N-th-Order Solutions for the Reverse Space-Time Nonlocal mKdV Equation: Riemann–Hilbert Approach

Abstract

This paper explores the reverse space-time mKdV equation through the application of the Riemann–Hilbert problem. Under the zero boundary condition, we derive the Jost solutions, examine their the analytic and symmetry properties alongside those of the scattering matrix, and formulate the corresponding Riemann–Hilbert problem. By assuming that the scattering coefficient has multiple simple zero points and one higher-order zero point, we obtain explicit solutions to the Riemann–Hilbert problem in a reflection-less situation and display two types of formulae for the N-th order solutions of the reverse space-time nonlocal mKdV equation, which correspond to multiple simple poles and one higher-order pole, respectively. As applications, we display two kinds of double-periodic solutions explicitly and graphically. Additionally, we display the conversation laws for the reverse space-time nonlocal mKdV equation.

1. Introduction

The modified Korteweg–de Vries (mKdV) equation, expressed as

$$u_t+u_{xxx}+6{\left|u\right|}^2{u}_x=0,$$

(1)

is a completely integrable system that models various nonlinear phenomena, such as the evolution of lower hybrid waves in homogeneous plasma [1], the transmission of few-cycle pulses in nonlinear optical media [2,3], the propagation of transverse waves in a molecular chain [4], and in a generalized elastic solid [5,6,7,8,9]. The exploration of explicit solutions to the mKdV equation, encompassing solitons, breathers, and rogue waves, has garnered significant attention over the past few decades [10,11,12,13,14,15].

In recent years, there has been a surge in research activity surrounding the $\mathcal{PT}$ symmetric deformation and extension of integrable systems [16,17,18]. Several years ago, Ablowitz and Musslimani introduced an integrable reverse space nonlocal NLS equation, which, despite being non-Hermitian, exhibits $\mathcal{PT}$ symmetry. In the same paper, they derived an explicit breathing one-soliton solution that develops a singularity within a finite time using inverse scattering transform (IST) [16,19]. Owing to its unusual scattering symmetries, this $\mathcal{PT}$ symmetric nonlocal integrable model holds special mathematical interest. Explicit solutions, including periodic, soliton, breather, and rogue wave (rational) solutions, have been obtained through various methods such as the IST [16,17,19], the Darboux transformation (DT) [18], the Riemann–Hilbert approach [20], and the Hirota bilinear method [21,22,23,24], and the physics-informed neural networks method [25].

In this paper, we study a reverse space-time nonlocal mKdV equation, expressed as

$$u_t(x,t)+u_{xxx}(x,t)+6u(x,t)u^{\ast }(-x,-t)u_x(x,t)=0,$$

(2)

which is derived as a nonlocal reduction, where $v(x,t)=u^{\ast }(-x,-t)$ and the asterisk means the complex conjugate of the coupled mKdV equations given by

$$\begin{array}{cc}\hfill u_t(x,t)+u_{xxx}(x,t)+6u(x,t)v(x,t)u_x(x,t)& =0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill v_t(x,t)+v_{xxx}(x,t)+6u(x,t)v(x,t)v_x(x,t)& =0.\hfill \end{array}$$

(4)

This reduction means that the evolution of solution $u(x,t)$ at location $(x,t)$ depends on both the nonlocal solution at $(-x,-t)$ and the local solution at $(x,t)$.

The coupled mKdV equations serve as the compatibility condition for the following linear differential equations:

$$\begin{array}{cc}\hfill {\partial }_x\varphi (x,t,\lambda )& =\left[-\mathrm{i}\lambda {\sigma }_3+Q(x,t)\right]\varphi (x,t,\lambda ),\hfill \end{array}$$

(5a)

$$\begin{array}{cc}\hfill {\partial }_t\varphi (x,t,\lambda )& =\left[-4\mathrm{i}{\lambda }^3{\sigma }_3+V(x,t,\lambda )\right]\varphi (x,t,\lambda ),\hfill \end{array}$$

(5b)

where ${\sigma }_3$ represents one of the Pauli matrices,

$$Q(x,t)=\left[\begin{array}{cc}0& u(x,t)\\ -v(x,t)& 0\end{array}\right],$$

(6)

and

$$V(x,t,\lambda )=4{\lambda }^{2}Q+2\mathrm{i}{\sigma }_3(Q_x-Q^2)+Q_{x}Q-Q{Q}_x-Q_{xx}+2Q^3.$$

(7)

In [26], the authors explore the DT of the reverse space-time nonlocal modified Korteweg–de Vries (mKdV) equation under a nonzero boundary condition, deriving explicit solutions that encompass soliton, breather, and rogue wave solutions. In this paper, we delve into the Riemann–Hilbert approach for the reverse space-time nonlocal mKdV equation, focusing on a zero boundary condition. Similar to [26], we obtain periodic and double periodic solutions; however, these solutions differ significantly. Specifically, the periodic and double periodic solutions in [26] are derived from a nonzero boundary condition, whereas those in this paper stem from a zero boundary condition.

The structure of this paper is as follows. In Section 2, we conduct a spectral analysis of the reverse space-time nonlocal mKdV equation with a zero boundary condition and construct the associated Riemann–Hilbert problem. In Section 3, we examine scenarios where the scattering coefficient has N simple zero points and one higher-order zero point, respectively, and derive the N-th-order solutions for the reverse space-time nonlocal mKdV equation. Finally, in Section 4, we summarize and discuss the findings of this paper.

2. The Spectral Analysis and Riemann–Hilbert Problem

We consider the initial condition $u(x,0)\to 0$ as $x\to \pm \infty$. In this case, the asymptotic Jost solution is given by $\varphi (x,t,\lambda )={\mathrm{e}}^{-\mathrm{i}(\lambda x+4{\lambda }^{3}t){\sigma }_3}$. Let us define $\varphi (x,t,\lambda )=\psi (x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}(\lambda x+4{\lambda }^{3}t){\sigma }_3}$. Substituting this expression into the Lax pair (5), we find that $\psi (x,t,\lambda )$ satisfies the following differential equations:

$$\begin{array}{c}\hfill {\psi }_x+\mathrm{i}\lambda [{\sigma }_3,\psi ]=Q\psi ,{\psi }_t+4\mathrm{i}{\lambda }^3[{\sigma }_3,\psi ]=V\psi .\end{array}$$

(8)

