Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions

Highlights

• The inverse scattering transforms are presented for the mKdV equation with NZBCs.

• The direct and inverse scattering problems are proposed on a complex plane.

• The inverse problems are solved via the matrix Riemann–Hilbert problems.

• The trace formulas and theta conditions are found.

• Some reflectionless potentials with distinct wave structures are illustrated.

Abstract

We explore the inverse scattering transforms with matrix Riemann–Hilbert problems for both focusing and defocusing modified Korteweg–de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity systematically. Using a suitable uniformization variable, the direct and inverse scattering problems are proposed on a complex plane instead of a two-sheeted Riemann surface. For the direct scattering problem, the analyticities, symmetries and asymptotic behaviors of the Jost solutions and scattering matrix, and discrete spectra are established. The inverse problems are formulated and solved with the aid of the matrix Riemann-Hilbert problems, and the reconstruction formulas, trace formulas, and theta conditions are also posed. In particular, we present the general solutions for the focusing mKdV equation with NZBCs and both simple and double poles, and for the defocusing mKdV equation with NZBCs and simple poles. Finally, some representative reflectionless potentials are in detail studied to illustrate distinct nonlinear wave structures containing solitons and breathers for both focusing and defocusing mKdV equations with NZBCs.

Introduction

We consider the inverse scattering transforms and general solutions for both focusing and defocusing modified Korteweg–de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity, namely

$$\left\{\begin{array}{c}{q}_t-6\sigma q^2{q}_x+q_{xxx}=0,(x,t)\in {\mathbb{R}}^2,\hfill \\ \underset{x\to \pm \infty }{lim}q(x,t)=q_{\pm },\left|q_{\pm }\right|=q_0>0\hfill \end{array}\right.$$

based on the generalized matrix Riemann–Hilbert problem, not the usual Gel’fand–Leviton–Machenko integral equation [1], [2], where $q=q(x,t)\in \mathbb{R}$, $\sigma =-1,1$ denote the focusing and defocusing mKdV equations, respectively. Eq. (1) possesses the scaling symmetry $q(x,t)\to \alpha q(\alpha x,{\alpha }^{3}t)$ with $\alpha$ being a non-zero parameter, and can be rewritten as a Hamiltonian system under the non-zero condition

$$q_t=\frac{\partial }{\partial x}\frac{\delta H\left[q\right]}{\delta q},H\left[q\right]=\frac{1}{2}{\int }_{-\infty }^{\infty }\left(\sigma q^4+q_x^2-\sigma q_0^4\right)dx.$$

Eq. (1) arises in many different physical contexts, such as acoustic wave and phonons in a certain anharmonic lattice [3], [4], Alfvén wave in a cold collision-free plasma [5], [6], thin elastic rods [7], meandering ocean currents [8], dynamics of traffic flow [9], [10], hyperbolic surfaces [11], slag-metallic bath interfaces [12], and Schottky barrier transmission lines [13]. The well-known Miura transform [2], [14]

$$u(x,t)=-\sigma q^2(x,t)+\sqrt{\sigma }{q}_x(x,t)$$

established the relation between Eq. (1) and the KdV equation

$$u_t+6u{u}_x+u_{xxx}=0.$$

The hodograph transform [15], [16]

$$r(y,t)=e^{1∕2{\int }^{x}q(x,t)dx},x={\int }^y{u}^{-1}(y,t)dy$$

was yielded between Eq. (1) and the Harry–Dym equation

$$u_t=u^3{u}_{yyy}.$$

Moreover, there exists a transformation [17]

$$p(x,t)=q(x-6\sigma {\nu }^{2}t,t)-\nu$$

between Eq. (1) and the Gardner equation (also called the combined KdV–mKdV equation) [2], [18], [19] with boundary conditions (BCs)

$$\left\{\begin{array}{c}{p}_t-6\sigma (p^2+2\nu p)p_x+p_{xxx}=0,\nu \in \mathbb{R}\setminus \left\{0\right\},(x,t)\in {\mathbb{R}}^2,\hfill \\ \underset{x\to \pm \infty }{lim}p(x,t)=p_{\pm },p_{\pm }=q_{\pm }-\nu .\hfill \end{array}\right.$$

Since the inverse scattering transform (IST) method was presented to solve the initial-value problem for the integrable KdV equation by Gardner, Greene, Kruskal and Miura [20], and for the nonlinear Schrödinger equation by Zakharov and Shabat [21], [22], there have been some results on the IST for the mKdV equation. For instance, Wadati studied the focusing mKdV equation with zero boundary conditions (ZBCs) and derived simple-pole, double-pole and triple-pole solutions [23], [24], after which the $N$-soliton and breather solutions for the focusing mKdV equation with ZBCs were proposed [25]. Deift and Zhou presented the long-time asymptotic behavior of the defocusing mKdV equation with ZBCs by using the powerful steepest descent method [26]. Recently, Germain et al. studied a full asymptotic stability for solitons of the Cauchy problem for the focusing mKdV equation [27]. The focusing mKdV equation with NZBCs was also studied such that the $N$-soliton solutions were obtained for the simple-pole case with pure imaginary discrete spectra [28], [29], and the breather solutions were found for the simple-pole case with a pair of complex conjugate discrete spectra [30]. The Hamiltonian formalism of the defocusing mKdV equation with special NZBCs $q_{+}=q_{-}$ was given [31]. Recently, the long-time asymptotic behavior of the simple-pole solution for the focusing mKdV equation with step-like NZBCs $q_{-}\ne q_{+}=0$ was studied [32].

