(1)}\) with carrier \(B^{1,3}\), the time evolution of the state

$$\begin{aligned} \begin{array}{|c|} \hline 3 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 3 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 2 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \cdots \end{aligned}$$

is the state

$$\begin{aligned} \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \begin{array}{|c|} \hline 3 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 3 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 2 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 1 \\ \hline \end{array} \otimes \cdots . \end{aligned}$$

Each \(\begin{array}{|c|} \hline 1 \\ \hline \end{array}\) represents a vacuum element (i.e. an empty box). The computation of this time evolution can be represented in the following diagram

where each crossing represents the application of the R-matrix, \(R:B^{1,3} \otimes B^{1,1} \rightarrow B^{1,1} \otimes B^{1,3}\). The top row represents the initial state, the bottom row represents the state after one time evolution, and the middle represents how the carrier changes during the evolution.

Within the BBS there exist states exhibiting solitonic behaviour; that is, states containing elements of \((B^{1,1})^{\otimes d}\) within \((B^{1,1})^{\otimes \infty }\) that move with speed corresponding to their length and are stable under collisions. (This stability is called scattering.) Such elements are called solitons. These solitons are the ultradiscrete analogue of the KdV solitons. In Example 1.1, \(\begin{array}{|c|} \hline 3 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 3 \\ \hline \end{array}\otimes \begin{array}{|c|} \hline 2 \\ \hline \end{array}\) is a soliton. For more details, we refer the reader to [4, 5].

In 2001, Hikami and Inoue generalised the BBS using crystals for the general linear Lie superalgebra \(\mathfrak {gl}(m|n)\) and showed that similar solitonic behaviour existed in this supersymmetric system [6]. Work by Yamada [16] generalised the system in a different manner by considering the crystal \(B^{r,s}\) of \(U'_q({\widehat{\mathfrak {sl}}}_n)\), producing a system with r rows. This paper uses the KR crystals for \({\widehat{\mathfrak {gl}}}(m|n)\) devised by Kwon and Okado [13] to generalise the Hikami–Inoue BBS analogously to Yamada’s generalisation of the \( U_q'({\widehat{\mathfrak {sl}}}_n) \) BBS. Each \( {\widehat{\mathfrak {gl}}}(m|n) \) KR crystal is parameterised by a Young diagram Y, and the crystal is identified with the set of semistandard Young tableaux (SSYT) of shape Y. We are primarily interested in \( B^{r,s} \); the crystal of rectangular SSYT of height r and width s. In our generalised BBS, we define states as elements of \(\bigl ( B^{r,1} \bigr )^{\otimes \infty } \). We similarly have an R-matrix giving a bijection of tensor products of crystals \( B^{r_1,s_1}\otimes B^{r_2,s_2} \rightarrow B^{r_2,s_2}\otimes B^{r_1,s_1} \). The R-matrix can be explicitly calculated with the RSK algorithm, using the modified Schensted’s bumping algorithm outlined in Sect. 2.3. This allows us to define the time evolution of the system analogously to the classical case. Taking \(r =1\) reduces our system to the Hikami–Inoue BBS [6]. The following example gives a two-soliton state within our generalised system.

Example 1.2

Consider the \( U_q({\widehat{\mathfrak {gl}}}(3|1)) \) crystal \( B^{2,1} \). In the following diagram is a state (in \( (B^{2,1})^{\otimes \infty } \)) evolved over four time steps starting at time \( t=0 \). The maximal weight element ( ) is represented as a dot.

At \( t=1 \), we observe both and have moved with speed proportional to their length. They collide at times \( t=2 \) and \( t=3 \), before separating back into two solitons at \(t = 4\) (stability under collisions). This demonstrates solitonic behaviour in our generalised system.

Note that, at \( t=4 \), is one step ahead (to the right) of where would be if there had been no collision. Similarly, is one step behind where would be. This phenomenon is called the phase shift and is a shadow of the nonlinearity. The phase shift is governed by the integer-valued energy function.

Example 1.2 shows that there are objects within our system exhibiting solitonic behaviour. Additionally, we can also find coupled solitons, which move with constant speed but contain many overlapping uncoupled solitons. Conjecture 4.4 proposes that these coupled solitons can be uncoupled upon collision. Conjecture 4.7 says that, given sufficient time, every state will separate into (potentially coupled) solitons.

We prove two main theorems. Theorem 4.8 provides sufficient conditions for a soliton in a BBS state to move with speed corresponding to its length. Conjecture 4.9 claims that these conditions are also necessary. In addition, Conjecture 4.11 provides a relationship between the conditions of Theorem 4.8 and the number of uncoupled solitons within a soliton. Theorem 4.14 provides a class of solitons which satisfy the conditions of Theorem 4.8 and also maintain their shape under collision (see Example 1.2). In addition, we characterise the phase shift of these solitons in terms of the energy function and show that the solitons after collision can be computed using the combinatorial R-matrix without needing to compute the entire sequence of time evolutions. In most cases, the proof of Theorem 4.14 reduces the behaviour of the system to modified version of the height 1 system in [6]. An interesting case occurs when, in a \( U_q(\mathfrak {gl}(m|n)) \) system, the height of the solitons is equal to m; in this case, the highest weight states have a form that is not analogous to any of the highest weight states in the non-super symmetric system of [16].

The paper is organised as follows: In Sect. 2, we quickly review the crystal base theory required for our purposes, including the computation of crystal operators, the combinatorial R-matrix and the energy function. In Sect. 3, we present the explicit structure of our generalised system and outline the process of time evolution. In Sect. 4, we present our two main theorems and our conjectures. The proofs of the theorems can be found in “Appendices A and B”, respectively. “Appendices C and D” contain the proofs of some technical lemmas used in “Appendix B”.

2 Background

2.1 The Affine General Linear Lie Superalgebra

The original BBS can be derived from the affine Lie algebra \( {\widehat{\mathfrak {sl}}}_n \). Our supersymmetric BBS is instead derived from the affine general linear Lie superalgebra \({\widehat{\mathfrak {gl}}}(m|n)\) and its corresponding quantum group \(U_q({\widehat{\mathfrak {gl}}}(m|n))\) (in the sense of [13]). Let \(I = I_{\text {even}}\sqcup I_{\text {odd}}\) be the indexing set of simple roots, where \( I_{\text {even}}= \{\overline{m-1}, \ldots ,\overline{1}, 1,\ldots , n-1 \}\) and \( I_{\text {odd}}= \{0,\overline{0}\}\). It is useful to set \( I_- = \{\overline{m-1},\ldots ,\overline{1}\} \) and \( I_+ = \{1,\ldots ,n-1\} \), so that \( I_{\text {even}}= I_-\sqcup I_+ \).

