Nondegenerate solitons in the integrable fractional coupled Hirota equation

Firstly, we consider the Ablowitz-Kaup-Newell-Segur (AKNS) scattering problem

(2.1a)		$$\Phi_{x}={\bf U}\Phi,$$
(2.1b)		$$\Phi_{t}={\bf V}\Phi,$$

where $\Phi(\lambda;x,t)$ is a wave function associated with $\lambda$, $\lambda\in\mathbb{C}$ is an eigenvalue parameter, and ${\bf U}(\lambda;x,t)$ can be expressed in the following polynomial with respect to $\lambda$

$$\begin{split}&{\bf U}=-\mathrm{i}\lambda\Lambda+{\bf U}_{0},\ \ \ \ \Lambda={\rm diag}\left(1,-1,-1\right),\ \ \ {\bf U}_{0}(x,t)=\begin{bmatrix}0&{\bf q}(x,t)\\[4.0pt] {\bf r}^{\top}(x,t)&0\\ \end{bmatrix},\\ &{\bf q}(x,t)=\begin{bmatrix}q_{1}(x,t)&q_{2}(x,t)\end{bmatrix},\ \ \ \ {\bf r}(x,t)=\begin{bmatrix}r_{1}(x,t)&r_{2}(x,t)\end{bmatrix},\end{split}$$

the superscript ^⊤ denotes the transpose, and

$${\bf V}(\lambda;x,t)=\begin{bmatrix}{\bf V}_{1}(\lambda;x,t)&{\bf V}_{2}(\lambda;x,t)\\[4.0pt] {\bf V}_{3}^{\top}(\lambda;x,t)&{\bf V}_{4}(\lambda;x,t)\end{bmatrix}.$$

Note that the matrix ${\bf V}(\lambda;x,t)$ has the same block structure as the matrix ${\bf U}(\lambda;x,t)$. In order to ensure the compatibility of system (2.1), the following zero curvature equation needs to be satisfied

(2.2)

$${\bf U}_{t}-{\bf V}_{x}+[{\bf U},{\bf V}]=0,$$

where the commutator is defined by $[{\bf U},{\bf V}]\equiv{\bf UV-VU}.$

Combining the equation (2.2) and making appropriate assumptions, the integrable hierarchy associated with the third-order scattering problem can be obtained

(2.3)

$$\mathrm{i}\begin{bmatrix}{\bf q}\\[4.0pt] -{\bf r}\end{bmatrix}_{t}=\mathcal{F}(\mathcal{L})\begin{bmatrix}{\bf q}\\[4.0pt] {\bf r}\end{bmatrix},$$

where $\mathcal{F}(\mathcal{L})$ is a function with respect to the operator $\mathcal{L}$, and here $\mathcal{F}(\mathcal{L})=\mathcal{L}^{n}\ (n=1,2,\cdots)$. The recursion operator $\mathcal{L}$ is

(2.4)

$$\begin{split}&\mathcal{L}\begin{bmatrix}\widetilde{{\bf q}}\\[4.0pt] \widetilde{{\bf r}}\end{bmatrix}=\mathrm{i}\begin{bmatrix}\widetilde{{\bf q}}_{x}-{\bf q}\partial_{-}^{-1}({\bf r}^{\top}\widetilde{{\bf q}}-\widetilde{{\bf r}}^{\top}{\bf q})-\big{(}\partial_{-}^{-1}(\widetilde{{\bf q}}{\bf r}^{\top}+{\bf q}\widetilde{{\bf r}}^{\top})\big{)}{\bf q}\\[4.0pt] -\widetilde{{\bf r}}_{x}-{\bf r}\partial_{-}^{-1}(\widetilde{{\bf q}}^{\top}{\bf r}-{\bf q}^{\top}\widetilde{{\bf r}})+\big{(}\partial_{-}^{-1}({\bf r}\widetilde{{\bf q}}^{\top}+\widetilde{{\bf r}}{\bf q}^{\top})\big{)}{\bf r}\end{bmatrix},\ \ \ \ \ \ \ \begin{split}\widetilde{{\bf q}}=&\begin{bmatrix}\widetilde{q}_{1}&\widetilde{q}_{2}\end{bmatrix},\\[4.0pt] \widetilde{{\bf r}}=&\begin{bmatrix}\widetilde{r}_{1}&\ \widetilde{r}_{2}\end{bmatrix},\end{split}\end{split}$$

where $\partial_{-}^{-1}=\int_{-\infty}^{x}\mathrm{d}y$. The adjoint operator of $\mathcal{L}$ is defined as

