Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation

In this paper, we consider pattern formation of higher-order lumps in the KP-I equation (3). General higher-order lump solutions have been derived by Ablowitz et al. Ablowitz2000 through Darboux transformation. Their solutions were given through determinants whose matrix elements involve differential operators with respect to the spectral parameter. For our analysis, those solution expressions are not explicit enough. Thus, we have derived these higher-order lumps again by the bilinear method. To present our solutions, we first introduce elementary Schur polynomials $S_{k}(\mbox{\boldmath$x$})$ with $\emph{{x}}=\left(x_{1},x_{2},\ldots\right)$, which are defined by the generating function

More explicitly,

Under these notations, our general higher-order KP-I lumps are given by the following theorem.

Theorem 1   General higher-order lumps of the KP-I equation (3) are

$\displaystyle u_{\Lambda}(x,y,t)=2\partial_{x}^{2}\ln\sigma,$

(7)

where

$$\sigma(x,y,t)=\det_{1\leq i,j\leq N}\left(m_{ij}\right),$$

(8)

$$m_{i,j}=\sum_{\nu=0}^{\min(n_{i},n_{j})}\left[\frac{|p|^{2}}{(p+p^{*})^{2}}\right]^{\nu}\hskip 1.70709ptS_{n_{i}-\nu}(\textbf{\emph{x}}^{+}+\nu\textbf{\emph{s}}+\textbf{\emph{a}}_{i})\hskip 1.70709ptS_{n_{j}-\nu}[(\textbf{\emph{x}}^{+})^{*}+\nu\textbf{\emph{s}}^{*}+\textbf{\emph{a}}_{j}^{*}],$$

(9)

$N$ is an arbitrary positive integer, $\Lambda\equiv(n_{1},n_{2},\cdots n_{N})$ is a vector of arbitrary positive integers, $p$ is an arbitrary non-imaginary complex number, the asterisk ‘*’ represents complex conjugation, the vector $\textbf{\emph{x}}^{+}=\left(x_{1}^{+},x_{2}^{+},\cdots\right)$ is defined by

$\displaystyle x_{k}^{+}=p\frac{1}{k!}x+p^{2}\frac{2^{k}}{k!}\textrm{i}y+p^{3}\frac{3^{k}}{k!}(-4)t,$

(10)

the vector $\textbf{\emph{s}}=(s_{1},s_{2},\cdots)$ is defined through the expansion

$\displaystyle\ln\left[\frac{1}{\kappa}\left(p+p^{*}\right)\left(\frac{e^{\kappa}-1}{p\hskip 1.42271pte^{\kappa}+p^{*}}\right)\right]=\sum_{j=1}^{\infty}s_{j}\hskip 1.42271pt\kappa^{j},$

(11)

vectors $\textbf{\emph{a}}_{i}$ are

$$\textbf{\emph{a}}_{i}=\left(a_{i,1},a_{i,2},\cdots,a_{i,n_{i}}\right),$$

(12)

and $a_{i,j}\hskip 1.42271pt(1\leq i\leq N,1\leq j\leq n_{i})$ are free complex constants.

The proof of this theorem will be given in the Appendix.

Remark 1. In this theorem, positive integers $(n_{1},n_{2},\cdots n_{N})$ do not have to be distinct if their corresponding vectors $\textbf{\emph{a}}_{i}$ are different. In such cases, by first rewriting the $\sigma$ determinant (8) as a larger determinant as was done in Ref. OhtaJY2012 , then linking Schur polynomials with different $\textbf{\emph{a}}_{i}$ vectors in that larger determinant by relations similar to Eq. (167) in Ref. YangYang3wave , and finally applying row operations and parameter redefinitions to the resulting determinant, we can show that this $\sigma$ determinant (8) with non-distinct integers $(n_{1},n_{2},\cdots n_{N})$ can be reduced to one where the new integers $(\hat{n}_{1},\hat{n}_{2},\cdots\hat{n}_{N})$ become distinct. Thus, in this paper, we will require positive integers $(n_{1},n_{2},\cdots n_{N})$ to be distinct without loss of generality. In this case, we will also arrange them in the ascending order, i.e., $n_{1}<n_{2}<\cdots<n_{N}$.