Simple calculations imply that $\psi (x,t,\lambda )$ satisfies

$$\mathrm{d}\left({\mathrm{e}}^{\mathrm{i}\theta \left(\lambda \right){\widehat{\sigma }}_3}\psi \right)={\mathrm{e}}^{\mathrm{i}\theta \left(\lambda \right){\widehat{\sigma }}_3}\left[\left(Q\mathrm{d}x+V\mathrm{d}t\right)\psi \right],\theta \left(\lambda \right)=\lambda x+4{\lambda }^{3}t,$$

meaning that ${\psi }_{\pm }(x,t,\lambda )$ can be represented as follows:

$$\begin{array}{cc}\hfill {\psi }_{+}(x,t,\lambda )& =I-{\int }_x^{+\infty }{\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\widehat{\sigma }}_3}Q(y,t){\psi }_{+}(y,t,\lambda )\mathrm{d}y,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\psi }_{-}(x,t,\lambda )& =I+{\int }_{-\infty }^x{\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\widehat{\sigma }}_3}Q(y,t){\psi }_{-}(y,t,\lambda )\mathrm{d}y.\hfill \end{array}$$

(10)

Since ${\psi }_{\pm }(x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\sigma }_3}$ solve the Lax pair and possess different boundary conditions, specifically, ${\psi }_{+}(x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\sigma }_3}$ and ${\psi }_{-}(x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\sigma }_3}$ are different solutions to the Lax pair, there exists a scattering matrix, $S\left(\lambda \right)$, such that

$${\psi }_{+}(x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\sigma }_3}={\psi }_{-}(x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}\theta \left(\lambda \right){\sigma }_3}S\left(\lambda \right),$$

(11)

where $S\left(\lambda \right)={\left(s_{jk}\left(\lambda \right)\right)}_{2\times 2}(j,k=1,2),$ and $s_{jk}\left(\lambda \right)$ can be represented as follows:

$$\begin{array}{cc}\hfill s_{11}& =W\left[{\psi }_{+,1},{\psi }_{-,2}\right],s_{21}=W\left[{\psi }_{-,1},{\psi }_{+,1}\right]{\mathrm{e}}^{-2\mathrm{i}\theta },\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill s_{22}& =W\left[{\psi }_{-,1},{\psi }_{+,2}\right],s_{12}=W\left[{\psi }_{+,2},{\psi }_{-,2}\right]{\mathrm{e}}^{2\mathrm{i}\theta }.\hfill \end{array}$$

(13)

Here, ${\psi }_{\pm ,j}(j=1,2)$ means the jth column of ${\psi }_{\pm }$.

Noting that ${\sigma }_1{Q}^{\ast }(-x,-t){\sigma }_1=-Q(x,t)$, simple calculations show that ${\sigma }_1{\varphi }^{\ast }(-x,-t,-{\lambda }^{\ast }){\sigma }_1$ also satisfies the Lax pair, sharing the same boundary condition $\varphi (x,t,\lambda )$. This implies that

$$\varphi (x,t,\lambda )={\sigma }_1{\varphi }^{\ast }(-x,-t,-{\lambda }^{\ast }){\sigma }_1.$$

(14)

Consequently,

$$s_{11}\left(\lambda \right)=s_{22}^{\ast }(-{\lambda }^{\ast }),s_{12}\left(\lambda \right)=s_{21}^{\ast }(-{\lambda }^{\ast }).$$

(15)

Following a similar procedure, we obtain the following theorem.

Theorem 1.

When $q(x,t)\in L^1\left(\mathbb{R}\right)$, the Jost solutions ${\psi }_{\pm }(x,t,\lambda )$ and the scattering coefficients have the following analytic properties:

• ${\psi }_{-,1}(x,t,\lambda )$ and ${\psi }_{+,2}(x,t,\lambda )$ are analytic when $\mathrm{Im}\lambda >0$.
• ${\psi }_{+,1}(x,t,\lambda )$ and ${\psi }_{-,2}(x,t,\lambda )$ are analytic when $\mathrm{Im}\lambda <0$.
• The scattering coefficient $s_{11}\left(\lambda \right)$ is analytic when $\mathrm{Im}\lambda <0$; The scattering coefficient $s_{22}\left(\lambda \right)$ is analytic when $\mathrm{Im}\lambda >0$.

Proof. 

Here, we prove the first column of ${\psi }_{-}(x,t,\lambda )$, denoted by ${\psi }_{-,1}(x,t,\lambda )$. To prove the analytic property of ${\psi }_{-,1}$, we set that ${\psi }_{-,1}$ has the following Legendre’s series,

$${\psi }_{-,1}=\sum _{k=0}^{+\infty }{\omega }^{\left(k\right)},{\omega }^{\left(0\right)}=\left[\begin{array}{c}1\\ 0\end{array}\right],$$

(16)

and ${\omega }^{(n+1)}$ and ${\omega }^{\left(n\right)}$ have the following relations,

$${\omega }^{(n+1)}={\int }_{-\infty }^{x}F{\omega }^{\left(n\right)}\mathrm{dy},$$

(17)

where $F=F(x-y,\lambda )=\mathrm{diag}(1,{\mathrm{e}}^{2\mathrm{i}\lambda (x-y)})Q$. This expression is derived directly from the Volterra integral of ${\psi }_{-}(x,t,\lambda )$, given by (10).

Define the $L^1$ norm of vector as $\left|\right|\omega \left|\right|=|{\omega }_1|+|{\omega }_2|$, then

$$\left|\right|{\omega }^{(n+1)}(x,t,\lambda )\left|\right|\le {\int }_{-\infty }^x\left|\right|F\left|\right|\left|\right|{\omega }^{\left(n\right)}(y,t,z)\left|\right|\mathrm{d}y.$$

Direct calculations yield that

$$\begin{array}{c}\hfill \left|\right|F\left|\right|=\left|\right|\mathrm{diag}(1,{\mathrm{e}}^{2\mathrm{i}\lambda (x-y)})\left|\right|\left|\right|Q\left|\right|,\end{array}$$

and when $\mathrm{Im}\lambda >0$, then

$$\left|\right|\mathrm{diag}(1,{\mathrm{e}}^{2\mathrm{i}\lambda (x-y)})\left|\right|\le 2,\left|\right|Q\left|\right|\le |u(x,t)|.$$

Thus

$$\left|\right|{\omega }^{(n+1)}\left|\right|\le 2{\int }_{-\infty }^x\left|u(y,t)\right|\left|\right|{\omega }^{\left(n\right)}\left|\right|\mathrm{d}y.$$

By applying the mathematical induction method,

$$\left|\right|{\omega }^{(n+1)}(x,t,\lambda )\left|\right|\le {\int }_{-\infty }^x{\rho }_x(x,t){\rho }^n(x,t)/n!\mathrm{d}x={\displaystyle \frac{{\rho }^{n+1}(x,t)}{(n+1)!}},$$

where

$$\rho (x,t)=2{\int }_{-\infty }^x\left|u(x,t)\right|\mathrm{d}x.$$

Since $u(x,t)\in L^1\left(\mathbb{R}\right)$, it means that $\rho (x,t)$ is bounded, that is, there exist a constant $\sigma$, such that $\rho (x,t)\le \sigma$.