(2.5)

$$\mathcal{L}^{A}\begin{bmatrix}\widetilde{{\bf q}}\\[4.0pt] \widetilde{{\bf r}}\end{bmatrix}=\mathrm{i}\begin{bmatrix}-\widetilde{{\bf q}}_{x}-{\bf r}\partial_{+}^{-1}({\bf q}^{\top}\widetilde{{\bf q}}+\widetilde{{\bf r}}^{\top}{\bf r})-\big{(}\partial_{+}^{-1}(\widetilde{{\bf q}}{\bf q}^{\top}+{\bf r}\widetilde{{\bf r}}^{\top})\big{)}{\bf r}\\[4.0pt] \widetilde{{\bf r}}_{x}+{\bf q}\partial_{+}^{-1}(\widetilde{{\bf q}}^{\top}{\bf q}+{\bf r}^{\top}\widetilde{{\bf r}})-\big{(}\partial_{+}^{-1}({\bf q}\widetilde{{\bf q}}^{\top}+\widetilde{{\bf r}}{\bf r}^{\top})\big{)}{\bf q}\end{bmatrix},$$

where $\partial_{+}^{-1}=\int_{x}^{+\infty}\mathrm{d}y$. In this case, given the value of $n$ and the symmetry relation between ${\bf q}(x,t)$ and ${\bf r}(x,t)$, the corresponding integer order integrable equation can be obtained. In order to introduce the fractional integrable equations, we need to make some appropriate changes to the function $\mathcal{F}(\mathcal{L})$. That is to say that we need to add the fractional power of the operator $\mathcal{L}$ to $\mathcal{F}(\mathcal{L})$. In this paper, we take $\mathcal{F}(\mathcal{L})=(\beta+\alpha\mathcal{L})\mathcal{L}^{2}|\mathcal{L}^{2}|^{\epsilon}$ with $\epsilon\in[0,1)$, $\alpha,\beta\in\mathbb{R}$, then

(2.6)

$$\begin{bmatrix}{\bf q}\\ {\bf r}\end{bmatrix}_{t}=\sigma_{3}|\mathcal{L}^{2}|^{\epsilon}\begin{bmatrix}\mathrm{i}\beta({\bf q}_{xx}-2{\bf q}{\bf r}^{\top}{\bf q})-\alpha({\bf q}_{xxx}-3{\bf q}_{x}{\bf r}^{\top}{\bf q}-3{\bf q}{\bf r}^{\top}{\bf q}_{x})\\[4.0pt] \mathrm{i}\beta({\bf r}_{xx}-2{\bf r}{\bf q}^{\top}{\bf r})+\alpha({\bf r}_{xxx}-3{\bf r}_{x}{\bf q}^{\top}{\bf r}-3{\bf r}{\bf q}^{\top}{\bf r}_{x})\end{bmatrix},$$

where $\sigma_{3}={\rm diag}(1,-1)$. In particular, when $\beta=0$, ${\bf r}^{\top}=\sigma{\bf q}^{\dagger},\ \sigma=\pm 1$, we can derive the fractional coupled complex mKdV equation, and when $\alpha=0$, ${\bf r}^{\top}=\sigma{\bf q}^{\dagger},\ \sigma=\pm 1$, we can derive the fractional coupled NLS equation. Without loss of generality, we take $\beta=\frac{1}{2}$, ${\bf r}^{\top}=-{\bf q}^{\dagger}$, then we can get the fcHirota equation which can be regarded as the fractional type to the coupled Hirota equation (1.1). Under this case, the corresponding operator $\mathcal{F}(\mathcal{L})$ becomes $\mathcal{F}_{h}(\mathcal{L})=(\frac{1}{2}+\alpha\mathcal{L})\mathcal{L}^{2}|\mathcal{L}^{2}|^{\epsilon}$. The explicit form of the fcHirota equation will be given in subsection 2.4.

Considering the linearization of the fcHirota equation (2.6) (i.e. when $\beta=\frac{1}{2},\ {\bf r}^{\top}=-{\bf q}^{\dagger}$)

(2.7)

$$\mathrm{i}{\bf q}_{t}+|-\partial^{2}|^{\epsilon}\Big{(}\frac{1}{2}{\bf q}_{xx}+\mathrm{i}\alpha{\bf q}_{xxx}\Big{)}=0,$$

where $|-\partial^{2}|^{\epsilon}$ is the Riesz fractional derivative [13]. We substitute the formal solution

(2.8)

$${\bf q}=\begin{bmatrix}q_{1}&q_{2}\end{bmatrix}\thicksim\begin{bmatrix}\mathrm{e}^{\mathrm{i}\big{(}k_{1}x-\omega_{1}(k_{1})t\big{)}}&\mathrm{e}^{\mathrm{i}\big{(}k_{2}x-\omega_{2}(k_{2})t\big{)}}\end{bmatrix}$$

into the equation (2.7), then the anomalous dispersive relations can be obtained

(2.9)

$$\omega_{j}(k_{j})=\mathcal{F}_{h}(-k_{j})=(\frac{1}{2}-\alpha k_{j})k_{j}^{2}|k_{j}^{2}|^{\epsilon},\ \ \ j=1,2,$$

in which we have $\mathcal{F}_{h}(k_{j})=\omega_{j}(-k_{j})$.

