2.1. Special class of rational NSE solutions

Let $k$ be a complex parameter and $\theta(k) := kx+k^2y+\gamma(k)$ where $\gamma(k)$ is an arbitrary function, differentiable to all orders, and satisfies $\overline{\gamma(\bar{k})} = \gamma(k), \, \gamma(0)=0$. We choose the vacuum solutions of (2.1) as follows

where ‘bar’ denotes complex conjugate, $k_0 := ib, \, b \gt 0$ so that the integrals in (2.2) converge, and the differential operator $D_k$ is defined by

By expanding the differential operator $D_k^{2j-1}$ one can re-express $\psi_j$ in (2.4a) as

where the constants $b_{jl}(k_0)$ will be deduced later, and $p_l$ are the generalized Schur polynomials defined by

These are polynomials in $x, y$ although they can be expressed in a more natural form as $p_l = p_l(\theta_1, \theta_2, \cdots, \theta_l)$ where $\theta_j := \textstyle\frac{i}{j!}\partial_k^j\theta$, that is

Thus only $\theta_1, \theta_2$ depend on $x,y$ and the rest are independent complex parameters. In terms of the $\theta_j$s the generalized Schur polynomials have the explicit form [Reference Chakravarty and Zowada10]

The first few generalized Schur polynomials are thus given by

It follows from (2.7) that the $p_n$ is a weighted homogeneous polynomial of degree $n$ in $\theta_j, \, j=1,\ldots,n$, i.e., $p_n(a\theta_1,a^2\theta_2,\cdots,a^n\theta_n)= a^np_n(\theta_1,\theta_2,\cdots,\theta_n)$, with weight $(\theta_j)=j$. They also satisfy the properties:

Next we proceed to construct the NSE solution $\psi$ from (2.3) and (2.2) using the particular vacuum solutions in (2.4a). First notice that using (2.5)

where the last equality is obtained by repeated integration by parts. It shows that $M_{0j}$ is meromorphic in $k$ with a pole of order $2j$ at $k=\bar{k}_0$ and the coefficients $\hat{b}_{jl}$ depend on $\bar{p}_l$ and its $x$-derivatives. Hence, these are polynomials in $x,y$. Consequently, the NSE solution and its adjoint in this special case are meromorphic in $k$ with poles of order $2n$ at $\bar{k}_0$ and $k_0$, respectively, and can be expressed as follows

The expression for $\psi(k)$ above should be evident from the determinant formula in (2.3) by expanding $\det \tilde{M}$ using the first row. It should also be clear that the coefficients of the polynomial $P_n(k)$, namely, $a_j=a_j(x,y; k_0)$ are rational in $x,y$. The symmetry $\phi(k) = \overline{\psi(\bar{k})}$ follows from the choices $\phi_0(k) = \overline{\psi_0(\bar{k})}, \phi_i=\bar{\psi}_i$ in (2.4a) which leads to $\overline{M_{0i}(\bar{k})} = M_{i0}(k)$ and $\bar{M}=M^T$. Furthermore, $M$ is a positive definite matrix so that the potential $u(x,y)$ given by (2.3) is real and non-singular.

2.2. Reduction of special NSE to NLS

The special solutions of NSE in (2.9) constructed using the vacuum solutions defined in (2.4a) possess certain unique properties arising from their meromorphicity in the $k$-space. We describe these properties in this subsection and analyse them to deduce a simple algebraic relation that connects a given NSE solution from this special class to its associated potential $u$.

Note that by construction the solutions in (2.9) satisfy the relations

which follows from their expressions in (2.3) and by noting that the above operations transform the first row to be identical to the $j^{\mathrm{th}}$ row in $\tilde{M}$ and $\hat{M}$, respectively, so that the resulting determinants vanish. Applying (2.4b) to the above expressions and since $z(k_0)=z(\bar{k}_0)=0$ with $k_0=ib$, one obtains

which implies that near $z=0$,

Next, consider the function $\zeta(k)=\textstyle\frac12(k-b^2/k)$ which satisfies $z^2-\zeta^2 = b^2$ so that $\zeta(z) = \sqrt{z^2-b^2}$ and near $z=0, \,\, \zeta(z) = \pm ib \big(1-\textstyle\frac12\textstyle\frac{z^2}{b^2}-\textstyle\frac{1}{8}(\textstyle\frac{z^2}{b^2})^2 - \ldots\big)$. It follows that the functions $\psi(z), \phi(z), \zeta(z)$ have no poles at $z=0$, hence the residues, namely

Equivalently, one has the conditions that the sum of residues at $k=k_0, \, \bar{k}_0$

From (2.9), one has the rational (in $k$) expression

In order to compute the residues at $k_0, \bar{k}_0$, the following relations are useful

which are easily verified by partial fraction decomposition. These relations in conjunction with (2.11) lead immediately to the following

where $k_0=ib$ is also used. Note that $k=0$ is a simple pole of $\textstyle\frac{1}{k}\psi(k)\phi(k)$ with residue $|a_{2n}|^2/b^{4n} = |\psi(0)|^2$. Hence, one needs to subtract this contribution from the third equality above in order to obtain the sum of residues $\sum\mathop{\mathrm{Res}}\limits_{k_0,\bar{k}_0}\textstyle\frac{1}{k}\psi(k)\phi(k)$, i.e., only at $k_0, \bar{k}_0$. Collecting all these results together yields

Moreover, since $\psi$ in (2.9) is a solution of the NSE, $P_n(k)$ satisfies

Equating the coefficients of $k^{2n}$ of this equation, yields $u=-2ia_{1x}$. Combining this relation with its complex conjugate and the fact that the potential $u$ is real, one obtains $(a_1+\bar{a}_1)_x=0$ and $iu = (a_1-\bar{a}_1)_x$. The first relation is consistent with the fact that $a_1+\bar{a}_1=0$ which follows from the first equations in (2.10) and (2.12). The coefficient of $k^{2n-1}$ of the equation for $P_n(k)$ gives $ia_{1y}=a_{1xx}+2ia_{2x}+ua_1$. Eliminating the $y$-derivative by using the complex conjugate of this equation, integrating once with respect to $x$, and using $a_1+\bar{a}_1=0$, yields the following relation between the potential $u$ and the complex coefficients $a_1, a_2$

where $C(y)$ is an arbitrary function to be determined later. Substituting the above relation in the second equation of (2.12), the vanishing residue condition from the second equality in (2.10) leads to

Finally, the boundary conditions: $u \to 0, \, |\psi(0)| \to 1$ as $|x| \to \infty$ (see remark below) imply that $C(y) = 2nb^2$ is a constant. Therefore, one obtains the following significant relation connecting $\psi(0)$ and the associated potential

Since $\psi(0)$ satisfy (2.1), one obtains from (2.13)

with the boundary conditions for $\psi(0)$ and $u$ mentioned above. The transformation $r(x,y) = b\psi(0)e^{-2ib^2y}$ leads to the NLS equation

with nonvanishing boundary condition $r(x,y) \to be^{-2ib^2y}$ as $|x| \to \infty$. Equation (2.13) is a self-consistent rational potential associated with the NSE satisfied by the function $\psi(0)$ which is also a solution of the NLS equation with non-vanishing boundary condition. Moreover, the obtained solutions for $\psi(0)$ are rational in $x,y$.