Therefore, we can conclude that the series given by (16) is absolutely convergent, which means ${\psi }_{-,1}$ is analytic when $\mathrm{Im}\lambda >0$. □

The Riemann–Hilbert Problem

To solve the Jost solutions explicitly, we introduce a meromorphic function,

$$M(x,t,\lambda )=\left\{\begin{array}{cc}\left[\begin{array}{cc}{\psi }_{-,1}& {\displaystyle \frac{{\psi }_{+,2}}{s_{22}}}\end{array}\right],\hfill & \mathrm{Im}\lambda >0,\hfill \\ \left[\begin{array}{cc}{\displaystyle \frac{{\psi }_{+,1}}{s_{11}}}& {\psi }_{-,2}\end{array}\right],\hfill & \mathrm{Im}\lambda <0.\hfill \end{array}\right.$$

(18)

According to the properties of ${\psi }_{\pm }(x,t,\lambda )$, it follows that

• Normalization: $M(x,t,\lambda )\to I$ when $\left|\lambda \right|\to \infty$.
• Symmetry: $M_{\pm }(x,t,\lambda )={\sigma }_1{M}_{\pm }^{\ast }(-x,-t,-{\lambda }^{\ast }){\sigma }_1.$
• Jump condition:

  $$M_{+}(x,t,\lambda )=M_{-}(x,t,\lambda ){\mathrm{e}}^{-\mathrm{i}\theta {\widehat{\sigma }}_3}J\left(\lambda \right),$$

  (19)

  where $\rho \left(\lambda \right)=s_{12}\left(\lambda \right)/s_{11}\left(\lambda \right)$ and

  $$J\left(\lambda \right)=\left[\begin{array}{cc}1& \rho \left(\lambda \right)\\ -{\rho }^{\ast }(-\lambda )& 1-\rho \left(\lambda \right){\rho }^{\ast }(-\lambda )\end{array}\right].$$
• The inverse problem: the potential $u(x,t)$ can be expressed as

  $$u(x,t)=\underset{\lambda \to \infty }{lim}2\mathrm{i}\lambda M_{12}(x,t,\lambda ).$$

  (20)

In the next section, we consider the reflection-less situation where $s_{12}\left(\lambda \right)=s_{21}\left(\lambda \right)=0$ for $\lambda \in \mathbb{R}$.

3. The Cases of Simple Zero Points and Higher-Order Zero Point

In this subsection, we consider the scenario where the scattering coefficient $s_{22}\left(\lambda \right)$ has N simple zero points, denoted as ${\lambda }_j$ for $j=1,2,\cdots ,N$. Meanwhile, based on the symmetry property (15), $s_{11}\left(\lambda \right)$ has N simple zero points at ${\lambda }_j^{\ast }$ for $j=1,2,\cdots ,N$. Several calculations reveal that $s_{11}\left(\lambda \right)$ and $s_{22}\left(\lambda \right)$ can be expressed as follows:

$$s_{11}=\prod _{j=1}^N{\displaystyle \frac{\lambda +{\lambda }_j^{\ast }}{\lambda -{\lambda }_j}},s_{22}=\prod _{j=1}^N{\displaystyle \frac{\lambda -{\lambda }_j}{\lambda +{\lambda }_j^{\ast }}}.$$

(21)

When $\lambda ={\lambda }_j$, $s_{22}\left({\lambda }_j\right)=0$, which means that there exists a constant $c_j$, such that

$${\psi }_{-,1}\left({\lambda }_j\right)=c_j{\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_j\right)}{\psi }_{+,2}\left({\lambda }_j\right).$$

(22)

Similarly, since $s_{11}(-{\lambda }_j^{\ast })=0$, there exists a constant ${\tilde{c}}_j$ such that

$${\psi }_{-,2}(-{\lambda }_j^{\ast })={\tilde{c}}_j{\mathrm{e}}^{-2\mathrm{i}\theta (-{\lambda }_j^{\ast })}{\psi }_{+,1}(-{\lambda }_j^{\ast }).$$

(23)

Furthermore, by invoking the symmetry property (15), it follows that ${\tilde{c}}_j=-c_j^{\ast }$.

Recalling the properties of $M(x,t,\lambda )$ and applying the Plemelj’s formula [20,27], $M(x,t,\lambda )$ has the following expression:

$$M(x,t,\lambda )=I+\sum _{j=1}^N{\displaystyle \frac{\underset{\lambda ={\lambda }_j}{\mathrm{Res}}{M}^{+}}{\lambda -{\lambda }_j}}+\sum _{j=1}^N{\displaystyle \frac{\underset{\lambda =-{\lambda }_j^{\ast }}{\mathrm{Res}}{M}^{-}}{\lambda +{\lambda }_j^{\ast }}}.$$

(24)

Element-wise, $M_{11}(x,t,\lambda )$ and $M_{12}(x,t,\lambda )$ possess the following expressions,

$$\begin{array}{c}\hfill M_{11}(x,t,\lambda )=1+\sum _{j=1}^N{\displaystyle \frac{f_j}{\lambda -{\lambda }_j}},M_{12}=\sum _{j=1}^N{\displaystyle \frac{g_j}{\lambda +{\lambda }_j^{\ast }}},\end{array}$$

(25)

where $f_j$ and $g_j$ are residues corresponding to $\lambda ={\lambda }_j$ and $\lambda =-{\lambda }_j^{\ast }$, respectively. To consider the residue conditions, we recall the parallelization conditions given by (26) and (27), and it yields that

$$\begin{array}{cc}\hfill \underset{\lambda ={\lambda }_j}{\mathrm{Res}}{M}_{11}(x,t,\lambda )& =f_j=\sum _{i=1}^N{\displaystyle \frac{d_j{\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_j\right)}}{{\lambda }_j+{\lambda }_i^{\ast }}}{g}_i,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \underset{\lambda =-{\lambda }_j^{\ast }}{\mathrm{Res}}{M}_{12}(x,t,\lambda )& =g_j=-d_j^{\ast }{\mathrm{e}}^{-2\mathrm{i}\theta (-{\lambda }_j^{\ast })}+\sum _{i=1}^N{\displaystyle \frac{d_j^{\ast }{\mathrm{e}}^{-2\mathrm{i}\theta (-{\lambda }_j^{\ast })}}{{\lambda }_j^{\ast }+{\lambda }_i}}{f}_i.\hfill \end{array}$$