We assume that the potential ${\bf q}(x,t)$ is sufficiently smooth and rapidly tends to zero as $|x|\to\infty$. In integer order integrable equations, the matrix ${\bf V}(\lambda;x,t)$ can be expressed as a matrix function concerning $\lambda$, while for the fractional integrable equations in the sense of this paper, the matrix ${\bf V}(\lambda;x,t)$ usually cannot be given precisely. Nevertheless, here we need to give the constraints [16]

(2.10)

$$\begin{cases}{\bf V}_{j}(\lambda;x,t)\to 0,\ \ j=1,2,3,\\[4.0pt] {\bf V}_{4}(\lambda;x,t)\to\mathrm{i}\mathcal{F}_{h}(2\lambda)\mathbb{I}_{2},\end{cases}\ \ |x|\to\infty.$$

Therefore, we can derive the asymptotic behavior

(2.11)

$$\Phi^{\pm}(\lambda;x,t)\sim\mathrm{e}^{-\mathrm{i}\lambda\Lambda x+\mathrm{i}\mathcal{F}_{h}(2\lambda)\Omega t},\ \ |x|\to\infty,$$

where $\Omega={\rm diag}(0,1,1)$, $\Phi^{\pm}(\lambda;x,t)=\begin{bmatrix}\phi^{\pm}_{1},&\phi^{\pm}_{2},&\phi^{\pm}_{3}\end{bmatrix}$, and the superscripts ^± refer to the cases $x\to\pm\infty$. Then the modified Jost solutions $\Psi^{\pm}(\lambda;x,t)$ can be defined as

(2.12)

$$\Psi^{\pm}(\lambda;x,t)=\Phi^{\pm}(\lambda;x,t)\mathrm{e}^{\mathrm{i}\lambda\Lambda x-\mathrm{i}\mathcal{F}_{h}(2\lambda)\Omega t}\rightarrow\mathbb{I}_{3},\ \ \ |x|\to\infty.$$

And we can get an equation of $\Psi(\lambda;x,t)$, which is equivalent to the equation (2.1a)

(2.13)

$$\Psi_{x}=-\mathrm{i}\lambda[\Lambda,\Psi]+{\bf U}_{0}\Psi.$$

In addition, the Volterra integral equations for $\Psi^{\pm}(\lambda;x,t)$ can be given as

(2.14)

$$\Psi^{\pm}(\lambda;x)=\mathbb{I}_{3}+\int_{\pm\infty}^{x}\mathrm{e}^{-\mathrm{i}\lambda{\rm ad}\widehat{\Lambda}(x-y)}{\bf U}_{0}(y)\Psi^{\pm}(\lambda;y)dy,$$

where $\mathrm{e}^{{\rm ad}\widehat{\Lambda}}{\bf A}=\mathrm{e}^{\Lambda}{\bf A}\mathrm{e}^{-\Lambda},$ $\Psi^{\pm}(\lambda;x,t)=\begin{bmatrix}\psi^{\pm}_{1},&\psi^{\pm}_{2},&\psi^{\pm}_{3}\end{bmatrix}$. Let $\mathbb{C^{+}}=\{\lambda|\Im\lambda>0\},\ \mathbb{C^{-}}=\{\lambda|\Im\lambda<0\}.$ Based on the equation (2.14), we can find that the first column of $\Psi^{+}$ (i.e., $\psi^{+}_{1}$) only contains the exponential factor $\mathrm{e}^{2\mathrm{i}\lambda(x-y)}$, which decays when $\lambda\in\mathbb{C^{-}}$ due to the condition $y>x$ in the integral. Similarly, the second and the third columns (i.e., $\psi^{+}_{2},\ \psi^{+}_{3}$) decay in the region $\lambda\in\mathbb{C^{+}}$. We can analyze the analyticity of $\Psi^{-}$ in the same way, and then we summarize the above analysis as follows: for any $x,t\in\mathbb{R}$, $\psi_{1}^{-},\ \psi_{2}^{+},\ \psi_{3}^{+}$ are analytic for $\lambda\in\mathbb{C}^{+}$, while $\psi_{1}^{+},\ \psi_{2}^{-},\ \psi_{3}^{-}$ are analytic for $\lambda\in\mathbb{C}^{-}$.