Remark 2. The higher-order lumps in Theorem 1 contain free complex parameters $p$ and $\textbf{\emph{a}}_{i}$ ($1\leq i\leq N$), totaling $1+n_{1}+n_{2}+\cdots n_{N}$. However, using techniques similar to that outlined in Remark 1, we can show that $N(N-1)/2$ of those parameters in $\{\textbf{\emph{a}}_{i}\}$ can be eliminated. Thus, the number of free complex parameters in these higher-order lumps can be reduced to $1+\rho$, where

This number of free parameters matches that given in Ref. Ablowitz2000 for solutions produced by Darboux transformation. In fact, from the derivation of Theorem 1 in the Appendix, we can see that our higher-order lumps given in this theorem by the bilinear method are equivalent to those derived in Ablowitz2000 by Darboux transformation, except that our expressions are more explicit.

Remark 3. The fundamental lump can be derived by taking $N=1$ and $n_{1}=1$ in Eq. (8). Through a shift of the $(x,y)$ axes, we can normalize $a_{1,1}=0$. Then, the resulting $\sigma_{1}(x,y,t)$ function can be reduced to

where $p_{r}$ and $p_{i}$ are the real and imaginary parts of the spectral parameter $p$. The corresponding solution $u_{1}(x,y,t)$ through Eq. (7) moves at $x$-direction velocity of $12|p|^{2}$ and $y$-direction velocity of $12p_{i}$. By applying the Galilean invariance (4) with $k=p_{i}$, we can remove the $y$-direction velocity $12p_{i}$ and reduce $\sigma_{1}(x,y,t)$ to

This means that, under Galilean invariance, we can take $p$ in the original fundamental lump to be purely real without loss of generality. Then, by utilizing the scaling invariance (5) with $\alpha=p_{r}$, we can further normalize $p_{r}$ in the above $\sigma_{1}$ to be unity. The final simplified fundamental-lump expression is

This is a moving single lump with peak amplitude $16$, which is attained at the spatial location of $(x,y)=(12t,\hskip 1.42271pt0)$.

Remark 4. In the general higher-order lump of Theorem 1, the whole solution complex moves at $x$-direction velocity $12|p|^{2}$ and $y$-direction velocity $12p_{i}$, plus some possible slower motion relative to that moving frame. In this general case, we can also use the Galilean invariance (4) to remove the $y$-direction velocity $12p_{i}$ of the complex, i.e., $p_{i}$ can be made to be zero. In addition, we can use the scaling invariance (5) to normalize $p_{r}$ to unity. Thus, without any loss of generality, we can choose $p$ in the higher-order lump solution of Theorem 1 to be equal to one. For this reason, we will set $p=1$ in the remainder of this paper.

Remark 5. In Peli93a , a class of higher-order lump solutions that are stationary in a moving frame was reported. Those special solutions satisfy the Boussinesq equation. Thus, they are special cases of Boussinesq rogue waves YangYangBoussi . Those stationary higher-order lumps are also special cases of our solutions in Theorem 1 when the index vector $(n_{1},n_{2},\cdots n_{N})$ and internal parameters $\{\textbf{\emph{a}}_{i}\}$ are properly chosen. Indeed, rational solutions in Theorem 1 would be stationary if the $\sigma$ function in (8) satisfies the dimension-reduction condition $\sigma_{t}-V\sigma_{x}=0$, where $V$ is the velocity of the moving frame along the $x$-direction. In the bilinear derivation of Boussinesq rogue waves YangYangBoussi , one needs to solve the bilinear $\tau$ equation of KP-I, together with this $\tau$’s dimension reduction condition $\tau_{x_{3}}-3\tau_{x_{1}}=C\tau$, where $x_{1}$ is proportional to $x$, $x_{3}$ proportional to $t$, and $C$ is a constant. Since this $\tau$ function turns out to be equal to $\sigma$ multiplying an exponential of a linear function of $x$ and $t$ YangYangBoussi , we see that $\tau$’s dimension reduction condition is equivalent to $\sigma$’s dimension reduction condition after proper variable scalings. This means that constraints from $\tau$’s dimension-reduction condition can be borrowed over and imposed on solutions of Theorem 1 in order to obtain stationary higher-order KP lumps. One of such constraints is on the index vector $(n_{1},n_{2},\cdots n_{N})$, where $n_{i}=2i-1$ must be chosen YangYangBoussi . In addition, internal parameters $\{\textbf{\emph{a}}_{i}\}$ also need to be constrained. For a different choice of differential operators than those in Eq. (105) of the Appendix, this parameter constraint was derived in YangYangBoussi . For the present choice of differential operators in Eq. (105), this parameter constraint would be more complex. In this case, such a parameter constraint was worked out in ChenJunChao2018 for another integrable system under a different parameterization.