3.1. The Gramian construction

We first obtain an explicit formula for the vacuum solutions given in (2.5) by determining the coefficients $b_{jl}(k_0)$. Using these, we will derive a Gramian form for the matrix $M$ and a similar form for $\tilde{M}$ associated with the NSE solution $\psi$ and potential $u$ in (2.3). We will show that the leading order expansions for $\det M$ and $\det \tilde{M}$ will provide insights into the structures of the underlying rational solutions of NLS.

From the expression for $z(k)$ in (2.4b) one can express the branches of inverse function $k(z)$ near $z=0$ as

where the coefficients $k_n$ are given by

Then the integral powers of the differential operator $D_k$ in (2.4b) can be obtained from Faá di Bruno’s formula [Reference Johnson23]

where the coefficients $k_0, k_1, \ldots, k_n$ are given in (3.1b). The polynomials $B^n_{l}(k_1,\cdots,k_n)$ are called Bell polynomials which are associated with set partitions. The expression $r_1+2r_2+\cdots+nr_n$ represents the partition of the positive integer $n$ into blocks of $1$’s, $2$’s, etc., $r_i$ denotes the number of blocks of $i$, and $l$ denotes the number of parts (total number of blocks) in the partition. For example, $1+1+1+2+2+3$ is a partition of $n=10$ into $l=6$ parts with $r_1=3, r_2=2, r_3=1, r_j=0, \, j \gt 3$. It is easy to verify from the expression above that $B^n_{n}=k_1^n = 1, \, B^n_1=n!k_n$. Thus from (3.1b), $B^n_1=0$ for odd $n$ except when $n=1$.

The vacuum solutions of NSE given in (2.5) now can be explicitly written as

$j=1,2,\ldots,n$. For example, $q_1=p_1$ since $b_{11}=B^1_1=1$, $q_2=b_{22}p_2+p_3 = 2k_2p_2+p_3$ since $B^3_1=0=b_{21}, b_{23}=B^3_3=1$. Similarly, $q_3=p_5+4k_2p_4+3k_2^2p_3+2k_4p_2$, and in general for $j \gt 1$, $q_j$ is a linear combination of the generalized Schur polynomials $p_2, \ldots, p_{2j-1}$.

We now proceed to compute the matrix $M$ in (2.2). Since from (2.4a) $\phi_j=\bar{\psi}_j$, the matrix $M$ is positive definite. Indeed, for any non-zero vector $v \in \mathbb{C}^n$

Next, substituting $\psi_j$ from (3.2) and $\phi_j=\bar{\psi}_j$ into (2.2) the matrix elements $M_{ij}$ can be readily computed by integration by parts, as follows:

where $k_0=ib, \overline{\theta(k)} = \theta(\bar{k})$ and Leibnitz rule of derivatives is used to expand the matrix elements $H_{ij}$. Note that the sums in the expressions for $H_{ij}$ terminate because $q_j$ in (3.2) is a linear combination of the generalized Schur polynomials which satisfy (2.8a) and the fact that $\partial_x=i\partial_{\theta_1}$ from (2.6). In fact, since the indices $1 \leq i,j \leq n$ both sums in the last expression for $H_{ij}$ above, can be extended to $2n-1$ without affecting the sums as $\partial_x^s q_i =0$ if $s \gt 2i-1$. Now consider the complex vectors

whose last $2(n-i)$ components are $0$ whenever $i \lt n$. Then the matrix elements $H_{ij}$ are given by the inner products [Reference Chakravarty and Zowada11]

where $C$ is a $2n \times 2n$, symmetric matrix. The matrix $H$ is Hermitian and has the form

where $Q$ is a $2n \times n$ matrix of column vectors $Q_i, \, 1\leq i\leq n$. The matrix $H$ is known as the Gram matrix, and $\det H$ is called the Gramian of the vectors $Q_1,Q_2,\ldots,Q_n$ which are linearly independent since $\det H \neq 0$. In fact $H$ is positive definite because $M$ is, hence $\det H \gt 0$. Consequently, the NSE potential in (2.3) is given by

as the factor ${\textstyle\frac{e^{in(\theta-\bar{\theta})}}{(2b)^n}}$ arising in $\det M$ is annihilated by $\ln(\cdot)_{xx}$. Since the generalized Schur polynomial $p_j$ is of degree $j$ in $x,y$, the polynomial $q_j$ is of degree $2j-1$, hence $\det H$ is also a polynomial. Therefore, $u(x,y)$ is a nonsingular, rational function of its arguments and decays as $(x^2+y^2)^{-1}$. Then it is evident from (2.13) that $|\psi(0)|$ and hence, the modulus $|r(x,y)|=b|\psi(0)|$ of the NLS solution in (2.14b) is also rational and non-singular.

Next we will derive an expression for the NSE solution $\psi(k=0)$ from the first of the formulas in (2.3) after using (2.2) to compute the elements $M_{0j}$ of the matrix $\tilde{M}$. Substituting $\psi_0$ from (2.4a) and $\phi_j=\bar{\psi}_j$ from (3.2) in (2.2), and applying integration by parts as before

after using $k_0 = ib$. The matrix $\tilde{M}$ is a $(n+1) \times (n+1)$ matrix bordered by the elements $M_{0j}$ in its first row and $\psi_0, \psi_i$ in its first column. Applying the expansion formula of bordered determinants, yields

Furthermore, as in (3.3), the matrix $\tilde{H}$ can also be expressed as

i.e., the only non-zero elements of $D$ are $2b^{-r}$ in the first column, and $C$ is given above (3.3). Then the NSE solution $\psi(0)$ of (2.14a) is obtained as a ratio of two Gram determinants, namely

Thus $\psi(0)$ is a nonsingular, rational function in $x,y$ since it is the ratio of polynomials of the same degree and $\det H \gt 0$. Moreover, by expanding the leading terms of the determinants in (3.5) one can verify that $|\psi(0)| \to 1$ as $\sqrt{x^2+y^2} \to \infty$ as mentioned in Remark (b) of Section 2. We will discuss the structure of $|\psi(0)|$ and $u$ given by (2.13) later, but first we provide some examples of the rational solutions below.