The Dynkin diagram for \( {\widehat{\mathfrak {gl}}}(m|n) \) is:

figure a

where

figure b

denotes an isotropic simple root. The Dynkin diagram for the finite-dimensional \( \mathfrak {gl}(m|n) \) can be obtained from the above Dynkin diagram by removing the \( \overline{0}\) node.

The fundamental representation of \(U_q({\widehat{\mathfrak {gl}}}(m|n))\) is an \((m+n)\)-dimensional super vector space \( {\textbf{V}}= {\textbf{V}}_+ \oplus {\textbf{V}}_- \). The fundamental representation admits a crystal base \( \{v_b \mid b\in B\} \) with \( B=B_-\sqcup B_+ \) where \( B_- = \{\overline{m},\overline{m-1},\ldots , \overline{1}\} \) and \( B_+ = \{1,\ldots ,n-1,n\} \), which gives rise to the following crystal graph:

figure c

where \(\begin{array}{|c|} \hline b' \\ \hline \end{array}\xrightarrow {i}\begin{array}{|c|} \hline b \\ \hline \end{array}\) if and only if \( f_iv_{b'}=v_b \) (equivalently, \( e_i v_b = v_{b'} \)). If we instead consider the fundamental representation of the finite type \( U_q(\mathfrak {gl}(m|n)) \), the crystal graph is the same as above but without the \( \overline{0}\) arrow. We can interpret the finite type crystal graph (with arrow labels removed) as a total ordering; explicitly, \( \overline{m}< \cdots< \overline{1}< 1< \cdots < n \).

For a more detailed explanation of crystals for \( U_q({\widehat{\mathfrak {gl}}}(m|n)) \), see [13].

2.2 Finite Type Crystals and Tableaux

Now we restrict our attention to the finite type crystal. Let \({\textbf{V}}^{\otimes N}\) be the N-th tensor power of the fundamental representation of \( U_q(\mathfrak {gl}(m|n)) \). It can be shown that all tensor powers with \(N\ge 1\) are completely reducible. Moreover, the irreducible subrepresentations (up to isomorphism) are in bijection with Young diagrams of (m|n) -hook shape [2, 3]. This bijection is derived using a map from crystal base elements to semistandard Young tableaux. In this context, a tableau is called semistandard if the rows are weakly (resp. strictly) increasing for letters in \( B_- \) (resp. \( B_+ \)) and the columns are weakly (resp. strictly) increasing for letters in \( B_+ \) (resp. \( B_- \)).

We map crystal base elements to Young diagrams using a modified version of Schensted’s bumping algorithm. For inserting \( i\in B \) into a tableau \( {\mathcal {T}}\), which we will denote \( i\rightarrow {\mathcal {T}}\), the bumping algorithm is as follows:

  1. 1.

    For \( i\in B_{+} \), (resp. \( i\in B_{-} \)): if none of the boxes in the first column of \( {\mathcal {T}}\) are strictly larger than i (resp. larger than or equal to i) then add a box containing i at the bottom of the column.

  2. 2.

    Otherwise, for the topmost \(\begin{array}{|c|} \hline j \\ \hline \end{array}\) with \( j>i \) (resp. \( j\ge i \)) in the first column, replace \(\begin{array}{|c|} \hline j \\ \hline \end{array}\) with \(\begin{array}{|c|} \hline i \\ \hline \end{array}\). Then, insert j into the second column following analogous steps 1 and 2.

  3. 3.

    Repeat until the bumped number can be put in a new box.

Example 2.1

The following is an example computation of the bumping algorithm:

Let \( v_{b_1}\otimes v_{b_2} \otimes \cdots \otimes v_{b_N}\in {\textbf{V}}^{\otimes N} \) be a crystal base element with \( b_1,\ldots , b_N\in B \). The SSYT associated with this crystal base element is the insertion

$$\begin{aligned} b_N\rightarrow (\cdots \rightarrow (b_3\rightarrow (b_2\rightarrow \begin{array}{|c|} \hline b_1 \\ \hline \end{array}))\cdots ) \end{aligned}$$

which, for brevity, we will denote \(b_2 \cdots b_{N}\rightarrow \begin{array}{|c|} \hline b_1 \\ \hline \end{array}\).

Example 2.2

For \( {\textbf{V}}\) the fundamental representation of \( U_q(\mathfrak {gl}(3|5)) \), the crystal base element \( v_3\otimes v_5 \otimes v_1 \otimes v_{\overline{3}} \otimes v_2 \otimes v_{\overline{3}} \otimes v_1 \otimes v_{\overline{2}} \otimes v_{\overline{2}} \in {\textbf{V}}^{\otimes 9}\) is mapped to the tableau

Note that the map from crystal base elements to SSYT is not injective (for example, \( v_3\otimes v_2\otimes v_5 \otimes v_1\otimes v_1 \otimes v_{\overline{3}}\otimes v_{\overline{2}} \otimes v_{\overline{3}} \otimes v_{\overline{2}} \) is mapped to the same tableau in Example 2.2). However, this map sends crystal base elements of isomorphic irreducible subrepresentations to the same set of SSYT. (In particular, this map gives a bijection between irreducible subrepresentations (up to isomorphism) and Young diagrams.)

Note also that it is possible to construct a bijection between crystal base elements and ordered pairs of tableaux using the RSK algorithm (which makes use of Schensted’s bumping algorithm) [3].

Using SSYT allows us to give combinatorial descriptions of \( U_q(\mathfrak {gl}(m|n)) \) crystals. We will now restrict our attention to rectangular tableaux, but much of the discussion in this section applies more generally.

Let \( B^{r,s} \) be the set of rectangular SSYT with height r and width s. Take, an arbitrary tableau,

We define a function, \( {{\,\textrm{col}\,}}\) by reading the tableau from top-to-bottom, right-to-left; explicitly,

$$\begin{aligned} {{\,\textrm{col}\,}}({\mathcal {T}}) = \underbrace{t_{1s}\ldots t_{rs}}_{t_{*s}} \cdots \underbrace{t_{12}\ldots t_{r2}}_{t_{*2}} \underbrace{t_{11}\ldots t_{r1}}_{t_{*1}}. \end{aligned}$$

Moreover, for \( {\mathcal {T}}_1,{\mathcal {T}}_2\in B^{r,s} \), we define \( {{\,\textrm{col}\,}}({\mathcal {T}}_1\otimes {\mathcal {T}}_2)={{\,\textrm{col}\,}}({\mathcal {T}}_1){{\,\textrm{col}\,}}({\mathcal {T}}_2) \).