We will show in later text that patterns of certain higher-order lump solutions at large time are described by root structures of the Yablonskii–Vorob’ev polynomials and Wronskian-Hermit polynomials. Thus, these polynomials and their root structures will be introduced first.

Our first theorem is for the case where the index vector $\Lambda$ is equal to $(1,3,5,\dots,2N-1)$.

Theorem 2. If the index vector $\Lambda=(1,3,5,\dots,2N-1)$, then, when $|t|\gg 1$, the higher-order lump solution $u_{\Lambda}(x,y,t)$ asymptotically separates into $N(N+1)/2$ fundamental lumps $u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)$, where $u_{1}(x,y,t)$ is given in Eq. (16),

$$x_{0}=\Re(z_{0})\hskip 1.42271pt(12t)^{1/3},\quad y_{0}=\frac{\Im(z_{0})}{2}(12t)^{1/3},$$

(36)

$z_{0}$ is each of the $N(N+1)/2$ simple roots of the Yablonskii–Vorob’ev polynomial $Q_{N}(z)$, and $\Re$, $\Im$ represent the real and imaginary parts of a complex number. The peak of each fundamental lump is spatially located at $(x,y)=(12t+x_{0},y_{0})$. The absolute error of this fundamental-lump approximation is $O(|t|^{-1/3})$ when $z_{0}\neq 0$ and $O(|t|^{-1})$ when zero is a root and $z_{0}=0$. Expressed mathematically, when $(x,y)$ is in the neighborhood of each of these fundamental lumps, i.e., $(x-12t-x_{0})^{2}+(y-y_{0})^{2}=O(1)$, we have the following solution asymptotics for $|t|\gg 1$,

$$u_{\Lambda}(x,y,t)=\left\{\begin{array}[]{ll}u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)+O\left(|t|^{-1/3}\right),&\mbox{if}\hskip 1.9919ptz_{0}\neq 0,\\ u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)+O\left(|t|^{-1}\right),&\mbox{if}\hskip 1.9919ptz_{0}=0.\end{array}\right.$$

(37)

When $(x,y)$ is not in the neighborhood of any of these $N(N+1)/2$ fundamental lumps, $u_{\Lambda}(x,y,t)$ asymptotically approaches zero as $|t|\to\infty$.

The proof of this theorem will be provided in Sec. V.

This theorem indicates that, wave patterns at large times are formed by $N(N+1)/2$ fundamental lumps. Relative to the moving frame of $x$-direction velocity $12$, positions $(x_{0},y_{0})$ of these fundamental lumps are just a simple linear transformation of the root structure of the Yablonskii–Vorob’ev polynomial $Q_{N}(z)$, i.e.,

Since the transformation matrix is diagonal, this transformation is simply a stretching along both horizontal and vertical directions. Recall that the Yablonskii–Vorob’ev root structure is triangular (see Fig. 1). The resulting lump pattern is then triangular as well. When $t\gg 1$, this triangular lump pattern preserves the same orientation of the original triangle of the Yablonskii–Vorob’ev root structure. But when $t\ll-1$, the triangular lump pattern would be oriented opposite of the Yablonskii–Vorob’ev root structure. Indeed, it is easy to see from Eq. (38) that, when time changes from large negative to large positive, i.e., from $-t$ to $+t$, their lump positions would be related as

Thus, these triangular lump patterns have reversed directions along the $x$-axis (the $y$-direction reversal does not matter since the pattern is symmetric in $y$). This $x$-direction reversal of triangular lump patterns when time changes from large negative to large positive is a dramatic pattern transformation in the KP-I equation. This phenomenon has been graphically reported in Dubard2013 on several low-order solution examples. Here, we established this fact for the general case.