(27)

with

$$\begin{array}{c}\hfill d_j={\displaystyle \frac{c_j}{{\dot{s}}_{11}\left({\lambda }_j\right)}}.\end{array}$$

Solving Equations (26) and (27), we can obtain the $f_j$ and $g_j(j=1,2,\cdots ,N)$. Also, we obtain the expressions of $M_{11}(x,t,\lambda )$ and $M_{12}(x,t,\lambda )$. After simple calculations, it yields that

$$\begin{array}{c}\hfill M_{11}(x,t,\lambda )={\displaystyle \frac{det(I-\tilde{\mathrm{\Omega }}\mathrm{\Omega }+|\eta \rangle \langle Y\left(\lambda \right)|\mathrm{\Omega })}{det(I-\tilde{\mathrm{\Omega }}\mathrm{\Omega })}},\end{array}$$

(28)

$$\begin{array}{c}\hfill M_{12}(x,t,\lambda )={\displaystyle \frac{det(I-\tilde{\mathrm{\Omega }}\mathrm{\Omega }+|\eta \rangle \langle Y\left(\lambda \right)|)}{det(I-\tilde{\mathrm{\Omega }}\mathrm{\Omega })}}-1,\end{array}$$

(29)

where $\langle Z\left(\lambda \right)|$ and $\langle Y\left(\lambda \right)|$ are row vectors with

$$\begin{array}{c}\hfill \langle Z\left(\lambda \right)|=\left[\begin{array}{cccc}{\displaystyle \frac{1}{\lambda -{\lambda }_1}}& {\displaystyle \frac{1}{\lambda -{\lambda }_2}}& \cdots & {\displaystyle \frac{1}{\lambda -{\lambda }_N}}\end{array}\right],\\ \hfill \langle Y\left(\lambda \right)|=\left[\begin{array}{cccc}{\displaystyle \frac{1}{\lambda +{\lambda }_1^{\ast }}}& {\displaystyle \frac{1}{\lambda +{\lambda }_2^{\ast }}}& \cdots & {\displaystyle \frac{1}{\lambda +{\lambda }_N^{\ast }}}\end{array}\right].\end{array}$$

Based on the inverse problem (20), the potential $u(x,t)$ is expressed as

$$\begin{array}{c}\hfill u(x,t)=2\mathrm{i}\left[{\displaystyle \frac{det(I-\tilde{\mathrm{\Omega }}\mathrm{\Omega }+|\eta \rangle \langle 1_N|)}{det(I-\tilde{\mathrm{\Omega }}\mathrm{\Omega })}}-1\right].\end{array}$$

(30)

Here, $\mathrm{\Omega }$ and $\tilde{\mathrm{\Omega }}$ are $N\times N$ matrices defined by

$${\mathrm{\Omega }}_{j,i}={\displaystyle \frac{d_j{\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_j\right)}}{{\lambda }_j+{\lambda }_i^{\ast }}},{\tilde{\mathrm{\Omega }}}_{j,i}={\displaystyle \frac{d_j^{\ast }{\mathrm{e}}^{-2\mathrm{i}\theta (-{\lambda }_j^{\ast })}}{{\lambda }_j^{\ast }+{\lambda }_i}},i,j=1,2,\cdots ,N.$$

(31)

$|\eta \rangle$ is a column vector with elements

$${\eta }_j=-d_j^{\ast }{\mathrm{e}}^{-2\mathrm{i}\theta (-{\lambda }_j^{\ast })},j=1,2,\cdots ,N.$$

(32)

Let $N=1$ and ${\lambda }_1={\alpha }_1+\mathrm{i}{\beta }_1$, according to Formula (30); the first-order solution is derived as follows,

$$u_1(x,t)=2\mathrm{i}{\alpha }_1{d}_1^{\ast }/\left|d_1\right|{\mathrm{e}}^{\mathrm{\Theta }}{sinh}^{-1}(\mathrm{i}\mathrm{\Gamma }+{\phi }_1),$$

(33)

with

$$\mathrm{\Theta }=2{\beta }_1\left[x+(12{\alpha }_1^2-4{\beta }_1^2)t\right],\mathrm{\Gamma }=2{\alpha }_1\left[x+(4{\alpha }_1^2-12{\beta }_1^2)t\right],{\phi }_1=ln\left(2{\alpha }_1\right).$$

Noting that $exp\left(\mathrm{\Theta }\right)$ becomes very large as $\mathrm{\Theta }$ approaches infinity, we set ${\beta }_1=0$, i.e., ${\lambda }_1\in \mathbb{R}$, to obtain a solution with finite amplitude. Consequently, the first-order solution is given by

$$u_1(x,t)=2\mathrm{i}{\alpha }_1{d}_1^{\ast }/\left|d_1\right|{sinh}^{-1}\left[2\mathrm{i}{\alpha }_1(x+4{\alpha }_1^{2}t)+ln\left(2{\alpha }_1\right)\right].$$

(34)

This is a periodic solution, with characteristic lines given by $2{\alpha }_1(x+4{\alpha }_1^{2}t)=k\pi$ and $2{\alpha }_1(x+4{\alpha }_1^{2}t)=k\pi +{\displaystyle \frac{\pi }{2}}$ for $(k\in \mathbb{Z})$. When $2{\alpha }_1(x+4{\alpha }_1^{2}t)=k\pi$, $u_1(x,t)$ reaches its maximum amplitude:

$$|u_1|={\displaystyle \frac{8{\alpha }_1^2}{4{\alpha }_1^2-1}}.$$

When $2{\alpha }_1(x+4{\alpha }_1^{2}t)=k\pi +{\displaystyle \frac{\pi }{2}}$, $u_1(x,t)$ arrives at its minimum amplitude:

$$|u_1|={\displaystyle \frac{8{\alpha }_1^2}{4{\alpha }_1^2+1}}.$$

Here, we have set $d_1=1$. The dynamics evolution of first-order solution is shown in Figure 1.