For \( i\in I_{\text {even}}\), the action of the crystal operators \( e_i \) and \( f_i \) can be computed by a signature rule similar to that for \( U_q'({\widehat{\mathfrak {sl}}}_n) \)-crystals [16].

Definition 2.3

For some positive integer d, let \( {\mathcal {T}}\in (B^{r,s})^{\otimes d} \) and let \( i\in I_{\text {even}}\). If \( i=\overline{k}\in I_- \), we denote \( i+1 = \overline{k+1} \). We define the i-signature, denoted \( {{\,\textrm{sg}\,}}_i({\mathcal {T}}) \), to be the sequence of \( + \) and \( - \) obtained by deleting all letters in \( {{\,\textrm{col}\,}}({\mathcal {T}}) \) that are not i or \( i+1 \), and then replacing all i with a \( - \) symbol and replacing all \( i+1 \) with a \( + \) symbol.

We define the reduced i-signature, denoted \( {{\,\textrm{rsg}\,}}_i({\mathcal {T}}) \), to be equal to the i-signature, except with \( +- \) pairs (in that order) successively deleted, so that \( {{\,\textrm{rsg}\,}}_i({\mathcal {T}}) \) is of the form

$$\begin{aligned} \underbrace{-\cdots -}_{a}\underbrace{+\cdots +}_{b} \end{aligned}$$

(where a or b can be zero).

For a tableau \( {\mathcal {T}}\in B^{r,s} \) and for \( i\in I_{\text {even}}\) where \( i\in I_+ \) (resp. \( i\in I_- \)):

  • To evaluate \( f_i({\mathcal {T}}) \) (resp. \( e_i({\mathcal {T}}) \)), find the rightmost \( - \) symbol in \( {{\,\textrm{rsg}\,}}_i({\mathcal {T}}) \) and change the corresponding in \( {\mathcal {T}}\) to . If there are no \( - \) symbols, then \( f_i({\mathcal {T}})=0 \) (resp. \( e_i({\mathcal {T}})=0 \)).

  • To evaluate \( e_i({\mathcal {T}}) \) (resp. \( f_i({\mathcal {T}}) \)), find the leftmost \( + \) symbol in \( {{\,\textrm{rsg}\,}}_i({\mathcal {T}}) \) and change the corresponding in \( {\mathcal {T}}\) to . If there are no \( + \) symbols, then \( e_i({\mathcal {T}})=0 \) (resp. \( f_i({\mathcal {T}})=0 \)).

The \( f_0 \) and \( e_0 \) operators have a different algorithm:

  • If the first occurrence of \( \overline{1}\) in \( {{\,\textrm{col}\,}}({\mathcal {T}}) \) is before the first occurrence of 1, then \( e_0({\mathcal {T}})=0 \) and \( f_0({\mathcal {T}}) \) replaces the corresponding in \( {\mathcal {T}}\) with .

  • If the first occurrence of 1 in \( {{\,\textrm{col}\,}}({\mathcal {T}}) \) is before the first occurrence of \( \overline{1}\), then \( f_0({\mathcal {T}})=0 \) and \( e_0({\mathcal {T}}) \) replaces the corresponding in \( {\mathcal {T}}\) with .

Example 2.4

We will compute \( e_{\overline{3}}({\mathcal {T}}) \) for

We have that

$$\begin{aligned} \begin{array}{rcccccccccc} {{\,\textrm{col}\,}}({\mathcal {T}}) &{} = &{} \overline{3}&{} 3 &{} 3 &{} \overline{3}&{} 1 &{} 2 &{} \overline{4}&{} \overline{3}&{} 1\\ {{\,\textrm{sg}\,}}_{\overline{3}}({\mathcal {T}}) &{} = &{} - &{}&{}&{} - &{}&{}&{} + &{} - &{}\\ {{\,\textrm{rsg}\,}}_{\overline{3}}({\mathcal {T}}) &{} = &{} - &{}&{}&{} - &{}&{}&{}&{}&{} \end{array}. \end{aligned}$$

The rightmost \( - \) corresponds to the bolded number below,

so we replace this with to get

We can also use SSYT to describe the weights (in the representation theoretic sense) of the crystal elements. Weights are linear combinations in the set \( \bigoplus _{b\in B}{\mathbb {Z}}\varepsilon _b \) (for our purposes, \( \varepsilon _b \) can be treated as formal symbols). In the weight of a SSYT \( {\mathcal {T}}\), the coefficient corresponding to \( \varepsilon _b \) is equal to the number of appearances of b in \( {\mathcal {T}}\) [2].

Example 2.5

Let \( {\mathcal {T}}\) be as in Example 2.4. Then, the weight of \( {\mathcal {T}}\) is \( \varepsilon _{\overline{4}}+3\varepsilon _{\overline{3}}+2\varepsilon _1+\varepsilon _2+2\varepsilon _3 \).

We define arbitrary weights \(\mu \) and \(\nu \) as follows:

$$\begin{aligned} \mu&= \mu _1\varepsilon _{\overline{m}}+\cdots +\mu _m\varepsilon _{\overline{1}}+\mu _{m+1}\varepsilon _1+\cdots +\mu _{m+n}\varepsilon _n\,,\\ \nu&= \nu _1\varepsilon _{\overline{m}}+\cdots +\nu _m\varepsilon _{\overline{1}}+\nu _{m+1}\varepsilon _1+\cdots +\nu _{m+n}\varepsilon _n \,. \end{aligned}$$

We can define a partial ordering on the set of weights by saying \( \mu \ge \nu \) if the following hold:

$$\begin{aligned} \mu _1+\cdots +\mu _{m+n}&=\nu _1+\cdots +\nu _{m+n} \\ \mu _1+\cdots +\mu _j&\ge \nu _1+\cdots +\nu _j{} & {} \text {for all}~j=1,\ldots ,m+n. \end{aligned}$$

Note that the operators \( e_i \) (\( i\in I\setminus \{\overline{0}\} \)) raise the weight and the operators \( f_i \) (\( i\in I\setminus \{\overline{0}\} \)) lower the weight. We say that \( {\mathcal {T}}\) is a highest weight element if \( e_i {\mathcal {T}}= 0 \) for all \( i\in I\setminus \{\overline{0}\} \).

Definition 2.6

A crystal element \( {\mathcal {T}}\) with weight \( \lambda \) is a genuine highest weight element if

  1. (i)

    given any other crystal element with some weight \( \mu \), the expression \( \lambda -\mu \) has only positive coefficients; and

  2. (ii)

    no other crystal element has weight \( \lambda \).