Theorem 2 also indicates that, at large time, fundamental lumps in the solution complex separate from each other in proportion to $|t|^{1/3}$. This rate of separation is very slow, relative to the overall (linear) speed $12$ of the whole complex.

One more feature of Theorem 2 is that, positions $(x_{0},y_{0})$ in Eq. (36) for individual fundamental lumps in the solution complex are independent of the solution’s internal parameters a. This means that, when $|t|\to\infty$, solutions $u_{\Lambda}(x,y,t)$ with different internal parameters a would approach the same limit solution.

When $\Lambda\neq(1,3,5,\dots,2N-1)$, the Wronskian-Hermite polynomial $W_{\Lambda}(z)$ has a zero root of multiplicity $d(d+1)/2$, with $d$ given in Eq. (26), as well as nonzero roots that are conjectured to be all simple (see Sec. II.2.2). Note that the zero root would be absent if $d=0$ or $-1$; but nonzero roots always exist. In this case, our results on solution patterns at large time are summarized in the following theorem.

Theorem 3. Suppose the index vector $\Lambda\neq(1,3,5,\dots,2N-1)$, and all nonzero roots of the Wronskian-Hermite polynomial $W_{\Lambda}(z)$ are simple. Then, for $|t|\gg 1$, the following asymptotics for the solution $u_{\Lambda}(x,y,t)$ holds.

1. 1.

   In the outer region — the region that is $O\left(|t|^{1/2}\right)$ away from the wave center $(x,y)=(12t,0)$, or $\sqrt{(x-12t)^{2}+y^{2}}=O\left(|t|^{1/2}\right)$, the higher-order lump $u_{\Lambda}(x,y,t)$ asymptotically separates into $N_{W}$ fundamental lumps $u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)$, where $N_{W}$ is given in Eq. (27), $u_{1}(x,y,t)$ is given in Eq. (16),

   $$x_{0}=\Re\left[z_{0}(-12t)^{1/2}\right]+O(1),\quad y_{0}=\frac{\Im\left[z_{0}(-12t)^{1/2}\right]}{2}+O(1),$$

   (40)

   and $z_{0}$ is each of the $N_{W}$ nonzero simple roots of $W_{\Lambda}(z)$. The absolute error of this fundamental-lump approximation is $O(|t|^{-1/2})$. Expressed mathematically, when $(x,y)$ is in the neighborhood of each of these fundamental lumps, i.e., $(x-12t-x_{0})^{2}+(y-y_{0})^{2}=O(1)$, we have the following solution asymptotics

   $$u_{\Lambda}(x,y,t)=u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)+O\left(|t|^{-1/2}\right),\quad|t|\gg 1.$$

   (41)

2. 2.

   If zero is a root of $W_{\Lambda}(z)$, i.e., $d\neq 0$ and $-1$, then in the inner region — the region that is within $O(|t|^{1/3})$ of the wave center $(x,y)=(12t,0)$, or $\sqrt{(x-12t)^{2}+y^{2}}\leq O\left(|t|^{1/3}\right)$, lies $d(d+1)/2$ fundamental lumps $u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)$, where $u_{1}(x,y,t)$ is given in Eq. (16),

   $$x_{0}=\Re(z_{0})\hskip 1.42271pt(12t)^{1/3}+O(1),\quad y_{0}=\frac{\Im(z_{0})}{2}(12t)^{1/3}+O(1),$$