3.2. Examples of rational solutions

The solutions $\psi(0)$ can be constructed explicitly by expanding the determinants in the numerator and denominator of the last equality in (3.5) by applying Binet–Cauchy formula. When $n=1$, the vacuum solution from (3.2) $\psi_1 = q_1e^{i \theta(k_0)}=p_1e^{i \theta(k_0)}$ where $p_1$ is the generalized Schur polynomial of degree $1$. In this case, the matrix $Q$ has a single column and $C, \tilde{C}$ are $2 \times 2$ matrices defined above (3.3) and (3.5), respectively. These are given by

The matrices $H=Q^{\dagger}CQ$ and $\tilde{H} = Q^{\dagger}\tilde{C}Q$ are both $1 \times 1$, so that

where $p_1 = \theta_1 = i(x+2k_0y+\gamma_1)$ from (2.6) and $\partial_x p_1=i\partial_{\theta_1} p_1 = p_0 =1$ from (2.8a). Introducing the coordinates $r=x, \, s=2by$ and using $k_0=ib$, we can express $\theta_1 = i(r+is+\gamma_1)$. Then the corresponding NLS solution is given by

with parameters $r_0= -\operatorname{\rm Re}\nolimits(\gamma_1), \, s_0=-\operatorname{\rm Im}\nolimits (\gamma_1)$ and $b$. Switching $\psi(0) \to -\psi(0)$ which does not change (2.14a), one recovers the well-known Peregrine solution [Reference Peregrine37]. Then the corresponding NSE potential is

which can be calculated from (2.13) or directly from (2.3) as $u=2(\log \det H)_{xx}$.

The solutions $|\psi(0)|$ and $u$ are illustrated in Figure 1, which shows a single peak of height $3$ for $|\psi(0)|$ and $16b^2$ for $u$, both at $(r_0,s_0)$. Moreover, the vertical sections (at $s=s_0$) indicate that $|\psi(0)|$ and $u$ have two local minima of depth $0$ and $-2b^2$, respectively, symmetrically located from the peak at $(r_0 \pm \textstyle\frac{\sqrt{3}}{2b})$. Notice also that $\psi(0)| \to 1$ while $u \to 0$ for large $|r|, |s|$.

Figure 1.

$|\psi(0)|$ for the Peregrine solution and the corresponding NSE potential $u$. Parameters: $\gamma_1=0, b=\textstyle\frac34$. Right panels: Vertical cross-section ( $s=0$) showing the maximum and two minima.

Figure 1 Long description

A showing a 3D surface plot labeled, absolute value of psi 0. The horizontal axes are labeled, s and r, each ranging from minus 5 to 5. The vertical axis has tick labels 0, 1, 2, 3. The surface has one tall central peak and two lower depressions on opposite sides of the peak. B showing a line graph labeled, absolute value of psi 0. The horizontal axis is labeled, r, with tick labels minus 4, minus 2, 0, 2, 4. The vertical axis is labeled, absolute value of psi 0, with tick labels 0, 1, 2, 3. The curve rises to a single peak near r equals 0 at about 3 and has two symmetric minima near r equals minus 2 and r equals 1 at about 0. C showing a 3D surface plot labeled, u. The horizontal axes are labeled, s and r, each ranging from minus 5 to 5. The vertical axis has tick labels minus 5, 0, 5. The surface has one tall central peak and two lower depressions on opposite sides of the peak. D showing a line graph labeled, u. The horizontal axis is labeled, r, with tick labels minus 4, minus 2, 0, 2, 4. The vertical axis is labeled, u, with tick labels 0, 5, 10, 15. The curve rises to a single peak near r equals 0 at about 9 and dips below 0 on both sides near r equals minus 2 and r equals 1.

The higher ( $n \gt 1$) order rational solutions of NLS can be generated in a similar way. For example, one starts with 2 vacuum solutions of NSE $\psi_1, \psi_2$ when $n=2$, and $\psi_1, \psi_2, \psi_3$ for $n=3$. From (3.2), these are given as $\psi_i=q_ie^{i\theta(k_0)}, \, i=1,2,3$ where

and $p_j$s are the generalized Schur polynomials given by (2.7). Then one constructs the $2n \times n$ matrix $Q$ and the $2n \times 2n$ matrices $C$ and $\tilde{C}$ as described in Section 3.1, and finally applies Binet–Cauchy formula to compute $\det H$ and $\det \tilde{H}$ to construct $\psi(0)$ in (3.5). The corresponding expressions are too lengthy even for $n=2,3$, hence they are omitted here. Instead, these solutions are illustrated in Figures 2 and 3. They show that both $|\psi(0)|$ and $u$ are localized around multiple peaks whose numbers are 3 and 6 for $n=2$ and $n=3$, respectively. The peaks can be well-separated by increasing the magnitude of the parameter $\gamma_3$. Moreover, in the neighbourhood of each local maximum, the solution is effectively a Peregrine solution with the same peak height and with two local minima symmetrically placed around the peak. We will discuss the structure of these rational NLS solutions for arbitrary values of $n$ including the number and location of the peaks as well as peak patterns in the next section where we connect these solutions to the multi-lump solutions of the KPI equation.

Figure 2.

$3$-lump solutions of the NLS equation. Parameters: $\gamma_1=\gamma_2=0, \gamma_3 = 100, b=\textstyle\frac12$. Left panel: $|\psi(0)|$, right panel: $u$.

Figure 3.

$6$-lump solutions of the NLS equation. Parameters: $\gamma_1=\gamma_2=0, \gamma_3 = 80, \gamma_5=0, b=\textstyle\frac12$. Left panel: $|\psi(0)|$, right panel: $u$.

4.1. KPI multi-lumps

The binary Darboux transformation of Section 2 can be extended to construct the solutions of the KPI hierarchy by considering the linear system

which includes the vacuum NSE when $n=2$ and $t_2=y$. A particular solution to this system is given by $\psi_0(k) = e^{i\theta(k)}$ as in (2.4a) with

Furthermore, the functions $\{\psi_j, \phi_j\}$ defined in (2.4a) (as well as in subsequent equations (2.5) and (3.2)) in terms of the generalized Schur polynomials $p_n(\theta_1, \ldots, \theta_n)$, also solve the above linear system and its adjoint. The generalized Schur polynomial variables $\theta_j$ given by (2.6) now depend on the multi-time variables through the parameters $\gamma_j=\gamma_j(t_3, t_4, \ldots)$. Then the solution $\psi$ and the potential $u$ given by (2.2) and (2.3) constitute the multi-time KPI hierarchy. In particular, the KPI equation (4.1) is the first non-trivial member of the hierarchy with $t_3=t \in \mathbb{R}$. Consequently, the Gramian construction in Section 3 leads to a special class of rational solutions of KPI given by (3.4) which is also related to the NLS equation (2.14a) via (2.13).