Every genuine highest weight element is a highest weight element, but not every highest weight element is a genuine highest weight element [2].

For crystals whose elements are the SSYT of the same shape, the genuine highest weight element exists and is unique [2]. Each connected component of \( B^{r_1,s_1}\otimes B^{r_2,s_2} \) is isomorphic to such a crystal (this isomorphism is given by \( ({\mathcal {T}}_1\otimes {\mathcal {T}}_2) \mapsto ({{\,\textrm{col}\,}}({\mathcal {T}}_2)\rightarrow {\mathcal {T}}_1) \) for \( {\mathcal {T}}_1\otimes {\mathcal {T}}_2 \) in the connected component of interest). Thus, each connected component of \( B^{r_1,s_1}\otimes B^{r_1,s_1} \) has a unique genuine highest weight element. This property is of great utility in the proofs of the main theorems.

2.3 Combinatorial R-Matrix

Consider two \(U_q({\widehat{\mathfrak {gl}}}(m|n))\)-crystals \(B^{r_1,s_1}\) and \(B^{r_2,s_2}\). Then there exists [13] a unique isomorphism that commutes with \( e_i \) and \( f_i \) (for all \( i\in I \)) called the combinatorial R-matrix:

$$\begin{aligned} R:B^{r_1,s_1} \otimes B^{r_2,s_2} \rightarrow B^{r_2,s_2} \otimes B^{r_1,s_1}. \end{aligned}$$

We can describe the action of the combinatorial R-matrix using Schensted’s bumping algorithm.

Theorem 2.7

[13, Theorem 7.9]. The combinatorial R-matrix maps \( {\mathcal {T}}_1\otimes {\mathcal {T}}_2 \) to \( {\widetilde{{\mathcal {T}}}}_2\otimes {\widetilde{{\mathcal {T}}}}_1 \) if and only if \( {{\,\textrm{col}\,}}({\mathcal {T}}_2)\rightarrow {\mathcal {T}}_1 = {{\,\textrm{col}\,}}({\widetilde{{\mathcal {T}}}}_1)\rightarrow {\widetilde{{\mathcal {T}}}}_2 \)

Example 2.8

Set

Then, \( R({\mathcal {T}}_1\otimes {\mathcal {T}}_2)={\widetilde{{\mathcal {T}}}}_2\otimes {\widetilde{{\mathcal {T}}}}_1 \). Indeed, let us first compute

We similarly find that

There is a more explicit method of determining the R-matrix of two tableaux. This involves an inversion of the modified bumping algorithm outlined previously. Let \({\mathcal {T}}_1 \in B^{r_1,s_1}\) and \({\mathcal {T}}_2 \in B^{r_2,s_2}\). Then, \(R({\mathcal {T}}_1 \otimes {\mathcal {T}}_2)\) is determined by the following process. Begin with \( P = {{\,\textrm{col}\,}}({\mathcal {T}}_2) \rightarrow {\mathcal {T}}_1\). Let \({\widehat{Q}}\) be a rectangular reverse semi-standard tableau of height \( r_1 \) and width \( s_1 \). We construct \({\widehat{Q}}\) using the weight vector given by \(\mu = ({\widetilde{d}}_i - d_i, \cdots , {\widetilde{d}}_s - d_s)\) where \(\widetilde{d_i}\) and \(d_i\) are the heights of the ith column of P and \({\mathcal {T}}_2\), respectively, and s is the width of P. Then, \({\widehat{Q}}\) is given by the unique reverse conjugate semi-standard tableau of shape \({\mathcal {T}}_1\) and weight \(\mu \).

We then perform a reverse insertion on P by reading \({\widehat{Q}}\) bottom-to-top, left-to-right. Each element in the reading of \({\widehat{Q}}\) gives the next column on which we perform the bumping algorithm in reverse. The elements removed from P are placed bottom-to-top and left-to-right into a rectangular tableau of height \( r_1 \) and width \( s_1 \). This tableau is \({\widetilde{{\mathcal {T}}}}_1\). Continuing until \({\widehat{Q}}\) is empty, we obtain the resultant tableaux as \(P = {\widetilde{{\mathcal {T}}}}_2\) and \({\widetilde{{\mathcal {T}}}}_1\).

Example 2.9

This example will demonstrate the explicit R-matrix computation for \({\mathcal {T}}_1\) and \({\mathcal {T}}_2\) defined as follows

From Example 2.8 we find that

The weight vector associated with \({\widehat{Q}}\) is then given by (2, 3, 3, 1). There exists a unique reverse conjugate semi-standard tableau of shape \({\mathcal {T}}_1\) and weight (2, 3, 3, 1) given as follows,

Now reading \({\widehat{Q}}\) from bottom-to-top, left-to-right. We first read the element 3, beginning in column 3 of P we pop the final row element 3, then performing reverse insertion from column 2, this 3 switches with the 2 which then subsequently switches with 1 in column 1. The 1 left over then begins filling a tableau \({\widetilde{{\mathcal {T}}}}_1\) bottom-to-top, left-to-right. We are then left with the following P and \({\widehat{Q}}\), and \({\widetilde{{\mathcal {T}}}}_1\):

Continuing this process until \({\widehat{Q}}\) is empty we obtain the following tableaux

These are indeed the resultant tableaux satisfying the R-matrix as shown in Example 2.8.

2.4 Energy Function

Definition 2.10

We call a function \( H:B^{r_1,s_1}\otimes B^{r_2,s_2} \rightarrow {\mathbb {Z}}\) an energy function if, for all \( b=x\otimes y \in B^{r_1,s_1}\otimes B^{r_2,s_2} \), we have \( H(f_i b)=H(b) \) and \( H(e_i b)=H(b) \) for \( i\in I\setminus \{\overline{0}\} \), and if \( {\widetilde{y}}\otimes {\widetilde{x}} = R(b) \) then

$$\begin{aligned} H(e_{\overline{0}}b) = H(b) + {\left\{ \begin{array}{ll} 1 &{} \text {if}~e_{\overline{0}}b = (e_{\overline{0}}x)\otimes y~ \text {and}~e_{\overline{0}}R(b)=(e_{\overline{0}}{\widetilde{y}})\otimes {\widetilde{x}},\\ -1 &{} \text {if}~e_{\overline{0}}b = x\otimes (e_{\overline{0}}y)~ \text {and}~e_{\overline{0}}R(b)={\widetilde{y}}\otimes (e_{\overline{0}}{\widetilde{x}}),\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

The energy function exists and is unique up to additive constant [13]. Moreover, we can compute the energy function using the bumping algorithm.