When $N=2$ and ${\lambda }_j={\alpha }_j\in \mathbf{Z}(j=1,2)$, a double-periodic second-order solution is derived using Formula (30),

$$u_2(x,t)=8\mathrm{i}{({\alpha }_1+{\alpha }_2)}^2{\displaystyle \frac{N_2}{D_2}},$$

(35)

where

$$\begin{array}{cc}\hfill N_2& =4d_1\left({\mathrm{e}}^{{\delta }_1}+{\mathrm{e}}^{{\delta }_2}\right)-d_2\left({\alpha }_1^2{\mathrm{e}}^{{\delta }_3}+{\alpha }_2^2{\mathrm{e}}^{{\delta }_4}\right),\hfill \\ \hfill D_2& =4{\tilde{d}}_1\left({\alpha }_2^2{\mathrm{e}}^{2{\delta }_1}+{\alpha }_1^2{\mathrm{e}}^{2{\delta }_2}\right)+{\tilde{d}}_3{\mathrm{e}}^{{\tilde{\delta }}_3}-{\tilde{d}}_4{\mathrm{e}}^{2{\tilde{\delta }}_3}-{\tilde{d}}_5,\hfill \\ \hfill d_1& ={\alpha }_1^2{\alpha }_2^2{({\alpha }_1+{\alpha }_2)}^2,d_2={({\alpha }_1-{\alpha }_2)}^2,\hfill \\ \hfill {\tilde{d}}_1& ={({\alpha }_1+{\alpha }_2)}^4,{\tilde{d}}_2=32{\alpha }_1^2{\alpha }_2^2{({\alpha }_1+{\alpha }_2)}^2,\hfill \\ \hfill {\tilde{d}}_3& ={({\alpha }_1-{\alpha }_2)}^4,{\tilde{d}}_4=16{\alpha }_1^2{\alpha }_2^2{({\alpha }_1+{\alpha }_2)}^4,\hfill \\ \hfill {\delta }_1& =2\mathrm{i}{\alpha }_1(x+4{\alpha }_1^{2}t),{\delta }_2=2\mathrm{i}{\alpha }_2(x+4{\alpha }_2^{2}t),\hfill \\ \hfill {\delta }_3& =2\mathrm{i}\left[({\alpha }_1+2{\alpha }_2)x+4({\alpha }_1^3+2{\alpha }_2^3)t\right],\hfill \\ \hfill {\delta }_4& =2\mathrm{i}\left[(2{\alpha }_1+{\alpha }_2)x+4(2{\alpha }_1^3+{\alpha }_2^3)t\right],\hfill \\ \hfill {\tilde{\delta }}_3& =2\mathrm{i}({\alpha }_1+{\alpha }_2)\left[x+4({\alpha }_1^2-{\alpha }_1{\alpha }_2+{\alpha }_2^2)t\right].\hfill \end{array}$$

Like the first-order solution, this second-order solution is symmetric with respect to ${\lambda }_1$ and ${\lambda }_2$, namely, $u_2(x,t,{\lambda }_1,{\lambda }_2)=u_2(x,t,{\lambda }_2,{\lambda }_1)$. We now consider its dynamics. For ${\lambda }_1=1/3$ and ${\lambda }_2=1/4$, as well as for ${\lambda }_1=2/5$ and ${\lambda }_2=1/4$, two regular double-periodic solutions are displayed in Figure 2. Due to the interaction between the two first-order waves, the amplitude increases in some regions and decreases in others, resulting in a lattice-like appearance of the second-order solution’s trajectory. Ref. [26] also presents a similar solution, but our results differ from theirs. Their solutions are derived from a non-zero boundary condition, whereas our solutions are derived from a zero boundary condition.

An interesting phenomenon is discovered: the first-order solution becomes singular as $\lambda =1/2$, but the second-order solution remains regular if one of the $\lambda$ values is equal to $1/2$. For instance, as ${\lambda }_1=1/2$ and ${\lambda }_2=1/4$, the second-order solution $u_2(x,t)$ is given by

$$u_2(x,t)={\displaystyle \frac{-18\mathrm{i}\left[4{\mathrm{e}}^{\mathrm{i}/4\left(5t+8x\right)}+{\mathrm{e}}^{\mathrm{i}/8\left(17t+20x\right)}-9{\mathrm{e}}^{\left(t+x\right)\mathrm{i}}-9{\mathrm{e}}^{\mathrm{i}/8\left(t+4x\right)}\right]}{288{\mathrm{e}}^{3/8\mathrm{i}\left(3t+4x\right)}-4{\mathrm{e}}^{3/4\mathrm{i}\left(3t+4x\right)}+324{\mathrm{e}}^{\mathrm{i}/4\left(t+4x\right)}-81+81{\mathrm{e}}^{2\mathrm{i}\left(t+x\right)}}},$$

(36)

which is regular and depicted in Figure 3.

The Case of N-th-Order Zero Point

In this subsection, we consider the situation where both $s_{22}\left(\lambda \right)$ and $s_{11}\left(\lambda \right)$ possess only one N-th-order zero point. Analogous to the case of simple points, $s_{11}\left(\lambda \right)$ and $s_{22}\left(\lambda \right)$ can be expressed as follows:

$$s_{11}={\left({\displaystyle \frac{\lambda +{\lambda }_j^{\ast }}{\lambda -{\lambda }_j}}\right)}^N,s_{22}\left(\lambda \right)={\left({\displaystyle \frac{\lambda -{\lambda }_j}{\lambda +{\lambda }_j^{\ast }}}\right)}^N.$$

(37)

Noting that $\lambda ={\lambda }_1$ is the N-th-order zero point of $s_{22}\left(\lambda \right)$, meaning that

$$s_{22}\left({\lambda }_1\right)=0,{\dot{s}}_{22}\left({\lambda }_1\right)=0,\cdots ,{\displaystyle \frac{{\mathrm{d}}^{N-1}}{\mathrm{d}{\lambda }^{N-1}}}{s}_{22}\left({\lambda }_1\right)=0,$$

we have the following parallelization conditions related to $\lambda ={\lambda }_1$ and $\lambda =-{\lambda }_1^{\ast }$.

Lemma 1.

When $\lambda ={\lambda }_1$ is the N-th-order zero point of $s_{22}$, the generalization parallelization conditions related to the Jost solutions are given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^n}{\partial {\lambda }^n}}{\psi }_{-,1}\left({\lambda }_1\right)& ={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)c_{n+1-k}{\displaystyle \frac{{\partial }^k}{\partial {\lambda }^k}}{\left[{\psi }_{+,2}{\mathrm{e}}^{2\mathrm{i}\theta \left(\lambda \right)}\right]}_{\lambda ={\lambda }_1},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^n}{\partial {\lambda }^n}}{\psi }_{-,2}(-{\lambda }_1^{\ast })& ={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)c_{n+1-k}^{\ast }{\displaystyle \frac{{\partial }^k}{\partial {\lambda }^k}}{\left[{\psi }_{+,1}{\mathrm{e}}^{-2\mathrm{i}\theta \left(\lambda \right)}\right]}_{\lambda =-{\lambda }_1^{\ast }},\hfill \end{array}$$

(39)

for $n=1,2,\cdots ,N-1$. In other words, the derivative of ${\psi }_{-,1}$ related to ${\lambda }_1$ and the derivative of ${\psi }_{-,2}$ related to $-{\lambda }_1^{\ast }$ can be linearly expressed by the derivative of ${\psi }_{+,2}{\mathrm{e}}^{2\mathrm{i}\theta \left(\lambda \right)}$ related to ${\lambda }_1$ and the derivative of ${\psi }_{+,1}{\mathrm{e}}^{-2\mathrm{i}\theta \left(\lambda \right)}$ related to $-{\lambda }_1^{\ast }$, respectively.