From above, we know $\Phi^{\pm}(\lambda;x,t)$ can be regarded as the fundamental solution of the equation (2.1). By the theory of ordinary differential equations, there exists a matrix ${\bf S}(\lambda)=\left(s_{ij}(\lambda)\right)_{i,j=1,2,3}$ between them obeying the relation

(2.15)

$$\Phi^{-}=\Phi^{+}{\bf S}(\lambda),\ \ \ \ \lambda\in\mathbb{R},$$

and the matrix ${\bf S}(\lambda)$ is called the scattering matrix. Combined with the equation (2.13), the equation (2.15) can be rewritten as

(2.16)

$$\Psi^{-}=\Psi^{+}\mathrm{e}^{-\mathrm{i}\lambda{\rm ad}\widehat{\Lambda}x+\mathrm{i}\mathcal{F}_{h}(2\lambda){\rm ad}\widehat{\Omega}t}{\bf S}(\lambda),\ \ \ \lambda\in\mathbb{R},$$

where $\mathrm{e}^{{\rm ad}\widehat{\Omega}}{\bf A}=\mathrm{e}^{\Omega}{\bf A}\mathrm{e}^{-\Omega}.$ Now we assume $\det\Phi^{\pm}=\mu(\lambda)$ which implies $\det{\bf S}(\lambda)=1$. Using (2.15),

(2.17)

$$s_{1j}(\lambda)=\frac{1}{\mu}\big{|}\phi_{j}^{-},\phi_{2}^{+},\phi_{3}^{+}\big{|},\ \ s_{2j}(\lambda)=\frac{1}{\mu}\big{|}\phi_{1}^{+},\phi_{j}^{-},\phi_{3}^{+}\big{|},\ \ s_{3j}(\lambda)=\frac{1}{\mu}\big{|}\phi_{j}^{-},\phi_{1}^{+},\phi_{2}^{+}\big{|},\ \ \ \ j=1,2,3,$$

and we can find that $s_{11}(\lambda)$ is analytic in $\mathbb{C^{+}}$, $\{s_{jk}(\lambda)\}_{j,k=2,3}$ are analytic in $\mathbb{C^{-}}$. In general, $s_{12}(\lambda),\ s_{13}(\lambda)$, $s_{21}(\lambda),\ s_{31}(\lambda)$ cannot be extended off the real $\lambda$-axis.

In what follows, we shall formulate the RHP using the properties of $\Psi^{\pm}(\lambda;x,t)$. To this end, we combine the ideas in [30] to construct two matrix functions ${\bf P}^{\pm}(\lambda;x,t)$ which are analytic in $\mathbb{C}^{\pm}$, respectively,

(2.18)

$$\begin{split}&{\bf P}^{+}(\lambda;x,t)=\begin{bmatrix}\psi_{1}^{-}&\psi_{2}^{+}&\psi_{3}^{+}\end{bmatrix}=\Psi^{-}{\bf H}_{1}+\Psi^{+}{\bf H}_{2},\\ &{\bf P}^{-}(\lambda;x,t)=\begin{bmatrix}\widetilde{\psi}^{-}_{1}&\widetilde{\psi}^{+}_{2}&\widetilde{\psi}^{+}_{3}\end{bmatrix}^{\top}={\bf H}_{1}\widetilde{\Psi}^{-}+{\bf H}_{2}\widetilde{\Psi}^{+},\end{split}$$

where

(2.19)

$${\bf H}_{1}={\rm diag}(1,0,0),\ \ \ \ {\bf H}_{2}={\rm diag}(0,1,1),\ \ \ \widetilde{\Psi}^{\pm}=(\Psi^{\pm})^{-1}=\begin{bmatrix}\widetilde{\psi}^{\pm}_{1}&\widetilde{\psi}^{\pm}_{2}&\widetilde{\psi}^{\pm}_{3}\end{bmatrix}^{\top}.$$

There are some properties of ${\bf P}^{\pm}(\lambda;x,t)$ which can be derived through a series of complex analyses, and we summarize them in the proposition 1.