It is useful to examine the structure of the solution $u$ in (3.4) in some details in order to gain insight into the structure of the rational solutions of NLS. The real symmetric matrix $C$ in (3.3) admits a Cholesky decomposition $C=U^{\dagger}DU$ where $U$ is a real, upper-triangular matrix with 1’s along its main diagonal and $D$ is a diagonal matrix, i.e.,

Then in (3.3), $H=R^{\dagger}DR$ with $R=UQ$. Consequently, $\det H$ can be expressed as a sum of squares after using the Binet–Cauchy formula

where $R(l_1 \cdots l_n)$ is the principal minor of the $2n \times n$ matrix $R$ whose columns are $UQ_i, \, 1 \leq i \leq n$ and rows are indexed by $(l_1,\dots l_n)$. The $n \times n$ minor $U\tbinom{l_1 \cdots l_n}{r_1 \cdots r_n}$ of $U$ is obtained from the submatrix whose rows and columns are indexed by $(l_1,\dots l_n)$ and $(r_1,\dots r_n)$, respectively, and $Q(r_1 \cdots r_n)$ is the principal minor of $Q$ with rows indexed by $(r_1,\dots r_n)$. Since $U$ is upper triangular with $1$’s along its diagonal, it follows that the principal minors $U\tbinom{l_1 \cdots l_n}{l_1 \cdots l_n}=1$ and $U\tbinom{l_1 \cdots l_n}{r_1 \cdots r_n} \neq 0$ if and only if $l_j \leq r_j$ for each $j=1,\ldots,n$. Consequently, the determinant in (4.2a) can be re-expressed as

where $\textbf{r} := (r_1 \cdots r_n), \, 0 \leq r_1 \lt r_2 \lt \cdots \lt r_n \leq 2n-1$ denote lexicographically ordered multi-index sets whose earliest member is $\textbf{0} := (01 \cdots n-1)$, and $|\textbf{r}| := r_1+\cdots + r_n$. In terms of the polynomial $q_j$ in (3.2), these maximal minors can be expressed as

where the superscripts denote derivative with respect to $x$, and $q_j^{(r)}=0, \, r \gt 2j-1$ (cf. Section 3.1). In particular, note that $Q(\textbf{0}) = \operatorname{\rm Wr}\nolimits(q_1,q_2,\ldots,q_n)$ is a Wronskian.

Recall from Section 2 that the generalized Schur polynomials $p_l$ are weighted homogeneous polynomials of degree $l$ in $\theta_1, \theta_2, \ldots, \theta_l$ with weight $(\theta_j)=j, \, 1 \leq j \leq l$. Moreover, from (2.6), $\theta_1=i(r+is+\gamma_1)$ in terms of the physical coordinates $x=r, s=2by$ (note that for KPI solutions, $r=x-3b^2t$ where $x,y,t \in \mathbb{R}$ are the physical variables). Then the determinant $Q(\textbf{r})$ is a weighted homogeneous polynomial in $\theta_1, \theta_2, \ldots, \theta_{2n-1}$ of degree $(1+3+\cdots+2n-1)-|\textbf{r}|$. Particularly when $\textbf{r} = \textbf{0}$, the degree of the leading principal minor $Q(\textbf{0})$ is the highest in the coordinates $r,s$, and is given by

Thus, $\det H$ in (4.2) is a positive definite polynomial in $r$ and $s$ of degree $n(n+1)$ and with $2n-1$ complex parameters $\gamma_j, \, 1 \leq j \leq 2n-1$ which appear in the arguments $\theta_j, \, j=1,\ldots, 2n-1$ of the generalized Schur polynomials. However, it turns out that the minors $Q(\textbf{r})$ in (4.3) depend on only $n$ independent combinations of these parameters. The proof of this fact is somewhat technical, only the basic ideas are outlined below.

First, note that for each $j=1,2,\ldots,n$ the polynomial $q_j$ in (3.2) can be transformed to a single generalized Schur polynomial with shifted arguments by using the property (2.8b), i.e.,

The shifts $h^j_k$ can be suitably chosen such that the coefficients $b_{jl} = p_{2j-1-l}(h^j_1,h^j_2, \ldots, h^j_{2j-1-l})$. Note that the shifts do depend on the index $j$ of the polynomial $q_j$, that is, $h^j_k \neq h^{j'}_k, \, j \neq j'$. Then each column of the matrix $Q$ is expressed as

where $p_l(\hat{\theta}) := p_l(\hat{\theta}_1, \ldots, \hat{\theta}_l)$. Next, decompose the even-indexed argument $\hat{\theta}_{2k} = \hat{\theta}^0_{2k}+\gamma_{2k}, \, 1 \leq k \leq j-1$ of each generalized Schur polynomial $p_{2j-1}$ in the first row of the matrix $Q$ and expand using (2.8b) to yield

By elementary row and/or column operations of $Q$ (which does not change the determinants $Q(\textbf{r})$), one can eliminate almost all terms involving $\gamma_{2j}, \, 1 \leq j \leq n-1$ and absorb the remaining terms by redefining (if necessary) the arbitrary constants $\gamma_{2j-1}, \, 1 \leq j \leq n$. Thus, without any loss of generality, one can set the original parameters $\gamma_{2j}=0, \, 1 \leq j \leq n-1$ that appear in the generalized Schur polynomials in $Q$.

Then it follows from (4.2) that $\det H$ depends only on $n$ parameters namely, $\{\gamma_1, \gamma_3, \ldots, \gamma_{2n-1} \}$. The resulting NSE potential $u$ in (3.4) is a nonsingular, rational function of $r,s$ and the odd multi-time parameters $t_{2j-1}$ which arises through the parameters $\gamma_{2j-1}, \, 1 \leq j \leq n$. Specifically, $u=u(r,s,t_3=t)$ satisfies the KPI equation (4.1) and constitute a special family of multi-lump solutions [Reference Chakravarty and Zowada11].

In summary, the NSE potential given by (3.4) relates the rational solutions $\psi(0)$ of the NLS equation (2.14a) to a class of multi-lump solutions of KPI via (2.13). In fact, the NSE (2.1) with the self-consistent potential given by (2.13) is the Lax equation for KPI, and the solution $\psi(k)$ given by (2.9) is the corresponding KP eigenfunction associated with this special class of KPI multi-lumps. Furthermore, the NLS solutions depend on $n$ arbitrary parameters $\gamma_1, \gamma_3, \ldots, \gamma_{2n-1}$, and in particular, $\gamma_3$ can be identified with the evolution variable $t_3=t$ of the KPI equation (4.1). The remaining parameters may be interpreted as generating symmetries of the multi-lump solutions corresponding to the higher odd flows of the KPI hierarchy.