Proof. 

This lemma can be proven by applying the mathematical induction method. Here, we only prove the situations where ${\lambda }_1$ is the second-order zero point of $s_{22}\left(\lambda \right)$.

When $s_{22}\left({\lambda }_1\right)=0$, it means that $W[{\psi }_{-,1},{\psi }_{+,2}]{\mathrm{e}}^{2\mathrm{i}\theta \left(\lambda \right)}{|}_{\lambda ={\lambda }_1}=0$, which implies that there exist constant $c_0$, such that

$${\psi }_{-,1}\left({\lambda }_1\right)=c_0{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}.$$

(40)

When ${\dot{s}}_{22}\left({\lambda }_1\right)=0$, it means that

$$W\left[{\displaystyle \frac{\partial }{\partial \lambda }}{\psi }_{-,1}\left({\lambda }_1\right),{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}\right]+W\left[{\psi }_{-,1}\left({\lambda }_1\right),{\displaystyle \frac{\partial }{\partial \lambda }}{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}\right]=0.$$

(41)

Recalling (40), Equation (41) can be written as

$$W\left[{\displaystyle \frac{\partial }{\partial \lambda }}{\psi }_{-,1}\left({\lambda }_1\right)-c_0{\displaystyle \frac{\partial }{\partial \lambda }}{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)},{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}\right]=0,$$

(42)

which means that there exist a constant $c_1$, such that

$${\displaystyle \frac{\partial }{\partial \lambda }}{\psi }_{-,1}\left({\lambda }_1\right)=c_0{\displaystyle \frac{\partial }{\partial \lambda }}{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}+c_1{\psi }_{+,2}\left({\lambda }_1\right){\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}.$$

(43)

By applying the mathematical induction method, this lemma can be proved. □

By utilizing the above generalization parallelization conditions to $M(x,t,\lambda )$, we obtain the explicit relationships between $M_{+}(x,t,\lambda )$ and $M_{-}(x,t,\lambda )$ as follows.

Lemma 2.

When $\lambda ={\lambda }_1$ is the N-th-order zero points of $s_{22}$, $M_{+}(x,t,\lambda )$ and $M_{-}(x,t,\lambda )$ are associated as follows:

$$\begin{array}{cc}\hfill \underset{\lambda ={\lambda }_1}{P_{-j}}{M}_1(x,t,\lambda )& =\sum _{k=0}^{N-j}\sum _{m=0}^k{\displaystyle \frac{{({\lambda }_1+{\lambda }_1^{\ast })}^N{c}_{N+1-j-k}}{(N-j-k)!m!(k-m)!}}{\displaystyle \frac{{\partial }^{k-m}}{\partial {\lambda }^{k-m}}}{\mathrm{e}}^{2\mathrm{i}\theta \left({\lambda }_1\right)}{\displaystyle \frac{{\partial }^m}{\partial {\lambda }^m}}{M}_2(x,t,{\lambda }_1),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \underset{\lambda =-{\lambda }_1^{\ast }}{P_{-j}}{M}_2(x,t,\lambda )& =\sum _{k=0}^{N-j}\sum _{m=0}^k{\displaystyle \frac{{(-1)}^N{({\lambda }_1+{\lambda }_1^{\ast })}^N{c}_{N+1-j-k}^{\ast }}{(N-j-k)!m!(k-m)!}}{\displaystyle \frac{{\partial }^k}{\partial {\lambda }^k}}{M}_1(x,t,-{\lambda }_1^{\ast }){\displaystyle \frac{{\partial }^{k-m}}{\partial {\lambda }^{k-m}}}{\mathrm{e}}^{-2\mathrm{i}\theta (x,t,-{\lambda }_1^{\ast })}.\hfill \end{array}$$

(45)

Lemma 2 is crucial in solving the matrix $M(x,t,\lambda )$. Based on Plemelj’s formula, we can set $M_{11}(x,t,\lambda )$ and $M_{12}(x,t,\lambda )$ in the following forms:

$$M_{11}(x,t,\lambda )=1+\sum _{k=1}^N{\displaystyle \frac{f_k(x,t)}{{(\lambda -{\lambda }_1)}^k}},M_{12}(x,t,\lambda )=\sum _{k=1}^N{\displaystyle \frac{g_k(x,t)}{{(\lambda +{\lambda }_1^{\ast })}^k}}.$$

(46)

Proof. 

This lemma is only a direct application of Lemma 2, and we do not prove it here. □

where $f_k(x,t)$ and $g_k(x,t)$ are unknown functions to be determined. According to Lemma 2, $f_k(x,t)$ and $g_k(x,t)$ can be written as

$$\begin{array}{c}\hfill |f(x,t)\rangle =\mathrm{\Omega }(x,t)|g(x,t)\rangle ,|g(x,t)\rangle =|\eta (x,t)\rangle +\widehat{\mathrm{\Omega }}(x,t)|f(x,t)\rangle ,\end{array}$$

(47)

where

$$|f(x,t)\rangle ={\left[\begin{array}{ccc}{f}_1(x,t)& f_2(x,t)& \cdots f_N(x,t)\end{array}\right]}^T,|g(x,t)\rangle ={\left[\begin{array}{ccc}{g}_1(x,t)& g_2(x,t)& \cdots g_N(x,t)\end{array}\right]}^T,$$

and $|\eta (x,t)\rangle$ is a $N\times 1$ column vector with

$${\eta }_j(x,t)={(-1)}^N{({\lambda }_1+{\lambda }_1^{\ast })}^N\sum _{k=0}^{N-j}{\displaystyle \frac{c_{N+1-j-k}^{\ast }}{(N-j-k)!}}{w}_k(x,t),$$

$\mathrm{\Omega }$ and $\widehat{\mathrm{\Omega }}$ are $N\times N$ matrices with

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{js}(x,t)& =\sum _{k=0}^{N-j}\sum _{m=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{m+s-1}{m}}\right){\displaystyle \frac{{(-1)}^m{c}_{N+1-j-k}}{(N-j-k)!}}{({\lambda }_1+{\lambda }_1^{\ast })}^{N-s-m}{w}_{k-m}(x,t),\hfill \\ \hfill {\widehat{\mathrm{\Omega }}}_{js}(x,t)& =\sum _{k=0}^{N-j}\sum _{m=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{m+s-1}{m}}\right){\displaystyle \frac{{(-1)}^{N+m-s}{c}_{N+1-j-k}^{\ast }}{(N-j-k)!}}{({\lambda }_1+{\lambda }_1^{\ast })}^{N-s-m}{\widehat{w}}_{k-m}(x,t),\hfill \end{array}$$