Moreover, from the fact that ${\bf U}_{0}^{\dagger}(x,t)=-{\bf U}_{0}(x,t)$, we have $\Psi^{\dagger}(\lambda^{*};x,t)=\Psi^{-1}(\lambda;x,t)$. Then the symmetry relation of ${\bf S}(\lambda)$ can be obtained by combining the equation (2.16)

(2.20)

$${\bf S}^{\dagger}(\lambda^{*})={\bf S}^{-1}(\lambda),$$

which implies $s^{*}_{11}(\lambda^{*})=\widehat{s}_{11}(\lambda),\ \lambda\in\mathbb{C^{-}}$. Note that the inverse of the solution of the adjoint matrix spectral problem is exactly the solution of the matrix spectral problem. So we can define a sectional analytic solution ${\bf M}(\lambda;x,t)$

(2.21)

$$\begin{split}&{\bf M}^{+}(\lambda;x,t)={\bf P}^{+}(\lambda;x,t),\ \ \ \ \ \ \ \ \ \ \lambda\in\mathbb{C^{+}},\\ &{\bf M}^{-}(\lambda;x,t)=\big{(}{\bf P}^{-}(\lambda;x,t)\big{)}^{-1},\ \ \ \lambda\in\mathbb{C^{-}}.\end{split}$$

Then we can formulate the following RHP.
Riemann-Hilbert problem $1$. We can find a matrix ${\bf M}(\lambda;x,t)$ with the following properties:

• •

  ${\bf Analyticity:}$ ${\bf M}^{\pm}(\lambda;x,t)$ are analytic in $\mathbb{C}^{\pm}$, respectively.

• •

  ${\bf Jump\ condition:}$ The two matrix functions ${\bf M}^{\pm}(\lambda;x,t)$ can be linked by

  $${\bf M}^{+}(\lambda;x,t)={\bf M}^{-}(\lambda;x,t){\bf J}(\lambda;x,t),$$

  where the matrix ${\bf J}(\lambda;x,t)$ is

  $$\begin{split}{\bf J}(\lambda;x,t)&=\mathrm{e}^{-\mathrm{i}\lambda{\rm ad}\widehat{\Lambda}x+\mathrm{i}\mathcal{F}_{h}(2\lambda){\rm ad}\widehat{\Omega}t}\big{(}({\bf H}_{1}{\bf S}^{-1}+{\bf H}_{2})({\bf S}{\bf H}_{1}+{\bf H}_{2})\big{)}\\[4.0pt] &=\begin{bmatrix}1&\mathrm{e}^{-\mathrm{i}(2\lambda x+\mathcal{F}_{h}(2\lambda)t)}\widehat{s}_{12}&\mathrm{e}^{-\mathrm{i}(2\lambda x+\mathcal{F}_{h}(2\lambda)t)}\widehat{s}_{13}\\[4.0pt] \mathrm{e}^{\mathrm{i}(2\lambda x+\mathcal{F}_{h}(2\lambda)t)}s_{21}&1&0\\[4.0pt] \mathrm{e}^{\mathrm{i}(2\lambda x+\mathcal{F}_{h}(2\lambda)t)}s_{31}&0&1\end{bmatrix}.\end{split}$$

• •

  ${\bf Normalization:}$ ${\bf M}(\lambda;x,t)\rightarrow\mathbb{I}_{3}$, as $\lambda\to\infty$.

The RHP with zeros generates soliton solutions. The uniqueness of the RHP $1$ does not hold unless the zeros of $\det{\bf P}^{\pm}(\lambda;x,t)$ in $\mathbb{C}^{\pm}$ are specified and the kernel structures of ${\bf P}^{\pm}(\lambda;x,t)$ at these zeros are determined. In view of (2.20), it is clear that if $\lambda$ is a zero of $s_{11}(\lambda)$, $\lambda^{*}$ will be a zero of $\widehat{s}_{11}(\lambda)$. Based on these results, we suppose that $s_{11}(\lambda)$ has $n$ simple zeros $\{\lambda_{k}\}_{1}^{n}$. Accordingly, each zero of $\widehat{s}_{11}(\lambda)$ is complex conjugate to that of $s_{11}(\lambda)$, and we define them by $\{\widehat{\lambda}_{j}\}_{1}^{n}$. Under this assumption, the kernel of ${\bf P}^{+}(\lambda;x,t)$ contains a single column vector $|v_{k}\rangle=\begin{bmatrix}v_{1k},&v_{2k},&v_{3k}\end{bmatrix}^{\top}$ and the kernel of ${\bf P}^{-}(\lambda;x,t)$ contains a single row vector $\langle\widehat{v}_{j}|=\begin{bmatrix}\widehat{v}_{j1},&\widehat{v}_{j2},&\widehat{v}_{j3}\end{bmatrix}$,

(2.22)

$${\bf P}^{+}(\lambda_{k};x,t)|v_{k}\rangle=0,\ \ \ \ \langle\widehat{v}_{j}|{\bf P}^{-}(\widehat{\lambda}_{j};x,t)=0,\ \ \ \ j,k=1,2,\cdots,n.$$