4.2. The triangular peak patterns

In this subsection, we will use the theory of KP1 in order to analyse the peak patterns of the NLS lumps. It is useful to note that due to the relation (2.13), the local maxima of $|\psi(0)|$ for the rational solutions of NLS coincide with those of the NSE potential $u$, which are also the multi-lump solutions of KPI. Therefore, it suffices to analyse (3.4) with $\det H$ given by (4.2). The resulting expression

is a nonsingular, rational function of $x, y$ for fixed values of the $n$-parameters $\gamma_{2j-1}, \, 1 \leq j \leq n$. Thus, $u$ has finitely many local maxima in the $xy$-plane, which are complicated to find exactly for $n \gt 1$. However, it is possible to find them approximately. The above expression suggests that the local maxima of $u$ are near the local minima of $\det H$ where $\partial_x\det H =0$ and $\partial^2_{xx}\det H \gt 0$. Since $\det H$ is expressed as a sum of squares in (4.2b), the local minima of $\det H$ are located near the points where the leading-order polynomial in this expression given by the maximal minor $R(\textbf{0}) =0$. It should be clear from the second equality in (4.2b) and the expression for $Q(\textbf{0})$ in (4.3) that $R(\textbf{0})$ is a complex polynomial of degree $n(n+1)/2$ in the variable $\theta_1=i(x+2iby+\gamma_1)$ and depending on the complex parameters $\gamma_{2j-1}, \, j=1,\ldots,n$. Hence, $R(\textbf{0})$ has $n(n+1)/2$ roots in the $xy$-plane although some of them may be multiple roots. However, the multiplicity can be broken by choosing the parameters $\gamma_{2j-1}$ arbitrarily so that generically speaking, $R(\textbf{0})$ has simple roots. Consequently, $u$ has

distinct peaks in the $xy$-plane as were earlier noticed in the examples of Section 3.2.

Next we proceed to estimate the approximate location of the peaks. Note that the leading order term in the expression for the minor $R(\textbf{0})$ in (4.2b) is the Wronskian function $Q(\textbf{0})$ given by the second expression in (4.3), and which depends on the variables $\theta_k, \, 1\leq k \leq 2n-1$. By shifting the variables $\theta_k \to \theta_k+\delta_k := \tilde{\theta}_k$ it is possible to express $R(\textbf{0})$ as a single Wronskian function

using an identity similar to (2.8b) for appropriate choice of $\delta=(\delta_1, \delta_2, \ldots, \delta_N)$. In particular, it turns out that $\delta_1=\textstyle\frac{in}{2b}$. (The reader may consult [Reference Chakravarty and Zowada11] for more details and [Reference Macdonald33] for a proof). Therefore, the approximate location of the peaks is given by the zeros of the Wronskian $Q(\textbf{0})$ with shifted argument $\tilde{\theta}$ since

To determine the peak locations, one solves (4.5) asymptotically for $\tilde{\theta}_1$ when the parameters satisfy $|\gamma_{2j-1}| \gg 1, \, 2 \leq j \leq n$. At first, we will consider the case $j=2$, i.e., when $|\gamma_3| \gg 1$ and $|\gamma_{2j-1}| \sim O(1), \, j \gt 2$. Also, without loss of generality, we set $\gamma_3 = t \in \mathbb{R}$ since it is the time variable for KPI. We will only consider the leading-order estimate for the peak locations below.

Recall from (3.2) that the highest degree generalized Schur polynomial arising in $q_j$ is $p_{2j-1}$. Thus asymptotically as $|t| \gg 1$, the leading order contribution to the Wronskian in (4.4) is

It can be shown that the $p_l, \, l \lt 2j-1$ terms in $q_j$ contribute to lower order in the asymptotic expansion of $Q(\textbf{0})(\tilde{\theta})$. This function $\operatorname{\rm Wr}\nolimits(p_1, p_3, \ldots, p_{2n-1})(\tilde{\theta})$ is a special case of the Schur functions which play a fundamental role in the representation theory of symmetric groups. Specifically, it is a weighted homogeneous polynomial in the variables $\tilde{\theta}_j$ of degree $N=n(n+1)/2$ with weight $(\tilde{\theta}_j)=j$, and it is of the form

where the coefficients $c_N(\textbf{l}) \in \mathbb{Z}$ are called the group characters of the symmetric group $S_N$ (permutation group of $N$ elements), and $\textbf{l}=(l_1,l_2,\ldots, l_N)$. Since $\theta_j = i\gamma_j, \, j \gt 2$ (cf. (2.6)), the leading order terms in the asymptotic expansion involves only $\tilde{\theta}_1$ and $\tilde{\theta}_3$ in (4.6) when $|t| \gg 1$. These are of the form $\tilde{\theta}_1^{l_1}\tilde{\theta}_3^{l_3}$ with $l_1+3l_3 = N$. Thus, asymptotically as $|t| \gg 1$, the expression for $R(\textbf{0})$ in (4.4) reduces to

Since $N=n(n+1)/2$ is a triangular number, it can always be expressed as either $N=3m$ or $N=3m+1$ for some $m \in \mathbb{N}$. Then the above sum can be re-expressed as

Assuming that $c_N(N,0) \neq 0$ (see below), it is clear that the dominant balance required to asymptotically solve $R(\textbf{0}) = 0$ must be $\tilde{\theta}_1^N \sim \tilde{\theta}_1^{N-3r}\tilde{\theta}_3^r$ so that $\tilde{\theta}_1 \sim |t|^{1/3}$ for $|t| \gg 1$. Using the coordinates $r,s$ introduced in Section 3.2 one can set

where $\delta_1 = \textstyle\frac{n}{2b}$ (see above (4.5)), $z$ is rescaled using the dominant balance $z \sim O(t^{1/3})$, and $a \neq 0$ is a constant to be determined later. Then the above asymptotic expression of $R(\textbf{0})$ takes the form

where $Q_n(\xi)$ is a monic polynomial of degree $N=n(n+1)/2$, which can be identified with the well-known Yablonskii–Vorob’ev polynomials as follows.