$w_m(x,t)$ and ${\widehat{w}}_m(x,t)$ are functions defined by

$$w_m(x,t)={\displaystyle \frac{1}{m!}}{\left.{\displaystyle \frac{{\partial }^m}{\partial {\lambda }^m}}\right|}_{\lambda ={\lambda }_1}{\mathrm{e}}^{2\mathrm{i}\theta (x,t,\lambda )},{\widehat{w}}_m(x,t)={\displaystyle \frac{1}{m!}}{\left.{\displaystyle \frac{{\partial }^m}{\partial {\lambda }^m}}\right|}_{\lambda =-{\lambda }_1^{\ast }}{\mathrm{e}}^{-2\mathrm{i}\theta (x,t,\lambda )}.$$

The expressions of $|\eta \rangle$, $\mathrm{\Omega }$, and $\widehat{\mathrm{\Omega }}$ can be expressed in the form of matrices. Let $\mathrm{\Delta }$ be a $N\times N$ matrix and $|y\rangle$ be a column vector with N components,

$$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{jk}& =\left\{\begin{array}{cc}{({\lambda }_1+{\lambda }_1^{\ast })}^N{\displaystyle \frac{C_{N+1-j-k}^{\ast }}{(N-j-k)!}},\hfill & j\le k,\hfill \\ 0,\hfill & j>k.\hfill \end{array}\right.\hfill \\ \hfill |y\rangle & ={\left[\begin{array}{cccc}{w}_{N-1}& w_{N-2}& \cdots & w_0\end{array}\right]}^T,\hfill \end{array}$$

then $|\eta \rangle ={(-1)}^N\mathrm{\Delta }|y\rangle$. Let Y and Z be $N\times N$ matrices,

$$\begin{array}{cc}\hfill Y_{jk}& =\left\{\begin{array}{cc}0,\hfill & j<k\hfill \\ w_{j-k},\hfill & j\ge k\hfill \end{array}\right.,\hfill \\ \hfill Z_{jk}& ={\displaystyle \frac{{(-1)}^k}{{({\lambda }_1+{\lambda }_1^{\ast })}^{j+k}}}\left({\displaystyle \genfrac{}{}{0pt}{}{j+k-1}{j}}\right),\hfill \end{array}$$

then $\mathrm{\Omega }$ and $\widehat{\mathrm{\Omega }}$ can be expressed as

$$\mathrm{\Omega }=\mathrm{\Delta }YZ,\widehat{\mathrm{\Omega }}=-{\mathrm{\Delta }}^{\ast }YZ,$$

(48)

where ${\mathrm{\Delta }}^{\ast }$ denotes the complex conjugate of $\mathrm{\Delta }$. Based on inverse problem (20) and through straightforward calculations, the N-th-order solution related to one N-th-order zero point is

$$u(x,t)=2\mathrm{i}\left[{\displaystyle \frac{det(I-\widehat{\mathrm{\Omega }}\mathrm{\Omega }+|\eta \rangle \langle {\mathrm{e}}_1|)}{det(I-\widehat{\mathrm{\Omega }}\mathrm{\Omega })}}-1\right],$$

(49)

with $\langle {\mathrm{e}}_1|=\left[\begin{array}{cccc}1& 0& 0\cdots & 0\end{array}\right]$.

According to [15], we can consider the mixture of simple poles and higher-order poles and derive several new solutions. Let $\lambda ={\lambda }_1$ be a second-order pole and $\lambda ={\lambda }_2$ be a simple pole from [15]; it follows that the third-order solution is

$$u_3(x,t)=4\mathrm{i}{({\alpha }_1+{\alpha }_2)}^2{\displaystyle \frac{N_3}{D_3}},$$

(50)

where

$$\begin{array}{cc}\hfill N_3& =2{\alpha }_1^2{\alpha }_2^2{({\alpha }_1+{\alpha }_2)}^2(2{\mathrm{e}}^{{\delta }_1}+{\mathrm{e}}^{{\delta }_2})-({\alpha }_1^2-{\alpha }_2^2)(2{\alpha }_2^2{\mathrm{e}}^{{\delta }_3}+{\alpha }_1^2{\mathrm{e}}^{{\delta }_4}),\hfill \\ \hfill D_3& ={({\alpha }_1+{\alpha }_2)}^4(4{\alpha }_2^2{\mathrm{e}}^{{\delta }_1}+{\alpha }_1^2{\mathrm{e}}^{{\delta }_2})+16{\alpha }_1^2{\alpha }_2^2{({\alpha }_1+{\alpha }_2)}^2{\mathrm{e}}^{{\delta }_5}-{({\alpha }_1-{\alpha }_2)}^4{\mathrm{e}}^{2{\delta }_5}\hfill \\ \hfill & -4{\alpha }_1^2{\alpha }_2^2{({\alpha }_1+{\alpha }_2)}^4,\hfill \\ \hfill {\delta }_1& =2\mathrm{i}{\alpha }_1(x+4{\alpha }_1^{2}t),{\delta }_2=2\mathrm{i}{\alpha }_2(x+4{\alpha }_2^{2}t),{\delta }_3=2\mathrm{i}\left[({\alpha }_1+2{\alpha }_2)x+4({\alpha }_1^3+2{\alpha }_2^3)t\right],\hfill \\ \hfill {\delta }_4& =2\mathrm{i}\left[(2{\alpha }_1+{\alpha }_2)x+4(2{\alpha }_1^3+{\alpha }_2^3)t\right],{\delta }_5=2\mathrm{i}({\alpha }_1+{\alpha }_2)\left[x+4({\alpha }_1^2-{\alpha }_1{\alpha }_2+{\alpha }_2^2)t\right].\hfill \end{array}$$

Similar to $u_2(x,t)$, $u_3(x,t)$ is also related to ${\alpha }_2$ and ${\alpha }_3$, but it is not symmetric with regard to ${\alpha }_2$ and ${\alpha }_3$, that is, $u_3(x,t,{\alpha }_2,{\alpha }_3)\ne u_3(x,t,{\alpha }_3,{\alpha }_2)$. Graphically, the amplitude of $u_3(x,t)$ is very large while ${\alpha }_1>{\alpha }_2$, and we thus set ${\alpha }_2>{\alpha }_1$ to obtain several beautiful structures. When ${\alpha }_1=1/2,{\alpha }_2=2/3$, the evolution of $u_3(x,t)$ is displayed in Figure 4a. When ${\alpha }_1=2/5$ and ${\alpha }_2=1/2$, the evolution of $u_3(x,t)$ is displayed in Figure 4b. Both of them are double-periodic solutions. As the values of ${\alpha }_1$ and ${\alpha }_2$ increase, the period becomes small. The relationship between the amplitude and the parameters ${\alpha }_1$ and ${\alpha }_2$ is rather complicated, and we have not obtained the specific relationship.