Then we should eliminate the zero structures of ${\bf P}^{\pm}(\lambda;x,t)$ at $\lambda_{k},\ \widehat{\lambda}_{j}$ to ensure that the RHP $1$ can be solved via the Plemelj formula. Combining the idea in [30], we assume

(2.23)

$${\bf G}^{+}(\lambda;x,t)=\mathbb{I}_{3}-{\bf v}{\bf N}^{-1}\widehat{\Gamma}\widehat{{\bf v}},\ \ \ \ \ ({\bf G}^{-}(\lambda;x,t))^{-1}=\mathbb{I}_{3}+{\bf v}\Gamma{\bf N}^{-1}\widehat{{\bf v}},$$

where ${\bf N}(\lambda;x,t)$ is a matrix of order $n$ with its $(j,k)$-th element given by $n_{jk}=\frac{\langle\widehat{v}_{j}|v_{k}\rangle}{\lambda_{k}-\widehat{\lambda}_{j}},$ and

$$\begin{split}&{\bf v}=\big{(}|v_{1}\rangle,\cdots,|v_{n}\rangle\big{)},\ \ \ \ \Gamma={\rm diag}\big{(}(\lambda-\lambda_{1})^{-1},\cdots,(\lambda-\lambda_{n})^{-1}\big{)},\\ &\widehat{{\bf v}}=\big{(}\langle\widehat{v}_{1}|,\cdots,\langle\widehat{v}_{n}|\big{)}^{\top},\ \ \widehat{\Gamma}={\rm diag}\big{(}(\lambda-\widehat{\lambda}_{1})^{-1},\cdots,(\lambda-\widehat{\lambda}_{n})^{-1}\big{)}.\end{split}$$

Then there is ${\bf G}^{+}(\lambda;x,t)({\bf G}^{-}(\lambda;x,t))^{-1}=\mathbb{I}_{3}.$ From the above, we can construct two matrix functions

(2.24)

$$\widetilde{{\bf M}}^{+}(\lambda;x,t)={\bf M}^{+}(\lambda;x,t)\big{(}{\bf G}^{+}(\lambda;x,t)\big{)}^{-1},\ \ \ \ \ \widetilde{{\bf M}}^{-}(\lambda;x,t)={\bf M}^{-}(\lambda;x,t)\big{(}{\bf G}^{-}(\lambda;x,t)\big{)}^{-1}.$$

Then a new RHP can be constructed.
Riemann-Hilbert problem $2$. The matrix $\widetilde{{\bf M}}(\lambda;x,t)$ has the following properties:

• •

  ${\bf Analyticity:}$ $\widetilde{{\bf M}}^{\pm}(\lambda;x,t)$ is analytic in $\mathbb{C}^{\pm},$ respectively.

• •

  ${\bf Jump\ condition:}$ When $\lambda\in\mathbb{R}$, the jump relation between $\widetilde{{\bf M}}^{\pm}(\lambda;x,t)$ is as follows

  $$\widetilde{{\bf M}}^{+}(\lambda;x,t)=\widetilde{{\bf M}}^{-}(\lambda;x,t)\ \widetilde{{\bf J}}(\lambda;x,t),$$

  where the jump matrix is

  $$\widetilde{{\bf J}}(\lambda;x,t)={\bf G}^{-}(\lambda;x,t){\bf J}(\lambda;x,t)\big{(}{\bf G}^{+}(\lambda;x,t)\big{)}^{-1}.$$

• •

  ${\bf Normalization:}$ $\widetilde{{\bf M}}(\lambda;x,t)\rightarrow\mathbb{I}_{3}$, as $\lambda\to\infty$.

Then the solution to the RHP $2$ can be obtained by applying the Plemelj formula

(2.25)

$$\widetilde{\bf M}(\lambda)=\mathbb{I}_{3}+\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}}\frac{\widetilde{\bf M}^{-}(\xi)\left({\bf G}^{-}(\xi)({\bf J}(\xi)-\mathbb{I})({\bf G}^{+}(\xi))^{-1}\right)}{\xi-\lambda}d\xi.$$

Thus, when $\lambda\rightarrow+\infty,$

(2.26)

$${\bf M}^{+}(\lambda)=\mathbb{I}_{3}-\frac{1}{\lambda}\left(\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}}\widetilde{\bf M}^{-}(\xi)\Big{(}{\bf G}^{-}(\xi)({\bf J}(\xi)-\mathbb{I})({\bf G}^{+}(\xi))^{-1}\Big{)}d\xi+{\bf v}{\bf N}^{-1}\widehat{{\bf v}}\right)+\mathcal{O}(\lambda^{-2}).$$