The leading order contribution to the Wronskian $\operatorname{\rm Wr}\nolimits(p_1,p_3,\ldots, p_{2n-1})$ as $|t| \gg 1$, may be calculated from

and after rescaling $z$ as above. It follows from the definition of the generalized Schur polynomials in (2.7) that

Then the leading order term in $\operatorname{\rm Wr}\nolimits(p_1,p_3,\ldots,p_{2n-1})$ when $|t| \gg 1$, corresponds to the first term in (4.7) with

Finally, choosing the constant $a=\textstyle\frac{4}{3}$, the resulting polynomial $Q_n(\xi)$ has integer coefficients and coincides with the Yablonskii–Vorob’ev polynomials which were originally studied to obtain special rational solutions of the second Painlevé equation PII [Reference Vorob’ev41, Reference Yablonskii43]. These are of the form

where the coefficients $c_r$ are given in terms of the group characters $c(N-3r, r)$. These polynomials can also be obtained recursively from a nonlinear differential-difference system [Reference Vorob’ev41, Reference Yablonskii43] and are known to have $N$ distinct roots in the complex plane [Reference Fukutani, Okamoto and Umemura19]. The first few Yablonskii–Vorob’ev polynomials are given by

The roots of $Q_n(\xi)$ are either real or complex conjugate pairs. Moreover, the roots are arranged in the complex plane in a ‘triangular’ pattern due to the symmetry: $\xi \to \xi e^{2\pi i/3}$ since $Q(\xi)$ can be expressed as $Q(\xi) = \xi^{N-3m}P_m(\xi^3)$ where $P_m$ is a polynomial of degree $m$ with respect to its argument. There exists an extensive literature on the Yablonskii–Vorob’ev polynomials and patterns of its roots in the complex plane (see, e.g., [Reference Clarkson and Mansfield13] and references therein). They arise as the Painlevé II reductions of the rational solutions of the KdV and modified KdV equations [Reference Kajiwara and Ohta26], hence it is natural for them to appear in the context of the rational solutions of KPI (see Remark (b) below).

Note that the $O(t^{-1/3})$ term in (4.7) arises from all the subdominant terms in $R(\textbf{0})(\theta)$ after the rescaling $z(t) \sim O(t^{1/3})$. These arise from the lower degree generalized Schur polynomials neglected in the approximation $q_j \approx p_{2j-1}$ for the Wronskian in (4.4). Additionally, the terms in (4.6) that are not of the form $\tilde{\theta}_1^{l_1}\tilde{\theta}_3^{l_3}$ also contribute to the subdominant terms. Setting $R(\textbf{0})(r,s,t)=0$ in (4.7), one can solve for $\tilde{\theta}_1 =iz(t)$ asymptotically for $|t| \gg 1$ as

to obtain $\xi=\xi_j$ where $\xi_j$ is a root of $Q_n(\xi)=0$. Thus the leading-order approximation to the locations of the NLS and KPI $N$-lump peaks for $|t| \gg 1$ are obtained as

where $r_0+is_0 = -\gamma_1$.

If $n=3k+1$ for some $k \in \mathbb{N}$ then $Q_n(\xi)$ has a root at $\xi=0$ with multiplicity $1$. In this case, $N=3m+1$ for some $m \in \mathbb{N}$, and there is a single term in the Wronskian in (4.6) of the form $\tilde{\theta}_1\tilde{\theta}_3^m$ that dominates when $|t| \gg 1$. Rearranging all terms in the Wronskian in descending power of $\tilde{\theta}_3$, one finds that (4.5) admits one asymptotic solution $\tilde{\theta}_1 \sim 0 + O(t^{-1})$ which implies that the approximate peak location

This is the case which corresponds to the $\xi=0$ root of $Q_n(\xi)$. The first non-trivial solution of this type corresponding to $k=1$, i.e., $n=4$ is shown below in Figure 4. The simplest triangular lump solution corresponds to $n=2, \, N=3$ which was also discussed earlier in Section 3.2 and in the example given in Section 4.2. Figure 4 compares the exact and the approximate peak locations given by (4.8) for the cases $n=3,4,5$ corresponding to $N=6, 10, 15$, respectively. Recall from (2.13) that these also correspond to the peak locations of $|\psi(0)|^2$ of the NLS solution in (2.14a) where $t$ is a parameter.

Figure 4.

Triangular lump peak locations for $n=3,4,5$ at $t=-40$ (top panel) and $t=40$ (bottom panel). Total number of peaks $N=\textstyle\frac{n(n+1)}{2}$ (see text). Black dots represent locations $z_j$ where $R(\textbf{0})=0$, red open circles are exact locations. Under time reversal ( $t \to -t$), the peak locations are reflected across the dashed vertical line: $r=-\textstyle\frac{n}{2b}$. Parameters: $b=1$ and $\gamma_3 =t, \gamma_j=0, \, j \neq 3$.

Figure 4 Long description

A scatter plot with the text, t equals minus 40. The horizontal axis is labeled, r, with tick labels at negative 10, negative 5, 5. The vertical axis is labeled, s, with tick labels at negative 5 and 5. A dashed vertical line is drawn slightly left of the vertical axis. Plotted points appear near left side around r near negative 10 and s near 7, around r near negative 6 and s near 0, around r near negative 8 and s near negative 7, around r near 1 and s near 4, around r near 1 and s near negative 4 and around r near 8 and s near 0. The points form three vertical groupings around r near negative 10, r near 1 and r near 8. The image B showing a scatter plot with the text, t equals minus 40. The horizontal axis is labeled, r, with tick labels at negative 15, negative 10, negative 5, 5, 10. The vertical axis is labeled, s, with tick labels at negative 10, negative 5, 5, 10. A dashed vertical line is drawn slightly left of the vertical axis. Plotted points appear near r near negative 10 and s near 10, r near negative 10 and s near 3, r near negative 9 and s near negative 2, r near negative 8 and s near negative 9, r near 0 and s near 0, r near 0 and s near 7, r near 0 and s near negative 7, r near 4 and s near 4, r near 4 and s near negative 3 and r near 10 and s near 0. The points form groups around r near negative 10, r near 0, r near 4 and a single point near r near 10. The image C showing a scatter plot with the text, t equals minus 40. The horizontal axis is labeled, r, with tick labels at negative 20, negative 15, negative 10, negative 5, 5, 10, 15. The vertical axis is labeled, s, with tick labels at negative 15, negative 10, negative 5, 5, 10, 15. A dashed vertical line is drawn slightly left of the vertical axis. Plotted points appear near r near negative 12 and s near 13, r near negative 10 and s near 6, r near negative 10 and s near 0, r near negative 10 and s near negative 6, r near negative 10 and s near negative 13, r near negative 5 and s near 10, r near negative 5 and s near 2, r near negative 5 and s near negative 2, r near negative 5 and s near negative 10, r near 0 and s near 7, r near 0 and s near 0, r near 0 and s near negative 7, r near 6 and s near 4, r near 6 and s near negative 3 and r near 14 and s near 0. The points form multiple vertical groupings around r near negative 10, r near negative 5, r near 0, r near 6 and a single point near r near 14. The image D showing a scatter plot with the text, t equals 40. The horizontal axis is labeled, r, with tick labels at negative 10, negative 5, 5. The vertical axis is labeled, s, with tick labels at negative 5 and 5. A dashed vertical line is drawn slightly left of the vertical axis. Plotted points appear near r near negative 10 and s near 0, r near negative 6 and s near 4, r near negative 5 and s near negative 3, r near 2 and s near 0, r near 3 and s near 7 and r near 3 and s near negative 7. The points form groupings around r near negative 10, r near negative 6 to negative 5, r near 2 and r near 3. The image E showing a scatter plot with the text, t equals 40. The horizontal axis is labeled, r, with tick labels at negative 15, negative 10, negative 5, 5, 10. The vertical axis is labeled, s, with tick labels at negative 10, negative 5, 5, 10. A dashed vertical line is drawn slightly left of the vertical axis. Plotted points appear near r near negative 15 and s near 0, r near negative 10 and s near 3, r near negative 9 and s near negative 3, r near 0 and s near 0, r near 0 and s near 7, r near 0 and s near negative 7, r near 4 and s near 10, r near 4 and s near 4, r near 4 and s near negative 3 and r near 4 and s near negative 10. The points form groupings around r near negative 15, r near negative 10, r near 0 and r near 4. The image F showing a scatter plot with the text, t equals 40. The horizontal axis is labeled, r, with tick labels at negative 20, negative 15, negative 10, negative 5, 5, 10, 15. The vertical axis is labeled, s, with tick labels at negative 15, negative 10, negative 5, 5, 10, 15. A dashed vertical line is drawn slightly left of the vertical axis. Plotted points appear near r near negative 16 and s near 0, r near negative 12 and s near 3, r near negative 10 and s near negative 3, r near negative 6 and s near 0, r near 0 and s near 10, r near 0 and s near 4, r near 0 and s near negative 4, r near 0 and s near negative 10, r near 4 and s near 12, r near 4 and s near 7, r near 4 and s near 0, r near 4 and s near negative 7, r near 4 and s near negative 12 and r near 14 and s near 0. The points form groupings around r near negative 16, r near negative 12 to negative 10, r near negative 6, r near 0, r near 4 and a single point near r near 14. Across the six scatter plots, the dashed vertical line appears in each plot at a similar r position slightly left of the vertical axis. The number of plotted points increases from image A to image C and from image D to image F.