Proposition 2.11

[13, Theorem 7.9]. Up to additive constant, \( H(x\otimes y) \) is given by the number of boxes in \( {{\,\textrm{col}\,}}(y)\rightarrow x \) that are strictly to the right of the \( \max (s_1,s_2) \)-th column.

By convention, we will choose the additive constant so that the maximum value of H is zero. Explicitly, if \( {\widetilde{H}}(x\otimes y) \) is given by the number of boxes as in Proposition 2.11, with additive constant equal to 0, then we define \( H(x\otimes y) = {\widetilde{H}}(x\otimes y) - \min (r_1,r_2)\min (s_1,s_2). \)

Example 2.12

Set x and y as in Example 2.8. We know that

We have that \( \max (s_1,s_2)=\max (3,1)=3 \), and the number of boxes to the right of the third column is 1. So, \( H(x\otimes y) = 1 - \min (r_1,r_2)\min (s_1,s_2) = -2 \).

3 Super Box–Ball System

3.1 Box–Ball System Definition

A box–ball system possesses a vacuum element representing the absence of a ball. We require that the combinatorial R-matrix acts as an identity on the vacuum element; that is, if u is the vacuum element then \(R(u \otimes u) = u\otimes u\).

We define the vacuum element to be the genuine highest weight element of \( B^{r,1} \) as a finite-type \( U_q(\mathfrak {gl}(m|n)) \)-crystal where \( r\le m \) (see [2]). Such a vacuum element will have the desired property. More generally, the genuine highest weight element for \( B^{r,s} \) has the form

The vacuum element is then denoted by \( u_1 \).

We can think of the elements of \( B^{r,1} \setminus \{u_1\} \) as representing different balls in the system. Within the super box–ball system, a state consists of \(B^{r,1}\) elements in a one-dimensional lattice with only finitely many non-vacuum elements. More precisely, a state is of the form

$$\begin{aligned} b_0 \otimes b_1 \otimes \cdots \otimes b_K \otimes (u_1)^{\otimes \infty }\in (B^{r,1})^{\otimes \infty } \end{aligned}$$

where \( b_j \in B^{r,1} \) can be any element (including \( u_1 \)).

The state evolves in time via a carrier element which ‘picks up’ and ‘puts down’ \( B^{r,1} \) elements. The carrier is an element of \( B^{r,\ell } \) which changes based on its location in the state. It is initialised as the genuine highest weight element \( u_{\ell } \). The action of moving the carrier through the state is performed by the combinatorial R-matrix. In particular, this is performed by functions \( R_a \) where

$$\begin{aligned} R_a = \underbrace{{{\,\textrm{id}\,}}\otimes \cdots \otimes {{\,\textrm{id}\,}}}_a\otimes R\otimes {{\,\textrm{id}\,}}\otimes {{\,\textrm{id}\,}}\otimes \cdots . \end{aligned}$$

We can then define the time evolution operator, \( T_{\ell } \), by

$$\begin{aligned} T_{\ell }(b)\otimes u_{\ell }=\cdots R_3R_2R_1R_0(u_{\ell }\otimes b) \end{aligned}$$

for any state b. This is well-defined because there are finitely many non-vacuum elements in the state, so we eventually have \( R(u_{\ell }\otimes u_1)=u_1\otimes u_{\ell } \). The time evolution operator computes the state for the next time step. For convenience, we will write \( T_{\infty }=\lim _{\ell \rightarrow \infty }T_{\ell } \).

Proposition 3.1

\( T_{\infty } \) is well-defined

Proof sketch

A simple insertion argument shows that the action of the R-matrices in the definition of \( T_{K} \) and \( T_{\ell } \) are the same (under the inclusion ). This shows that \( T_{\ell }=T_{\ell '} \) for \( \ell ,\ell '\ge K \) and hence \( T_{\infty } \) is well-defined. \(\square \)

Pictorially, we can represent the computation of the time evolution \( T_\ell (b_1\otimes \cdots \otimes b_K\otimes (u_1)^{\otimes \infty }) = \bigotimes _{j=1}^\infty {\widetilde{b}}_j \) as follows:

where \( R(u_\ell ^{(j)}\otimes b_{j+1}) = {\widetilde{b}}_{j+1}\otimes u_\ell ^{(j+1)} \).

Example 3.2

For \( U_q({\widehat{\mathfrak {gl}}}(3|3)) \) crystals,

That is,

Remark 3.3

We only consider BBSs with \( r\le m \). This is because we encounter difficulties if we consider \( r>m \). For instance, the empty carrier \( u_\ell \) will contain fermionic boxes which will increase in value horizontally. So, we can no longer think of \( u_\ell \) as containing \( \ell \) vacuum elements. More concerning, the time evolution operator may no longer be well-defined. Consider a BBS defined from a \( U_q({\widehat{\mathfrak {gl}}}(1|3)) \)-crystal with \( r=2 \) and

Then,

which is not of the form

that is required for our above definition of \( T_2(p)\). Additionally, we do not want to define \( T_2(p) \) to be

because the boxes inside \( T_2(p) \) are different from p; intuitively, the ‘mass’ is no longer a conserved quantity. This indicates that other conserved quantities may not be present, which impacts the integrability of the system. Nevertheless, such a system with \( r>m \) may exhibit interesting behaviour or applications, but is beyond the scope of this article.

Remark 3.4

Many authors will define a state so that the tensor product extends infinitely in both directions (with only finitely many non-vacuum states). If we define a state in this way, then the system is time reversible, since the uniqueness of the combinatorial R-matrix implies \( R^{-1}_a = R_a \).

3.2 Properties of the Time Evolution Operator

Proposition 3.5

Time evolution operators commute: \( T_{\ell }T_{\ell '}(p) = T_{\ell '}T_{\ell }(p) \).

The proof of this fact is identical to [4, Theorem 3.1] and relies on the Yang–Baxter equation: \( (R\otimes 1)(1\otimes R)(R\otimes 1) = (1\otimes R)(R\otimes 1)(1\otimes R) \). The Yang–Baxter equation is proved for \( U_q({\widehat{\mathfrak {gl}}}(m|n)) \)-crystals in [13, Theorem 7.11].

The time evolution operator also respects the crystal structure, i.e. \( T_{\ell } \) commutes with some of the crystal operators, as outlined in the following Lemma:

Lemma 3.6

For all \( i\in I\setminus \{\overline{0},\overline{m-r}\} \), and for a state p, we have that \( T_{\ell }(e_i(p))=e_i(T_{\ell }(p)) \) and \( T_{\ell }(f_i(p))=f_i(T_{\ell }(p)) \).