To recover the potential functions $q_{1}(x,t)$ and $q_{2}(x,t)$, we expand ${\bf M}^{+}(\lambda;x,t)$ at large-$\lambda$ as

(2.27)

$${\bf M}^{+}(\lambda;x,t)=\mathbb{I}_{3}+\frac{1}{\lambda}\ {\bf M}^{+}_{1}(x,t)+\frac{1}{\lambda^{2}}\ {\bf M}^{+}_{2}(x,t)+\cdots.$$

Then the potential functions can be recovered by substituting (2.27) into (2.13) and comparing the coefficients of both sides of the equations to the same power of $\lambda$. From the coefficient of the highest power of $\lambda$, we can obtain

(2.28)

$${\bf U}_{0}=\mathrm{i}[\Lambda,{\bf M}^{+}_{1}]=\lim\limits_{\lambda\rightarrow+\infty}\mathrm{i}\lambda[\Lambda,{\bf M}^{+}].$$

Therefore

(2.29)

$$q_{1}(x,t)=2\mathrm{i}\big{(}{\bf M}^{+}_{1}(x,t)\big{)}_{12},\ \ \ \ q_{2}(x,t)=2\mathrm{i}\big{(}{\bf M}^{+}_{1}(x,t)\big{)}_{13}.$$

Based on the idea in [31, 30], we can construct the squared eigenfunctions $\{{\bf Z}_{1}^{\pm}(\lambda;x,t),\ {\bf Z}_{2}^{\pm}(\lambda;x,t)\}$ and the adjoint squared eigenfunctions $\{\Omega_{1}^{\pm}(\lambda;x,t),\ \Omega_{2}^{\pm}(\lambda;x,t)\}$, which are also the eigenfunctions of the recursion operator $\mathcal{L}$ and its adjoint $\mathcal{L}^{A}$, respectively,

(2.30)

$$\mathcal{L}{\bf Z}_{j}^{\pm}=2\lambda{\bf Z}_{j}^{\pm},\ \ \ \ \mathcal{L}^{A}\Omega_{j}^{\pm}=2\lambda\Omega_{j}^{\pm},\ \ j=1,2,$$

where

(2.31)

$$\begin{split}&{\bf Z}_{j}^{+}=\begin{bmatrix}\phi_{1,j+1}^{+}\ \widetilde{\phi}_{12}^{+}\\[4.0pt] \phi_{1,j+1}^{+}\ \widetilde{\phi}_{13}^{+}\\[4.0pt] \phi_{2,j+1}^{+}\ \widetilde{\phi}_{11}^{+}\\[4.0pt] \phi_{3,j+1}^{+}\ \widetilde{\phi}_{11}^{+}\end{bmatrix},\ \ {\bf Z}_{j}^{-}=\begin{bmatrix}\phi_{11}^{+}\ \widetilde{\phi}_{j+1,2}^{+}\\[4.0pt] \phi_{11}^{+}\ \widetilde{\phi}_{j+1,3}^{+}\\[4.0pt] \phi_{21}^{+}\ \widetilde{\phi}_{j+1,1}^{+}\\[4.0pt] \phi_{31}^{+}\ \widetilde{\phi}_{j+1,1}^{+}\end{bmatrix},\ \ \ \Omega_{j}^{+}=\begin{bmatrix}\ \ \ \widehat{\varphi}_{j1}\ \phi_{21}^{-}\\[4.0pt] \ \ \ \widehat{\varphi}_{j1}\ \phi_{31}^{-}\\[4.0pt] -\widehat{\varphi}_{j2}\ \phi_{11}^{-}\\[4.0pt] -\widehat{\varphi}_{j3}\ \phi_{11}^{-}\end{bmatrix},\ \ \Omega_{j}^{-}=\begin{bmatrix}-\widetilde{\phi}^{-}_{11}\ \varphi_{2j}\\[4.0pt] -\widetilde{\phi}^{-}_{11}\ \varphi_{3j}\\[4.0pt] \ \ \ \widetilde{\phi}^{-}_{12}\ \varphi_{1j}\\[4.0pt] \ \ \ \widetilde{\phi}^{-}_{13}\ \varphi_{1j}\end{bmatrix},\ \ j=1,2,\\[4.0pt] &\begin{bmatrix}\varphi_{1}&\varphi_{2}\end{bmatrix}=\begin{bmatrix}\phi_{2}^{-}&\phi_{3}^{-}\end{bmatrix}\begin{bmatrix}s_{33}&-s_{23}\\[4.0pt] -s_{32}&s_{22}\end{bmatrix},\ \ \ \ \ \ \ \begin{bmatrix}\widehat{\varphi}_{1}&\widehat{\varphi}_{2}\end{bmatrix}^{\top}=\begin{bmatrix}\widehat{s}_{33}&-\widehat{s}_{23}\\[4.0pt] -\widehat{s}_{32}&\widehat{s}_{22}\end{bmatrix}\begin{bmatrix}\widetilde{\phi}^{-}_{2}\\[4.0pt] \widetilde{\phi}^{-}_{3}\end{bmatrix},\\[4.0pt] &\ \ \widetilde{\Phi}^{\pm}=(\Phi^{\pm})^{-1}=\begin{bmatrix}\widetilde{\phi}^{\pm}_{1}&\widetilde{\phi}^{\pm}_{2}&\widetilde{\phi}^{\pm}_{3}\end{bmatrix}^{\top},\ \ \ \ \widetilde{\phi}^{\pm}_{k}=\begin{bmatrix}\widetilde{\phi}^{\pm}_{k1}&\widetilde{\phi}^{\pm}_{k2}&\widetilde{\phi}^{\pm}_{k3}\end{bmatrix},\ \ k=1,2,3,\\[4.0pt] &\ \ \ \varphi_{l}=\begin{bmatrix}\varphi_{1l}&\varphi_{2l}&\varphi_{3l}\end{bmatrix}^{\top},\ \ \ \ \widehat{\varphi}_{l}=\begin{bmatrix}\widehat{\varphi}_{l1}&\widehat{\varphi}_{l2}&\widehat{\varphi}_{l3}\end{bmatrix},\ \ l=1,2.\end{split}$$