4. The Finite Conversation Laws

In this section, we examine the finite conservation laws for the nonlocal mKdV equation. The Lax pair (5) indicates that ${\varphi }_1$ and ${\varphi }_2$ satisfy the following differential equations:

$$\begin{array}{cc}\hfill {\varphi }_{1x}& =-\mathrm{i}\lambda {\varphi }_1+u(x,t){\varphi }_2,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\varphi }_{2x}& =\mathrm{i}\lambda {\varphi }_2+u^{\ast }(-x,-t){\varphi }_1.\hfill \end{array}$$

(52)

From these equations, we can express ${\varphi }_2$ and ${\varphi }_{2x}$ as

$$\begin{array}{cc}\hfill {\varphi }_2& ={\displaystyle \frac{1}{u(x,t)}}\left[{\varphi }_{1x}+\mathrm{i}\lambda {\varphi }_1\right],\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\varphi }_{2x}& ={\displaystyle \frac{\mathrm{i}\lambda }{u}}({\varphi }_{1x}+\mathrm{i}\lambda {\varphi }_1)+u^{\ast }(-x,-t){\varphi }_1.\hfill \end{array}$$

(54)

Substituting Equations (53) and (54) into the first equation of the Lax pair, it follows that

$${\varphi }_{1xx}={\displaystyle \frac{u_x}{u}}({\varphi }_{1x}+\mathrm{i}\lambda {\varphi }_1)-{\lambda }^2{\varphi }_1-u(x,t)u^{\ast }(-x,-t){\varphi }_1.$$

(55)

Let ${\varphi }_1=exp(-\mathrm{i}\lambda x+{\widehat{\varphi }}_1)$ and substitute it into (55). This yields a differential equation involving ${\widehat{\varphi }}_1$, derived by

$${\widehat{\varphi }}_{1xx}+{\widehat{\varphi }}_{1x}^2-2\mathrm{i}\lambda {\widehat{\varphi }}_{1x}={\displaystyle \frac{u_x}{u}}{\widehat{\varphi }}_{1x}-u(x,t)u^{\ast }(-x,-t).$$

(56)

Setting ${\widehat{\varphi }}_{1x}=\mu (x,t)$, the equation for ${\widehat{\varphi }}_1$ transforms into an equation for $\mu (x,t)$, which is expressed as

$${\mu }_x+{\mu }^2-2\mathrm{i}\lambda \mu ={\displaystyle \frac{u_x}{u}}\mu +u(x,t)u^{\ast }(-x,-t).$$

(57)

Since $\widehat{\varphi }\to 0$, ${\widehat{\varphi }}_x\to 0$, as $\left|\lambda \right|\to +\mathrm{i}nfty$, then we set

$$\mu =\sum _{j=0}^{+\infty }{\displaystyle \frac{{\mu }_j}{{\left(2\mathrm{i}\lambda \right)}^j}}.$$

(58)

Substituting it into (56) yields

$$\sum _{l+m=j}{\displaystyle \frac{{\mu }_l{\mu }_m}{{\left(2\mathrm{i}\lambda \right)}^j}}+\sum _{j=0}^{+\infty }{\displaystyle \frac{{\mu }_{jx}}{{\left(2\mathrm{i}\lambda \right)}^j}}-\sum _{j=0}^{+\infty }{\displaystyle \frac{{\mu }_j}{{\left(2\mathrm{i}\lambda \right)}^{j-1}}}={\displaystyle \frac{u_x}{u}}\sum _{j=0}^{+\infty }{\displaystyle \frac{{\mu }_j}{{\left(2\mathrm{i}\lambda \right)}^j}}-u(x,t)u^{\ast }(-x,-t).$$

By comparing the coefficients of ${\left(2\mathrm{i}\lambda \right)}^j$, we have

$$\begin{array}{cc}\hfill 2\mathrm{i}\lambda :& {\mu }_0=0,\hfill \\ \hfill 1:& {\mu }_1=u(x,t)u^{\ast }(-x,-t),\hfill \\ \hfill {\displaystyle \frac{1}{2\mathrm{i}\lambda }}:& {\mu }_{1x}={\mu }_2+{\displaystyle \frac{u_x}{u}}{\mu }_1\Rightarrow {\mu }_2=-u(x,t)u^{\ast }(-x,-t),\hfill \\ \hfill {\displaystyle \frac{1}{{\left(2\mathrm{i}\lambda \right)}^j}}:& \sum _{l+m=j}{\mu }_l{\mu }_m+{\mu }_{jx}-{\mu }_{j+1}={\displaystyle \frac{u_x}{u}}{\mu }_j,\Rightarrow {\mu }_{j+1}={\mu }_{jx}-{\displaystyle \frac{u_x}{u}}{\mu }_j+\sum _{l+m=j}{\mu }_l{\mu }_m.\hfill \end{array}$$

This represents the finite conservation laws for the reverse space-time nonlocal mKdV equation.

5. Summary and Discussion

We examine the modified space-time nonlocal mKdV equation by implementing the Riemann–Hilbert problem with a zero boundary condition. Assuming the scattering coefficient possesses N simple zero points and one higher-order zero point, we derive two distinct types of N-th-order solutions, as presented in (30) and (49). Explicit expressions for the first-, second-, and third-order solutions are provided, corresponding to scenarios with one simple pole, two simple poles, and one second-order pole along with one simple pole, respectively. Graphical representations of their evolution are also presented. Furthermore, we investigate the finite conservation laws for the modified space-time nonlocal mKdV equation.

Owing to the effect of the nonlocal term, a majority of the solutions of the modified space-time nonlocal mKdV equation with a zero boundary condition are singular, whereas the regular solutions are double-periodic ones. By contrast, the solutions of the local mKdV equation under a zero boundary condition are regular and take the form of solitons or positons [15].

In the forthcoming period, we aim to explore the Riemann–Hilbert problem related to this equation under a non-zero boundary condition. Through the imposition of such conditions, we expect to witness several novel phenomena relevant to nonlocal nonlinear integrable equations. Besides this nonlocal mKdV equation, there exist other types of nonlocal mKdV equation, such as the generalized geophysical KdV equation [28], the $(2+1)$-dimensional KdV-mKdV equation [29], and so forth. We intend to investigate the analytical solutions of these nonlinear equations.