Cauchy matrix approach to the noncommutative Kadomtsev–Petviashvili equation with self-consistent sources

Abstract

We develop a direct method, the Cauchy matrix approach, to construct matrix solutions of noncommutative soliton equations. This approach is based on the Sylvester equation, and solutions can be presented without using quasideterminants. The matrix Kadomtsev–Petviashvili equation with self-consistent sources is employed as an example to demonstrate the approach. As a reduction, explicit solutions of the matrix Mel’nikov model for long–short wave interaction are obtained.

Appendix A. Proof of Lemma 1

We prove Lemma 1. Direct calculation shows that $\boldsymbol\rho$ and $\boldsymbol\sigma$ in (3.6) satisfy Eqs. (3.7b), (3.7c), and (3.7d). In what follows, we work with Sylvester equation (3.7a). Introducing

$$\boldsymbol\Xi=\mathbf F_a(\rho(l))^{-1} \boldsymbol\Theta(\mathbf F_b^{\mathrm T}(\sigma(k)))^{-1},$$

we can rewrite (3.7a) as

$$\boldsymbol\Gamma_a(l)\mathbf F_a(\rho(l)) \boldsymbol\Xi\,\mathbf F_b^{\mathrm T}(\sigma(k)) +\mathbf F_a(\rho(l))\boldsymbol\Xi \mathbf F_b^{\mathrm T}(\sigma(k))\boldsymbol\Gamma_b^{\mathrm T}(k) =\mathbf F_a(\rho(l))\mathbf e_a\mathbf e_b^{\mathrm T} \mathbf F_b^{\mathrm T}(\sigma(k)). $$

(A.1)

Because all lower-triangle Toeplitz matrices of the same order commute, we have

$$\boldsymbol\Gamma_a(l)\mathbf F_a(\rho(l)) =\mathbf F_a(\rho(l))\boldsymbol\Gamma_a(l),\qquad \mathbf F_b^{\mathrm T}(\sigma(k))\boldsymbol\Gamma_b^{\mathrm T}(k) =\boldsymbol\Gamma_b^{\mathrm T}(k)\mathbf F_b^{\mathrm T}(\sigma(k)).$$

It then follows that Eq. (A.1) reduces to

$$\boldsymbol\Gamma_a(l)\boldsymbol\Xi +\boldsymbol\Xi\boldsymbol\Gamma_b^{\mathrm T}(k) =\mathbf e_a\mathbf e_b^{\mathrm T}=\begin{pmatrix} 1 &0 &\dots &0 \\ 0 &0 &\ddots & \vdots \\ \vdots &\ddots &\ddots &0 \\ 0 &\dots &0 &0 \end{pmatrix}_{a\times b}.$$

The condition $k+l\ne 0$ leads to the unique solution (cf. [26])

$$\begin{aligned} \, &\boldsymbol\Xi=\{\mathbf G_{a,b}(l,k)\}_{ij} =\begin{pmatrix}{i-1}\\{i+j-2}\end{pmatrix} \frac{(-1)^{i+j}}{(l+k)^{i+j-1}}, \\ &\begin{pmatrix}{i-1}\\{i+j-2}\end{pmatrix} =\frac{(i+j-2)!}{(i-1)!\,(j-1)!}. \end{aligned} $$

(A.2)

Appendix B. Some examples of solutions

We present some examples of solutions of the matrix Mel’nikov model (4.6). For brevity, we take $2\times2$ matrix solutions as example and choose $\mathbf J=\mathbf I$, without giving illustrations for dynamics. However, we mention some typical features of the dynamics. We note that $\mathbf u$ is a Hermitian matrix (see (4.3)); in the $2\times 2$ case, there are three waves $u_{11}$, $u_{22}$, and $|u_{12}|$, which is a carrier wave. It follows from (B.7) that for a given $t$, the three waves behave in the same manner but with different initial phases: they are line solitons in the $(x,y)$ plane and an arbitrary function $\alpha_1(z)$ (due to the self-consistent source) can freely change the velocities of these line solitons.