and the eigenfunctions ${\bf Z}_{j}^{\pm}(\lambda;x,t),\ \Omega_{j}^{\pm}(\lambda;x,t)\ (j=1,2)$ are all complete. Here, the superscript ^± of ${\bf Z}_{j}^{\pm}(\lambda;x,t)$ and $\Omega_{j}^{\pm}(\lambda;x,t)$ mean that they are analytic for $\lambda\in\mathbb{C}^{\pm}$, respectively. Similarly,

(2.32)

$$\mathcal{F}_{h}(\mathcal{L}){\bf Z}_{j}^{\pm}=\mathcal{F}_{h}(2\lambda){\bf Z}_{j}^{\pm},\ \ \ \ \mathcal{F}_{h}(\mathcal{L}^{A})\Omega_{j}^{\pm}=\mathcal{F}_{h}(2\lambda)\Omega_{j}^{\pm},\ \ j=1,2.$$

Then we assume a sufficiently smooth and decaying vector function $\vartheta(x,t)=\begin{bmatrix}\vartheta_{1},&\vartheta_{2},&\vartheta_{3},&\vartheta_{4}\end{bmatrix}^{\top}$ which may be expanded in terms of the eigenfunctions (2.31) as

(2.33)

$$\vartheta(x,t)=-\frac{1}{\pi}\bigg{(}\int_{\Gamma_{\mathbb{R}}^{+}}s_{11}^{-2}(\xi)\int_{\mathbb{R}}{\bf F}_{1}(\xi;x,y,t)\vartheta(y,t)\mathrm{d}y\mathrm{d}\xi+\int_{\Gamma_{\mathbb{R}}^{-}}\widehat{s}_{11}^{\ -2}(\xi)\int_{\mathbb{R}}{\bf F}_{2}(\xi;x,y,t)\vartheta(y,t)\mathrm{d}y\mathrm{d}\xi\bigg{)},$$

where

(2.34)

$${\bf F}_{1}(\lambda;x,y,t)=\sum\limits_{j=1}^{2}{\bf Z}_{j}^{+}(\lambda;x,t)\big{(}\Omega_{j}^{+}(\lambda;y,t)\big{)}^{\top},\ \ \ {\bf F}_{2}(\lambda;x,y,t)=\sum\limits_{j=1}^{2}{\bf Z}_{j}^{-}(\lambda;x,t)\big{(}\Omega_{j}^{-}(\lambda;y,t)\big{)}^{\top},$$

and $\Gamma_{\mathbb{R}}^{\pm}$ are the semicircular contour in the upper and lower half plane evaluated from $-\infty$ to $+\infty$, respectively. Note that ${\bf F}_{1}(\lambda;x,y,t)$ is analytic in the upper half plane, while ${\bf F}_{2}(\lambda;x,y,t)$ is analytic in the lower half plane, and $s_{11}^{-2}(\lambda),\ \widehat{s}_{11}^{\ -2}(\lambda)$ are meromorphic in the upper and lower half-planes, respectively.