4.3. General peak patterns

The procedure outlined in Section 4.2 can be extended to determine the peak locations and associated patterns in the $rs$-plane for the function $u(r,s; \gamma_1, \gamma_3, \ldots, \gamma_{2n-1})$ asymptotically as $|\gamma_{2j-1}| \gg 1$ for any fixed $j \gt 2$, and where the magnitudes of the remaining parameters are $O(1)$. The resulting patterns are not triangular as for the $j=2$ case but vary with each $j$. For conveninence we set $\gamma_{2j-1} = t$ as in Section 4.2. Then as $|t| \gg 1$, the leading order behaviour of $R(\textbf{0})$ using (4.6) is obtained as

since $\theta_{2j-1} = i\gamma_{2j-1}$. Proceeding as in Section 4.2, one sets

where $a_j$ is a constant to be determined later. Substituting these into the expression for $R(\textbf{0})$ above, yields

where $Q^{[j]}_n(\xi)$ is a monic polynomial of degree $N=n(n+1)/2$ and is regarded as the $j^{\mathrm th}$-order Yablonskii–Vorob’ev polynomial with suitable choice of the scaling parameter $a_j$. The polynomial $Q^{[j]}_n(\xi)$ is derived from the Wronskian form $Q(\textbf{0})(\tilde{\theta}) \sim \operatorname{\rm Wr}\nolimits(p_1,p_3,\ldots, p_{2n-1})$ whose asymptotic form for $|\gamma_{2j-1}|=|t| \gg 1$ is obtained by taking

after rescaling $z=(t/a_j)^{1/(2j-1)}\xi$. The reduced Schur polynomials $S^{[j]}_r(\xi,a_j)$ are given by

Then to leading order

from which one can recover the expression for $Q_n^{[j]}(\xi)$ as

Here the coefficients $c_r$ are related to the characters of the symmetric group and the non-negative integers $s_j$ and $m$ are determined by (long division)

i.e., $s_j$ is the residue of $N$ modulo $2j-1$. Finally, if one sets $a_j=4^{j-1}/(2j-1)$, the corresponding polynomials $Q^{[j]}_n(\xi)$ are the $j \gt 2$ extensions of the Yablonskii–Vorob’ev polynomials as they are also associated with the rational solutions of the Painlevé II hierarchy [Reference Clarkson and Mansfield13]. For $j=2$, $Q^{[2]}_n(\xi) = Q_n(\xi)$ are the usual Yablonskii–Vorob’ev polynomials.

Recall that for $2j-1=3$, $N=3m, 3m+1$, i.e., $s_2 \in \{0,1\}$ and $s_2 \neq 2$ so that the Yablonskii–Vorob’ev polynomials have at most one root at $\xi=0$. However, for $j \gt 2$, the exact set of residues $N \equiv s_j$ mod $2j-1$ for triangular numbers $N=n(n+1)/2, \, n \in \mathbb{N}$ can not be enumerated in a simple way to the best of the author’s knowledge. If $n \equiv r$ mod $2j-1$ then $N=n(n+1)/2 \equiv r(r+1)/2$ mod $2j-1$. Hence, the residue $s_j$ can be found by only considering the values $0 \leq r \leq 2j-2$. Furthermore, the residues corresponding to $r=j-1+k$ and $r=j-1-k$ for $0 \leq k \leq j-1$ are the same. Thus it suffices to check the residues $r(r+1)/2$ mod $2j-1$ only for the values $0 \leq r \leq j-1$ in order to obtain the full set of $s_j$ for any given $j$. For example, when $j=3$, i.e., $2j-1=5$, the possible residues $s_3=0,1,3$ which are obtained by considering $r=0,1,2$. Similarly, $s_4=0,1,3,6$, $s_5=0,1,3,6$ and so on. The upshot is that the $Q^{[j]}_n(\xi)$ has a root at $\xi=0$ of multiplicity $s_j \gt 1$ depending on the value of $n$. As for the roots at $\xi \neq 0$, it was conjectured in [Reference Clarkson and Mansfield13] that they are simple.