One-soliton solution

To obtain a $2\times 2$ matrix one-soliton solution, we choose $N=1$, $n=2$, whence we have

$$\begin{aligned} \, &\mathbf L=l_1,\qquad \boldsymbol\alpha=\alpha_1,\qquad \mathbf r=(\tilde\rho_1,\hat\rho_1), \\ &\mathbf M=\tilde\rho_1\frac{1}{l_1+l_1^*}\tilde\rho_1^* +\hat\rho_1\frac{1}{l_1+l_1^*}\hat\rho_1^* =\frac{|\tilde\rho_1|^2+|\hat\rho_1|^2}{l_1+l_1^*}. \end{aligned} $$

(B.1)

The plane-wave factors are

$$\tilde\rho_1=e^{\ell(l_1)}\tilde\rho^{(0)}(l_1),\qquad \hat\rho_1=e^{\ell(l_1)}\hat\rho^{(0)}(l_1), $$

(B.2)

where $\ell(l_1)=\ell(x,t,y;l_1)$ is a linear function of $x$, $y$, and $t$ with parameters $l_1$ and $\hat\rho^{(0)}(l_1)$, defined by

$$\ell(l_1)=l_1x+\mathrm il_1^2y+4l_1^3t+\delta\int_0^t\alpha_1(z)\,dz. $$

(B.3)

We note that $\alpha_1(z)$ is an arbitrary function of $z$. Hence, we have

$$\mathbf v=\mathbf S^{(0,0)} =\mathbf r^\dagger(\mathbf I+\mathbf M)^{-1}\mathbf r=\begin{pmatrix} v_{11} &v_{12} \\ v_{21} &v_{22} \end{pmatrix}=\frac{1}{\tau} \begin{pmatrix} |\tilde\rho_1|^2 &\tilde\rho_1^*\hat\rho_1 \\ \hat\rho_1^*\tilde\rho_1 &|\hat\rho_1|^2 \end{pmatrix}, $$

(B.4)

where the $\tau$ function is defined as the determinant of $\mathbf I+\mathbf M$:

$$\tau=|\mathbf I+\mathbf M| =1+\frac{|\tilde\rho_1|^2+|\hat\rho_1|^2}{l_1+l_1^*} =1+\frac{|\tilde\rho_1^{(0)}|^2 +|\hat\rho_1^{(0)}|^2}{2\operatorname{Re}[l_1]} e^{2\operatorname{Re}[\ell(l_1)]}. $$

(B.5)

The matrix $\mathbf v$ can be written as

$$\mathbf v=\frac{2\operatorname{Re}[l_1]e^{2\operatorname{Re}[\ell(l_1)]}} {2\operatorname{Re}[l_1]+(|\tilde\rho_1^{(0)}|^2+|\hat\rho_1^{(0)}|^2) e^{2\operatorname{Re}[\ell(l_1)]}}\begin{pmatrix} |\tilde\rho_1^{(0)}|^2 &(\tilde\rho_1^{(0)})^*\hat\rho_1^{(0)} \\ \tilde\rho_1^{(0)}(\hat\rho_1^{(0)})^* &|\hat\rho_1^{(0)}|^2 \end{pmatrix}, $$

(B.6)

where the matrix part is composed of initial phase terms that are independent of $x$, $y$, and $t$. A solution of the matrix Mel’nikov model (4.6) is given by $\mathbf u=2\mathbf v_x$:

$$\mathbf u=\frac{16(\operatorname{Re}[l_1])^3e^{2\operatorname{Re}[\ell(l_1)]}} {(2\operatorname{Re}[l_1]+(|\tilde\rho_1^{(0)}|^2+|\hat\rho_1^{(0)}|^2) e^{2\operatorname{Re}[\ell(l_1)]})^2}\begin{pmatrix} |\tilde\rho_1^{(0)}|^2 &(\tilde\rho_1^{(0)})^*\hat\rho_1^{(0)} \\ \tilde\rho_1^{(0)}(\hat\rho_1^{(0)})^* &|\hat\rho_1^{(0)}|^2 \end{pmatrix}. $$

(B.7)