The proof is similar to [16, Lemma 2.8]. Let \( {\mathcal {B}}_{\overline{0},\overline{m-r}}\) be the \( U_q({\widehat{\mathfrak {gl}}}(m|n)) \)-crystal of BBS states where the operators \( f_{\overline{0}},\, e_{\overline{0}},\,f_{\overline{m-r}} \) and \( e_{\overline{m-r}} \) have been removed. Note that \( {\mathcal {B}}_{\overline{0},\overline{m-r}}\) is isomorphic to a \( U_q(\mathfrak {gl}(r))\otimes U_q(\mathfrak {gl}(m-r|n)) \)-crystal. Lemma 3.6 allows us to prove results by only considering a single element from each connected component of \( {\mathcal {B}}_{\overline{0},\overline{m-r}}\). In practice, this means that it is sufficient to consider genuine highest weight elements of \( {\mathcal {B}}_{\overline{0},\overline{m-r}}\).

4 Solitons

4.1 Properties of Solitons and Coupled Solitons

In this paper, a soliton is an element of \( (B^{r,1} {\setminus } \{u_1\})^{\otimes d} \) that moves with constant speed (not necessarily equal to its length).

Definition 4.1

We call an element \( v\in (B^{r,1} \setminus \{u_1\})^{\otimes d} \) a soliton if \( T_{\infty }(v\otimes u_1^{\otimes \infty }) = u_1^{\otimes c} \otimes v \otimes u_1^{\otimes \infty } \) for some positive integer c. We call c the speed of the soliton.

This definition is very broad, and these solitons do not satisfy many of the properties we want. However, this broad definition is convenient for our purposes. Many of the important properties will be satisfied by a specific type of soliton (uncoupled solitons) which is defined by conserved quantities \( N_{\ell } \).

Let \( p = p_1\otimes p_2 \otimes p_3 \otimes \cdots \) be a state. Let \( u_{\ell }^{(j)} \) be the carrier after applying the R-matrix j times; that is

$$\begin{aligned} R_{j-1}\cdots R_1R_0(u_{\ell }\otimes p) = {\widetilde{p}}_1\otimes \cdots \otimes {\widetilde{p}}_j \otimes u_{\ell }^{(j)}\otimes p_{j+1}\otimes \cdots . \end{aligned}$$

Define a function \( E_{\ell } \) by

$$\begin{aligned} E_{\ell }(p) = -\sum _{j=1}^{\infty }H(u_{\ell }^{(j-1)}\otimes p_j) \end{aligned}$$

where H is the energy function (note that the above sum is finite because we chose H such that \( H(u_{\ell }\otimes u_1) = 0 \)).

Proposition 4.2

For each \( \ell \), the number \( E_{\ell }(p) \) is a conserved quantity: \( E_{\ell }(T_{\ell '}(p)) = E_{\ell }(p) \) for every positive integer \( \ell ' \).

The proof of this proposition is the same as [4, Theorem 3.1]. Define \( N_{\ell } \) by

$$\begin{aligned} N_{\ell } = - E_{\ell - 1} + 2E_{\ell } - E_{\ell + 1} \end{aligned}$$

with \( E_0 = 0 \). We can now use \( N_{\ell } \) to define uncoupled solitons.

Definition 4.3

Let v be a soliton. If there exists a positive integer s for which \( N_{s}(v\otimes u_1^{\otimes \infty }) = 1 \) and \( N_{j}(v\otimes u_1^{\otimes \infty }) = 0 \) for all \( j\ne s \), then we call v an uncoupled soliton. Otherwise, we call v a coupled soliton.

Note that \( s \le d \). Indeed, the proof of Proposition 3.1 shows that \( T_{\ell } = T_{\ell '} \) for \( \ell ,\ell ' \ge d \). In fact, the same insertion argument shows that \( E_{\ell } = E_{\ell '} \) for \( \ell ,\ell '\ge d \) and hence \( N_{j} = 0 \) for \( j > d \). Moreover, we believe that \( s=d \) for every uncoupled soliton (see the discussion after Conjecture 4.11) so we can interpret s as the length of the uncoupled soliton.

Intuitively, a coupled soliton contains overlapping uncoupled solitons that are not interacting with one another. With this intuition in mind, we can interpret \( N_{\ell } \) as the total number of uncoupled solitons of length \( \ell \) in a state. With this interpretation, we should expect that a coupled soliton has \( N_s\ne 0 \) for exactly one positive integer s. Indeed, were this not the case, then a coupled soliton would contain overlapping uncoupled solitons of different speeds, and we would expect these uncoupled solitons to separate given enough time (contradicting the fact that the coupled soliton is a soliton—Definition 4.1).

We formalise this intuitive interpretation of \( N_{\ell } \) in the following conjecture, which claims that a coupled soliton v can be split into uncoupled solitons after collision.

Conjecture 4.4

Let v be a coupled soliton of speed s. There exist positive integers \( {\widetilde{t}}, A, c_1,\ldots , c_A \) and uncoupled solitons \( w_1,\ldots , w_A \) of speeds \( d_1\le \ldots \le d_A \) (respectively) greater than s such that if \( t>{\widetilde{t}} \) then

$$\begin{aligned}&(T_{\infty })^t(w_1\otimes u_1^{\otimes c_1}\otimes \cdots \otimes w_A\otimes u_1^{\otimes c_A} \otimes v \otimes u_1^{\otimes \infty })\\&\quad = u_1^{\otimes {\widetilde{c}}_1+st} \otimes {\widetilde{v}}_1 \otimes u_1^{\otimes {\widetilde{c}}_2} \otimes {\widetilde{v}}_2 \otimes \cdots \otimes u_1^{\otimes {\widetilde{c}}_B} \otimes {\widetilde{v}}_B\\&\qquad \otimes u_1^{\otimes {\widetilde{c}}_{B+1}+(d_1-s)t} \otimes {\widetilde{w}}_1 \otimes u_1^{\otimes {\widetilde{c}}_{B+2}+(d_2-d_1)t}\\&\qquad \otimes {\widetilde{w}}_2 \otimes \cdots \otimes u_1^{\otimes {\widetilde{c}}_{B+A}+(d_A-d_{A-1})t} \otimes {\widetilde{w}}_A \otimes u_1^{\otimes \infty } \end{aligned}$$

for some uncoupled solitons \( {\widetilde{v}}_1,{\widetilde{v}}_2,\ldots ,{\widetilde{v}}_B,{\widetilde{w}}_1,\ldots ,{\widetilde{w}}_A \), where \( B = N_{s}(v \otimes u_1^{\otimes \infty }) \), and for some integers \( {\widetilde{c}}_1,{\widetilde{c}}_2,\ldots ,{\widetilde{c}}_{B+A} \). Note that each \( {\widetilde{v}}_j \) has speed s and that each \( {\widetilde{w}}_j \) has speed \( d_j \).