To the best of our knowledge, there still exist the following open issues on the ISTs for the defocusing and focusing mKdV equations with NZBCs at infinity:

• Though some special simple-pole solutions of the focusing mKdV equation with NZBCs were given [28], [29], [30], there still exists a natural problem whether it admits a general simple-pole solution with mixed pairs of complex conjugate and pure imaginary discrete spectra, i.e., $N_1$-breather-$N_2$-soliton solutions ($N_1+N_2=N$).

• For the focusing mKdV equation with NZBCs, the multiple-pole solutions, i.e., the solutions corresponding to multiple-pole of the reflection coefficients, were not proposed yet. Especially, the general double-pole solutions with pairs of complex conjugate and pure imaginary discrete spectra were also unknown yet.

• The inverse problem of the focusing mKdV equation with NZBCs was solved by using the Gel’fand–Leviton–Machenko equation before [32], rather than formulated in terms of a matrix Riemann–Hilbert problem via a suitable uniformization variable.

• Though there are some partial results on the IST for the focusing mKdV equation with NZBCs, a more rigorous theory of the IST for the focusing mKdV equation with NZBCs remains open, such as the Riemann surface, uniformization variable, analyticity, symmetries, and the asymptotic of Jost solutions and scattering matrix, reconstruction formula, trace formulas, and theta conditions.

• There is almost no result on the IST for the defocusing mKdV equation with NZBCs.

Recently, Ablowitz, Biondini, Demontis, Prinary, et al. presented a powerful approach to study the ISTs for some nonlinear Schrödinger (NLS)-type equations with NZBCs at infinity [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], in which the inverse problems were formulated in terms of the suitable Riemann–Hilbert problems by defining uniformization variables. After that, the ISTs of the derivative NLS equation with NZBCs, the Hirota equation with NZBCs, and the nonlocal mKdV equation with NZBCs were also found [50], [51], [52]. Inspired by the above-mentioned idea, in this paper we would like to develop a more general theory to study systematically the ISTs for both focusing and defocusing mKdV equations with NZBCs (1) to solve affirmatively those above-mentioned open issues in turn.

It is well-known that the focusing or defocusing mKdV equation (1) is the compatibility condition, $X_t-T_x+[X,T]=0$, of the ZS-AKNS scattering problem (i.e., Lax pair) [53]

$${\Phi }_x=X\Phi ,X(x,t;k)=ik{\sigma }_3+Q,$$$${\Phi }_t=T\Phi ,T(x,t;k)=4k^{2}X-2ik{\sigma }_3\left(Q_x-Q^2\right)+2Q^3-Q_{xx},$$

where $k\in ℂ$ is a spectral parameter, the eigenfunction $\Phi =\Phi (x,t;k)$ is chosen as a 2 × 2 matrix, the potential matrix $Q$ is given by

$$Q=Q(x,t)=\left[\begin{array}{cc}\hfill 0\hfill & \hfill q(x,t)\hfill \\ \hfill \sigma q(x,t)\hfill & \hfill 0\hfill \end{array}\right],$$

and $\sigma =-1$ and $\sigma =1$ correspond to the focusing and defocusing mKdV equations, respectively.

Remark 1

The complex conjugate and conjugate transpose are denoted by $\ast$ and $†$, respectively. Three Pauli matrices are defined as

$${\sigma }_1=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right],{\sigma }_2=\left[\begin{array}{cc}\hfill 0\hfill & \hfill -i\hfill \\ \hfill i\hfill & \hfill 0\hfill \end{array}\right],{\sigma }_3=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right],$$

and $e^{\alpha {\widehat{\sigma }}_3}A≔e^{\alpha {\sigma }_3}A{e}^{-\alpha {\sigma }_3}$ with $A$ being a 2 × 2 matrix and $\alpha$ a scalar variable.

The rest of this paper is organized as follows. In Section 2, we present the IST for the focusing mKdV equation with NZBCs and simple poles. Moreover, the inverse problem of the focusing mKdV equation with NZBCs was formulated in terms of a matrix Riemann–Hilbert problem by a suitable uniformization variable. As a result, a general simple-pole solution with pairs of complex conjugate discrete spectra and pure imaginary discrete spectra, i.e., $N_1$-breather-$N_2$-soliton solutions, are found. In Section 3, we derive the IST for the focusing mKdV equation with NZBCs and double poles such that a general $N_1$-(breather, breather)-$N_2$-(bright, dark)-soliton solutions are found. In Section 4, the IST for the defocusing mKdV equation with NZBCs and simple poles is presented to generate multi-dark-soliton-kink solutions for some special cases. Finally, we give some conclusions and discussions in Section 5.