For the source part, we assume that $\operatorname{Re}[\alpha_1]>0$, whence we have

$$\Phi=\frac{\sqrt{\alpha_1+\alpha_1^*}}{\tau}\begin{pmatrix} \tilde\rho_1^* \\ \hat\rho_1^* \end{pmatrix} =\frac{2\operatorname{Re}[l_1]e^{2\operatorname{Re}[\ell(l_1)]+\ell^*(l_1)} \sqrt{2\operatorname{Re}[\alpha_1]}} {2\operatorname{Re}[l_1]+(|\tilde\rho_1^{(0)}|^2+|\hat\rho_1^{(0)}|^2) e^{2\operatorname{Re}[\ell(l_1)]}}\begin{pmatrix} (\tilde\rho_1^{(0)})^* \\ (\hat\rho_1^{(0)})^* \end{pmatrix}.$$

Two-soliton solution

To obtain a two-soliton solution, we write

$$\begin{aligned} \, &\mathbf L=\operatorname{diag}(l_1,l_2),\qquad \boldsymbol\alpha=\operatorname{diag}(\alpha_1,\alpha_2), \\ &\mathbf r=\begin{pmatrix} \tilde\rho_1 &\hat\rho_1 \\ \tilde\rho_2 &\hat\rho_2 \end{pmatrix},\qquad \mathbf M=\begin{pmatrix} \dfrac{|\tilde\rho_1|^2+|\hat\rho_1|^2}{l_1+l_1^*} &\dfrac{\tilde\rho_1\tilde\rho_2^*+\hat\rho_1\hat\rho_2^*}{l_1+l_2^*} \\ \dfrac{\tilde\rho_2\tilde\rho_1^*+\hat\rho_2\hat\rho_1^*}{l_2+l_1^*} &\dfrac{|\tilde\rho_2|^2+|\hat\rho_2|^2}{l_2+l_2^*} \end{pmatrix}, \end{aligned} $$

(B.8)

where

$$\begin{aligned} \, &\tilde\rho_i=e^{\ell(l_i)}\tilde\rho^{(0)}(l_i),\qquad \hat\rho_i=e^{\ell(l_i)}\hat\rho^{(0)}(l_i), \\ &\ell(l_i)=l_ix+\mathrm il_i^2y+4l_i^3t+\delta\int_0^t\alpha_i(z)\,dz,\qquad i=1,2. \end{aligned}$$

Similarly to the one-soliton case, we have

$$\begin{aligned} \, \tau&=1+\frac{|\tilde\rho_1|^2+|\hat\rho_1|^2}{l_1+l_1^*} +\frac{|\tilde\rho_2|^2+|\hat\rho_2|^2}{l_2+l_2^*}+{} \\ &\qquad{}+\frac{(|\tilde\rho_1|^2+|\hat\rho_1|^2)(|\tilde\rho_2|^2+|\hat\rho_2|^2)} {(l_1+l_1^*)(l_2+l_2^*)} -\frac{(\tilde\rho_1\tilde\rho_2^*+\hat\rho_1\hat\rho_2^*) (\tilde\rho_1^*\tilde\rho_2+\hat\rho_1^*\hat\rho_2)}{(l_1+l_2^*)(l_2+l_1^*)}= \\ &=1+e^{2\operatorname{Re}[\ell(l_1)]} \frac{|\tilde\rho_1^{(0)}|^2+|\hat\rho_1^{(0)}|^2}{2\operatorname{Re}[l_1]} +e^{2\operatorname{Re}[\ell(l_2)]} \frac{|\tilde\rho_2^{(0)}|^2+|\hat\rho_2^{(0)}|^2}{2\operatorname{Re}[l_2]}+{} \\ &\qquad{}+e^{2\operatorname{Re}[\ell(l_1+l_2)]} \biggl(\frac{(|\tilde\rho_1^{(0)}|^2+|\hat\rho_1^{(0)}|^2) (|\tilde\rho_2^{(0)}|^2+|\hat\rho_2^{(0)}|^2)} {4\operatorname{Re}[l_1]\operatorname{Re}[l_2]} -\frac{|\tilde\rho_1^{(0)}(\tilde\rho_2^{(0)})^* +\hat\rho_1^{(0)}(\hat\rho_2^{(0)})^*|^2} {|l_1+l_2^*|^2}\biggr) \end{aligned}$$