Example 4.5

Consider the \( U_q({\widehat{\mathfrak {gl}}}(3|1)) \) coupled soliton . Note that \( N_1(v \otimes u_1^{\otimes \infty })=2 \) and \( N_j(v \otimes u_1^{\otimes \infty })=0 \) for \( j\ne 1 \). So we expect that v contains two overlapping uncoupled solitons of length one. In the following diagram, we pass the uncoupled soliton through a state containing v. The maximal weight element is represented as a dot. We find that v splits into two copies of the uncoupled soliton after collision.

Example 4.6

Consider the \( U_q({\widehat{\mathfrak {gl}}}(4|1)) \) coupled soliton . Note that \( N_1(v \otimes u_1^{\otimes \infty })=3 \) and \( N_j(v \otimes u_1^{\otimes \infty })=0 \) for \( j\ne 1 \). So we expect that v contains three overlapping uncoupled solitons of length one. In the following diagram, we pass two copies of the uncoupled soliton through a state containing v. The maximal weight element is represented as a dot. We find that v splits into three copies of the uncoupled soliton after collision.

We know that \( N_{\ell } \) is a conserved quantity (since \( E_{\ell } \) is). We can interpret this fact as a form of stability under collision, which is one of the important properties of solitons.

In the height 1 BBS, every state separates into solitons given enough time. We conjecture that this is also true in our system (though the solitons might be coupled).

Conjecture 4.7

Let p be any state. There exists some positive integer \( {\widetilde{t}} \), some solitons \( v_1,v_2,\ldots ,v_D \) of speeds \( d_1,d_2,\ldots ,d_D \) (respectively) and some positive integers \( c_1,c_2,\ldots ,c_D \), such that for any \( t>{\widetilde{t}} \),

$$\begin{aligned} (T_{\infty })^t(p)= & {} u_1^{\otimes c_1+d_1(t-{\widetilde{t}})} \otimes v_1 \otimes u_1^{\otimes c_2+(d_2-d_1)(t-{\widetilde{t}})}\\{} & {} \otimes v_2 \otimes \cdots \otimes u_1^{\otimes c_D+(d_D-d_{D-1})(t-{\widetilde{t}})} \otimes v_D \otimes u_1^{\otimes \infty }. \end{aligned}$$

We have verified this conjecture for all \( U_q({\widehat{\mathfrak {gl}}}(2|2)) \) and \( U_q({\widehat{\mathfrak {gl}}}(3|3)) \) states with only the first five factors being non-vacuum elements.

There are two important properties that the solitons of the KdV equation and of height 1 BBSs satisfy: they move with speed corresponding to their length and are stable under collision. In general, the solitons of Definition 4.1 do not satisfy these two properties (even though \( N_{\ell } \) is conserved). However, we will show that uncoupled solitons do satisfy these two properties.

4.2 Solitons with Speed Equal to Their Length

One of the properties of the height 1 BBS is that the speed of the solitons is equal to their length. In general, this is not true of the solitons in our system. In this section, we provide a large class of solitons which move with speed corresponding to their length. We conjecture that this is the largest such class and that it contains all of the uncoupled solitons.

Theorem 4.8

Let \( B^{r,1} \) be the \( U_q(\mathfrak {gl}(m|n)) \)-crystal (with \( r\le m \)) of rectangular semistandard Young tableaux (SSYT) with height r and width 1. Let

Suppose the factors of the tensor product in reverse order:

form a SSYT and that there is a row number k (\( 1\le k\le r \)) such that

$$\begin{aligned} x_{ij}&<\overline{m-r} \qquad \text {for all } j \text { and for } i<k\\ x_{ij}&\ge \overline{m-r} \qquad \text {for all } j \text { and for } i\ge k. \end{aligned}$$

Then, \( (T_{\ell })^t(u_1^{\otimes c}\otimes x\otimes u_1^{\otimes \infty }) = u_1^{\otimes (c+t\min \{s,\ell \})}\otimes x\otimes u_1^{\otimes \infty } \) for all positive integers t.

We first use the R-matrix insertion algorithm to prove the theorem for a genuine highest weight state, and then generalise using Lemma 3.6. The proof is given in “Appendix A”.

Conjecture 4.9

The subset of \( \bigcup _{d=1}^{\infty }(B^{r,1})^{\otimes d} \) defined by Theorem 4.8 is the largest such subset of solitons which move with speed equal to their length.

We verified this conjecture experimentally for solitons with \( r=2,\, s=2 \), with \( r=2,\, s=3 \) and with \( r=3,\,s=2 \) for \( m=n=3 \) and \( m=n=4 \).

The value of k relates the structure of these solitons to the value of \( N_{s} \).

Proposition 4.10

Let \( x,\, k \) be as in Theorem 4.8. Then \( N_{s}(u_1^{\otimes c}\otimes x \otimes u_1^{\otimes \infty }) = r+1-k \) and \( N_{\ell } = 0 \) for \( \ell \ne s \).

Proof

Let \( p_1\otimes p_2 \otimes \cdots = u_1^{\otimes c}\otimes x \otimes u_1^{\otimes \infty } \). We can compute \( H(u_{\ell }^{(j-1)}\otimes u_1) = 0 \). Applying the bumping algorithm while moving the carrier through the soliton (c.f. “Appendix A”), we find that \( H(u_{\ell }^{(j-1)}\otimes p_j) = r+1-k \) unless the carrier \( u_{\ell }^{(j-1)} \) is full, in which case \( H(u_{\ell }^{(j-1)}\otimes p_j) = 0 \). Hence, \( E_{\ell }(p) = (r+1-k)\min (\ell ,s) \). It is then easily verified that \( N_{\ell }(p) = r+1-k \) if \( \ell =s \) and \( N_{\ell }(p) = 0 \) otherwise. \(\square \)

In particular, note that if \( k=r \) then x is an uncoupled soliton. We propose the following generalisation of the above proposition.