The peak locations for the lump solution $u(r,s,t)$ and $|\psi(0)|(r,s)$ for $|t| \gg 1$ is given in a similar way as in (4.8) from the expansion $z(t)=({\textstyle\frac t{a_j})^{1/(2j-1)}(\xi+\epsilon z_1+\epsilon^2z_2+\cdots), \,\,\epsilon=t^{-1/(2j-1)}.}$One obtains that

corresponding to the roots $\xi_p \neq 0$ for the polynomials $Q^{[j]}_n(\xi)$, assuming that they are simple. The peak locations in the $rs$-plane scale as $O(|t|^{1/(2j-1)})$ and are separated for large values of $|t|$. The peak patterns approximately follow the distribution of the zeros of the $Q^{[j]}_n(\xi)$ which exhibit the symmetry $\xi \to \xi e^{2\pi i/(2j-1)}$ in the complex plane since $Q^{[j]}_n(\xi)=\xi^{s_j}P^{[j]}(\zeta), \, \zeta=\xi^{2j-1}$. For each root $\zeta_r, \, 1 \leq r \leq m$ of $P^{[j]}(\zeta)$, there are $2j-1$ roots $\xi_i^{(r)}, \, 1 \leq i \leq 2j-1$ of $Q^{[j]}_n(\xi)$ arranged $2\pi/(2j-1)$ apart on a circle around the origin of radius $|\zeta_r|$.

The peak locations corresponding to the $\xi=0$ roots of $Q^{[j]}_n(\xi)$ are obtained as in the triangular case of Section 4.2. In this case, arranging the terms in the Wronskian of (4.6) in descending powers of $\tilde{\theta}_{2j-1}$, one finds that the term corresponding to highest power of $\tilde{\theta}_{2j-1}$ is $\tilde{\theta}_1^{s_j}\tilde{\theta}_{2j-1}^m, \, m=(N-s_j)/(2j-1)$. Then (4.5) implies that there is an asymptotic solution of the form $\tilde{\theta}_1^{s_j} \sim 0 + O(t^{-1}), \, \tilde{\theta}_{2j-1} \sim it$. This leads to the peak locations

Thus $z_p \sim O(1)$ and they are all located within a $O(|t|^{-1})$ neighbourhood, and it is necessary to retain the $O(t^{-1})$ term (which depends on the lower order terms contributing to $R(\textbf{0})$ in (4.4) and the parameters $\gamma_{2l-1}, \, l \neq j$) in order to distinguish them.

Figure 5.

$6$-lump solutions of the NLS equation. Parameters: $\gamma_1=\gamma_2=0=\gamma_3 = 0, \gamma_5=4000, b=\textstyle\frac12$. Left panel: $|\psi(0)|$, right panel: $u$.

Figure 6.

$6$-lump solutions of the NLS equation. Parameters: $\gamma_1=\gamma_2=\gamma_3 = 0, \gamma_5=4000, b=1$. Black dots represent locations $z_p$ where $R_3(0)=0$, red open circles are exact locations. Notice that the central peak is located on the $r$-axis near $r=-\textstyle\frac{n}{2b}$ with $n=3$, as prescribed by the asymptotic estimate (4.10b).

The special case of the $6$-lump solution and the corresponding peak locations for large $|\gamma_5|$ (i.e., $n=3, j=3$) are shown in Figures 5 and 6. Unlike the triangular patterns in Figures 3 and 4, these $6$-lump solutions form a pentagonal pattern in the $rs$-plane with a single peak at the centre of the pattern near the origin. The central peak appears since $N=6 \equiv 1$ mod $2j-1=5$ so that the residue $s_2=1$. Figure 6 shows both the exact peak locations as well as the estimated values obtained from (4.10) solving for $R_5(\textbf{0})=0$ asymptotically as $|\gamma_5| \gg 1$. The peak locations near the vertices of the pentagon correspond to the non-zero roots of the polynomial $Q^{[3]}_3(\xi)=\xi(\xi^5 -144)$.

Next we consider the case of $n=4$, which corresponds to a $10$-lump solution depending on the parameters $\{\gamma_1,\gamma_3,\gamma_5,\gamma_7\}$. The peak locations for this case when $|\gamma_3| \gg 1$ and the remaining parameters are $O(1)$ was already illustrated in Figure 4. Here, we depict the peak patterns when (i) $|\gamma_5| \gg 1$ and (ii) $|\gamma_7| \gg 1$ and the remaining parameters are $O(1)$ in each subcase. For the first subcase, $j=3$. Setting $R_3(\textbf{0})=0$ in (4.9), the leading-order peak locations are given in terms of the roots of the polynomial $Q^{[3]}_4(\xi)=\xi^{10}-1008\xi^5 -48384$. Note that $Q^{[3]}_4(\xi)$ has no roots at $\xi=0$ since $10$ is divisible by $2j-1=5$ and also that $Q^{[3]}_4(\xi) = P^{[3]}(\zeta), \, \zeta=\xi^5$. Since $P^{[3]}(\zeta)$ is quadradic in $\zeta$ with non-zero roots $\zeta_1, \zeta_2$, Figure 7 shows that the $10$ peaks are arranged in $2$ sets of pentagonal pattern spaced $2 \pi/5$ apart on circles of radii $|\zeta_1|$ and $|\zeta_2|$ around the origin in the $rs$-plane. These peak locations are obtained from (4.10a).

The second subcase corresponding to $j=4$ and $|\gamma_7| \gg 1$ is more interesting, and is obtained asymptotically from $R_4(\textbf{0})=0$ in (4.9). Since $10 \equiv 3$ mod $7$, i.e., $s_4=3$, there are $3$ peaks located near the origin as shown in Figure 8. These peaks correspond to the $\xi=0$ roots of multiplicity $3$ for the polynomial $Q^{[4]}_4(\xi) = \xi^3(\xi^7+14400)$ and the locations are given by (4.10b). Note that the peaks form a triangular pattern near the origin. This stems from considering the subdominant $O(t^{-1})$ terms in (4.10b) with the parameter $\gamma_3 \sim O(1)$. We do not pursue the analysis here. The remaining $7$ peaks are arranged $2 \pi/7$ apart on a circle around the origin, and their locations are given to leading order by (4.10a) for $|\gamma_7| \gg 1$.

Figure 7.

$10$-lump peak locations of the NLS equation. Parameters: $\gamma_j=0,\, 1 \leq j \leq 4, \gamma_5=10000, \gamma_6=\gamma_7=0, b=\textstyle\frac12$. Left panel: $|\psi(0)|$, right panel: $u$. The peaks are arranged in $2$ sets of pentagonal pattern given by (4.10a).

Figure 8.

$10$-lump solutions of the NLS equation. Parameters: $\gamma_j=0$ except for $\gamma_3=10, \gamma_7=10^6, b=\textstyle\frac{1}{2}$. Black dots represent locations $z_p$ where $R_4(0)=0$, red open circles are exact locations. Notice that the $3$ peaks near the centre form a triangular pattern due to $\gamma_3 \neq 0$. These locations can be obtained from further asymptotic estimate of the $O(t^{-1})$ term in (4.10b).