   (42)

   and $z_{0}$ is each of the $d(d+1)/2$ simple roots of the Yablonskii–Vorob’ev polynomial $Q_{\hat{d}}(z)$, with $\hat{d}$ defined as

   $$\hat{d}=\left\{\begin{array}[]{ll}d,&\mbox{when}\hskip 2.84544ptd\geq 0,\\ |d|-1,&\mbox{when}\hskip 2.84544ptd\leq-1.\end{array}\right.$$

   (43)

   Notice that $d(d+1)/2=\hat{d}(\hat{d}+1)/2$. The absolute error of this fundamental-lump approximation is $O(|t|^{-1/3})$ when $z_{0}\neq 0$ and $O(|t|^{-1})$ when zero is a root of $Q_{\hat{d}}(z)$ and $z_{0}=0$. Expressed mathematically, when $(x,y)$ is in the neighborhood of each of these fundamental lumps, i.e., $(x-12t-x_{0})^{2}+(y-y_{0})^{2}=O(1)$, with $(x_{0},y_{0})$ given in (42), we have the following solution asymptotics for $|t|\gg 1$,

   $$u_{\Lambda}(x,y,t)=\left\{\begin{array}[]{ll}u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)+O\left(|t|^{-1/3}\right),&\mbox{if}\hskip 1.9919ptz_{0}\neq 0,\\ u_{1}(x-x_{0},\hskip 1.13791pty-y_{0},t)+O\left(|t|^{-1}\right),&\mbox{if}\hskip 1.9919ptz_{0}=0.\end{array}\right.$$

   (44)

3. 3.

   When $(x,y)$ is not in the neighborhood of any of the above fundamental lumps, $u_{\Lambda}(x,y,t)$ asymptotically approaches zero as $|t|\to\infty$.

Remark 6. In this theorem, we assumed all nonzero roots of $W_{\Lambda}(z)$ simple, which is true for all examples we tested, such as the two in Eqs. (32)-(33). In view of Conjecture 1 in the previous section, this assumption is expected to hold in all cases. If this conjecture is false, i.e., some nonzero roots of $W_{\Lambda}(z)$ are not simple, then this theorem for the outer region, i.e., Eqs. (40)-(41), would still hold, but only for nonzero simple roots $z_{0}$ of $W_{\Lambda}(z)$.

Now, we explain what Theorem 3 says regarding solution patterns at large times when $\Lambda\neq(1,3,5,\dots,2N-1)$. In this case, Theorem 3 indicates that, the whole wave field is generically split up into two regions featuring different patterns.

1. 1.

   In the outer region — the region that is $O(|t|^{1/2})$ away from the wave center $(x,y)=(12t,0)$, the wave field at large time comprises $N_{W}$ fundamental lumps, whose positions are given through the nonzero roots of the Wronskian-Hermite polynomial $W_{\Lambda}(z)$. Specifically, relative to the moving frame of $x$-direction velocity $12$, positions $(x_{0},y_{0})$ of these fundamental lumps, to the leading order of large time, are just a linear transformation of $W_{\Lambda}(z)$’s nonzero-root structure. The reader is reminded from Sec. II.2.2 that when $\Lambda\neq(1,3,5,\dots,2N-1)$, nonzero roots of $W_{\Lambda}(z)$ always exist, and their shape in the $z$-plane is non-triangular. When $t$ is large negative, these fundamental-lump positions to the leading order are

   $$\left[\begin{array}[]{c}x_{0}^{-}\\ y_{0}^{-}\end{array}\right]=(12|t|)^{1/2}\left[\begin{array}[]{cc}1&0\\ 0&\frac{1}{2}\end{array}\right]\left[\begin{array}[]{c}\Re(z_{0})\\ \Im(z_{0})\end{array}\right],$$

   (45)

   where $z_{0}$ is any nonzero root of $W_{\Lambda}(z)$. However, when $t$ is large positive, these lump positions become

   $$\left[\begin{array}[]{c}x_{0}^{+}\\ y_{0}^{+}\end{array}\right]=(12|t|)^{1/2}\left[\begin{array}[]{cc}0&-1\\ \frac{1}{2}&0\end{array}\right]\left[\begin{array}[]{c}\Re(z_{0})\\ \Im(z_{0})\end{array}\right].$$

   (46)

   In the former case, the wave pattern formed by these fundamental lumps is simply a stretching of the Wronskian-Hermite nonzero-root structure along both horizontal and vertical directions. But in the latter case, on top of this stretching, the horizontal and vertical directions are also swapped. In both cases, the resulting wave patterns from transformations (45)-(46) are non-triangular since the root structure of $W_{\Lambda}(z)$ is non-triangular.