and

$$\mathbf v=\begin{pmatrix} v_{11} &v_{12} \\ v_{21} &v_{22} \end{pmatrix}=\frac{1}{\tau}\begin{pmatrix} \tilde\rho_1^* &\tilde\rho_2^* \\ \hat\rho_1^* &\hat\rho_2^* \end{pmatrix}\begin{pmatrix} 1+\dfrac{|\tilde\rho_2|^2+|\hat\rho_2|^2}{l_2+l_2^*} &-\dfrac{\tilde\rho_1\tilde\rho_2^*+\hat\rho_1\hat\rho_2^*}{l_1+l_2^*} \\ -\dfrac{\tilde\rho_2\tilde\rho_1^*+\hat\rho_2\hat\rho_1^*}{l_2+l_1^*} &1+\dfrac{|\tilde\rho_1|^2+|\hat\rho_1|^2}{l_1+l_1^*} \end{pmatrix}\begin{pmatrix} \tilde\rho_1 &\hat\rho_1 \\ \tilde\rho_2 &\hat\rho_2 \end{pmatrix}. $$

(B.9)

Assuming that $\operatorname{Re}[\alpha_1]>0$, $\operatorname{Re}[\alpha_2]>0$, we have

$$\begin{alignedat}{2} &T_{11}=1+\frac{|\tilde\rho_2|^2+|\hat\rho_2|^2}{l_2+l_2^*},&\qquad &T_{12}=-\frac{\tilde\rho_1\tilde\rho_2^*+\hat\rho_1\hat\rho_2^*}{l_1+l_2^*}, \\ &T_{21}=-\frac{\tilde\rho_2\tilde\rho_1^*+\hat\rho_2\hat\rho_1^*}{l_2+l_1^*},&\qquad &T_{22}=1+\frac{|\tilde\rho_1|^2+|\hat\rho_1|^2}{l_1+l_1^*}. \end{alignedat}$$

The four entries of $\mathbf v$ are then given by

$$\begin{aligned} \, &v_{11}=|\tilde\rho_1|^2T_{11}+|\tilde\rho_2|^2T_{22} +2\operatorname{Re}[\tilde\rho_1\tilde\rho_2^*T_{21}], \\ &v_{12}=\tilde\rho_1^*\hat\rho_1T_{11} +\hat\rho_1\tilde\rho_2^*T_{21}+\tilde\rho_1^*\hat\rho_2T_{12} +\tilde\rho_2^*\hat\rho_2T_{22}, \\ &v_{21}=\tilde\rho_1\hat\rho_1^*T_{11} +\tilde\rho_1\hat\rho_2^*T_{21}+\hat\rho_1^*\tilde\rho_2T_{12} +\tilde\rho_2\hat\rho_2^*T_{22}=v_{12}^*, \\ &v_{22}=|\hat\rho_1|^2T_{11}+|\hat\rho_2|^2T_{22} +2\operatorname{Re}[\hat\rho_1\hat\rho_2^*T_{21}], \end{aligned} $$

(B.10)

and for the source part, we have

$$\begin{aligned} \, \Phi&=\frac{1}{\tau}\begin{pmatrix} \tilde\rho_1^* &\tilde\rho_2^* \\ \hat\rho_1^* &\hat\rho_2^* \end{pmatrix}\begin{pmatrix} T_{11} &T_{12} \\ T_{21} &T_{22} \end{pmatrix}\begin{pmatrix} \sqrt{2\operatorname{Re}[\alpha_1]} &0 \\ 0 &\sqrt{2\operatorname{Re}[\alpha_2]} \end{pmatrix}= \\ &=\frac{1}{\tau}\begin{pmatrix} \sqrt{2\operatorname{Re}[\alpha_1]}(\tilde\rho_1^*T_{11}+\tilde\rho_2^*T_{21}) &\sqrt{2\operatorname{Re}[\alpha_2]}(\tilde\rho_1^*T_{12}+\tilde\rho_2^*T_{22}) \\ \sqrt{2\operatorname{Re}[\alpha_1]}(\hat\rho_1^*T_{11}+\hat\rho_2^*T_{21}) &\sqrt{2\operatorname{Re}[\alpha_2]}(\hat\rho_1^*T_{12}+\hat\rho_2^*T_{22}) \end{pmatrix}. \end{aligned}$$

As we can see, the source function $\Phi$ is no longer a row vector but a matrix.