Conjecture 4.11

Let \( v = v_1\otimes v_2\otimes \cdots \otimes v_{d} \in (B^{r,1})^{\otimes d} \) be a soliton of speed s. Let \( k_j \) be the value of k (defined in Theorem 4.8) for each \( v_j \). Then

$$\begin{aligned} \sum _{j=1}^{d} (r+1- k_j) = s N_{s}(v\otimes u^{\otimes \infty }). \end{aligned}$$
(4.1)

Intuitively, we interpret \( r+1-k_j \) as the number of overlapping solitons at the j-th position of the soliton.

Proposition 4.12

If we assume Conjecture 4.11, then the solitons of Theorem 4.8 with \( k=r \) are the only uncoupled solitons.

Proof

Since \( r+1-k_j \ge 1 \), the left-hand side of (4.1) is at least d, and for an uncoupled soliton, the right-hand side of (4.1) is s. But we already know \( s \le d \). We deduce that \( s=d \) and hence \( k_j = r \) for all j. \(\square \)

4.3 Scattering of Two Solitons

One of the main properties of the height 1 BBS is that solitons are stable under collision. This behaviour is also called scattering.

Definition 4.13

Let v, w be uncoupled solitons of lengths \( {\mathfrak {d}}_1,\,{\mathfrak {d}}_2 \), respectively, with \( {\mathfrak {d}}_1>{\mathfrak {d}}_2 \). We say that v and w scatter if there exist non-negative integers \( {\mathfrak {c}}_2,{\widetilde{t}} \) such that for any \( t>{\widetilde{t}} \) and \( {\mathfrak {c}}_1\in {\mathbb {Z}}_{\ge 0} \),

$$\begin{aligned} (T_{\infty })^t(u_1^{\otimes {\mathfrak {c}}_1}\otimes v \otimes u_1^{\otimes {\mathfrak {c}}_2}\otimes w \otimes u_1^{\otimes \infty }) = u_1^{\otimes {\mathfrak {c}}_3} \otimes {\widetilde{w}} \otimes u_1^{\otimes {\mathfrak {c}}_4}\otimes {\widetilde{v}} \otimes u_1^{\otimes \infty } \end{aligned}$$

for some non-negative integers \( {\mathfrak {c}}_3,{\mathfrak {c}}_4 \) (dependent on t) and some uncoupled solitons \( {\widetilde{w}},{\widetilde{v}} \) of lengths \( {\mathfrak {d}}_2,{\mathfrak {d}}_1 \), respectively.

We can interpret this definition as saying that the longer soliton, v, eventually collides and interacts with the shorter soliton, w, after which the states separate into two solitons again. However, it is important to note that \( {\widetilde{w}} \) and \( {\widetilde{v}} \) are generally different from w and v, respectively. We have already seen an example of scattering for \( {\widehat{\mathfrak {gl}}}(m|n) \) in Example 1.2.

Using the same notation in Definition 4.13, let \( j_v={\mathfrak {c}}_1+1,\, j_w={\mathfrak {c}}_1+{\mathfrak {d}}_1+{\mathfrak {c}}_2+1,\, j_{{\widetilde{w}}}={\mathfrak {c}}_3+1,\,j_{{\widetilde{v}}}={\mathfrak {c}}_3+{\mathfrak {d}}_2+{\mathfrak {c}}_4+1\) be the positions of \( v,w,{\widetilde{w}},{\widetilde{v}} \), respectively. If there exists an integer \( \delta \) such that \( j_{{\widetilde{v}}}=j_v+t{\mathfrak {d}}_1+\delta \) and \( j_{{\widetilde{w}}}=j_w+t{\mathfrak {d}}_2-\delta \) then we call \( \delta \) the phase shift.

Let V be a SSYT. Let \( {V_\downarrow } \) denote the bottom row of V, and \( {V_\uparrow } \) denote the other rows of V. We will just consider the case where \( {V_\downarrow } \) only has entries greater than or equal to \( \overline{m-r} \), and \( {V_\uparrow } \) only has entries strictly less than \( \overline{m-r} \) (where r is the height of V). In the notation from Theorem 4.8, we are only considering the case where \( k=r \).

Theorem 4.14

Consider uncoupled solitons composed of elements of \( U_q({\widehat{\mathfrak {gl}}}(m|n)) \)-crystals with height \( r\le m \). Then any two uncoupled solitons of the form given in Theorem 4.8 (i.e. with \( k=r \)) scatter.

Moreover, let v, w be uncoupled solitons and let \( {\widetilde{w}},{\widetilde{v}} \) be obtained from v, w as in Definition 4.13. The elements v and w are related to \( {\widetilde{w}} \) and \( {\widetilde{v}} \) via their semistandard Young tableaux (SSYT). Let \( V,W,{\widetilde{W}},{\widetilde{V}} \) be the SSYT corresponding to \( v,w,{\widetilde{w}},{\widetilde{v}} \), respectively. Then,

$$\begin{aligned} {{\widetilde{W}}_\uparrow }\otimes {{\widetilde{V}}_\uparrow } = R({V_\uparrow }\otimes {W_\uparrow }) \qquad \text {and} \qquad {{\widetilde{W}}_\downarrow }\otimes {{\widetilde{V}}_\downarrow } = R({V_\downarrow }\otimes {W_\downarrow }). \end{aligned}$$

The phase shift is given by \(\delta = 2{\mathfrak {d}}_2+H({V_\downarrow }\otimes {W_\downarrow })+H({V_\uparrow }\otimes {W_\uparrow }).\)

The proof of the above theorem is given in “Appendix B”

Note that the assumption of uncoupled is not a necessary condition, and some other coupled solitons also scatter as in Definition 4.13.

Example 4.15

Consider the following time evolution of a BBS composed of elements from the \(U_q(\widehat{\mathfrak {gl}}(4|1))\)-crystal with \( r=2 \).

We observe that the two objects and satisfy Theorem 4.8 with \(k = 1\) and are stable upon collision. However, and are not uncoupled. Calculating \( N_{\ell } \), we find that \( N_1 = 2,\, N_2 = 2 \) and \( N_{\ell } = 0 \) for \( \ell >2 \).

However, in general, the collisions of coupled solitons are more complicated.

Example 4.16

Consider the following time evolution of a BBS composed of elements from the \(U_q(\widehat{\mathfrak {gl}}(3|3))\)-crystal with \( r=2 \).

Calculating \( N_{\ell } \), we find that \( N_1 = 2,\, N_2 = 2 \) and \( N_{\ell } = 0 \) for \( \ell >2 \).

Remark 4.17

States with an arbitrary number of uncoupled solitons can be reduced to multiple collisions of two solitons. Moreover, it is a consequence of the Yang–Baxter equation that the states after all collisions have occurred are independent of the order of collisions.