   From the above two transformations, we see that fundamental lumps at large negative time $-t$ and large positive time $+t$ in the outer region are related as

   $$\left[\begin{array}[]{c}x_{0}^{+}\\ y_{0}^{+}\end{array}\right]=\left[\begin{array}[]{cc}0&-2\\ \frac{1}{2}&0\end{array}\right]\left[\begin{array}[]{c}x_{0}^{-}\\ y_{0}^{-}\end{array}\right].$$

   (47)

   Thus, when time goes from large negative to large positive, outer-region lump patterns in the $(x,y)$ plane have swapped horizontal and vertical directions. In addition, stretching of different amounts has also occurred along the two directions. This swapping of horizontal and vertical directions is another type of dramatic pattern transformation, and it is very different from the triangular $x$-direction reversal that occurs when $\Lambda=(1,3,5,\dots,2N-1)$. For certain single-line patterns of fundamental lumps, a change from a vertical line to a horizontal line in the $(x,y)$ plane has been graphically reported in Chen2016 and analytically explained in Chang2018 . Here, we proved this fact for the general case, where patterns of fundamental lumps based on Wronskian-Hermite root structures can be arbitrary, not just lines (see the next section for examples).

   In this outer region, fundamental lumps at large time separate from each other in proportion to $|t|^{1/2}$. This is another big difference between the present solutions and those with $\Lambda=(1,3,5,\dots,2N-1)$ in the previous subsection, where lumps separate in proportion to $|t|^{1/3}$ instead.

2. 2.

   In the inner region — the region that is within $O(|t|^{1/3})$ of the wave center $(x,y)=(12t,0)$, if $d\neq 0$ and $-1$, then the solution $u_{\Lambda}(x,y,t)$ at large time would comprise $d(d+1)/2$ fundamental lumps, whose positions are given through roots of the Yablonskii–Vorob’ev polynomial $Q_{\hat{d}}(z)$, with $\hat{d}$ defined in Eq. (43). The reader is reminded that $\hat{d}(\hat{d}+1)/2=d(d+1)/2$. Relative to the moving frame of $x$-direction velocity $12$, positions $(x_{0},y_{0})$ of these fundamental lumps, to the leading order of large time, are just a linear transformation of $Q_{\hat{d}}(z)$’s root structure, i.e.,

   $$\left[\begin{array}[]{c}x_{0}\\ y_{0}\end{array}\right]=(12t)^{1/3}\left[\begin{array}[]{cc}1&0\\ 0&\frac{1}{2}\end{array}\right]\left[\begin{array}[]{c}\Re(z_{0})\\ \Im(z_{0})\end{array}\right],$$

   (48)

   where $z_{0}$ is each of the $d(d+1)/2$ simple roots of $Q_{\hat{d}}(z)$. This lump-position formula in the inner region is very similar to (38) of the $\Lambda=(1,3,5,\dots,2N-1)$ case. Thus, the pattern of these $d(d+1)/2$ fundamental lumps in the inner region at large time is a simple stretching of $Q_{\hat{d}}(z)$’s root structure, and the resulting pattern is triangular if $\hat{d}>1$. In addition, as time evolves from large negative to large positive, these triangular lump patterns would reverse direction along the $x$-axis. Furthermore, fundamental lumps in this inner region separate from each other in proportion to $|t|^{1/3}$ at large time, similar to the $\Lambda=(1,3,5,\dots,2N-1)$ case in Theorem 2. If $\hat{d}=0$, i.e., $d=0$ or $-1$, this inner region would be absent.