1 Introduction

Abstract

We explore systematically a rigorous theory of 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. 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 Riemann-Hilbert problems, and the reconstruction formulae, trace formulae, and theta conditions are also found. 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 wave structures for both focusing and defocusing mKdV equations with NZBCs.

Keywords: focusing and defocusing mKdV equations; non-zero boundary conditions; direct and inverse scattering; Riemann-Hilbert problem; solitons; breathers

Inverse scattering transforms and solutions for the focusing and defocusing mKdV equations with non-zero boundary conditions

[0.2in]

Guoqiang Zhang and Zhenya Yan1

[0.03in] Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science,

Chinese Academy of Sciences, Beijing 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

(Date:   August 15, 2019)

1 Introduction

This paper focuses on the inverse scattering transforms and general solutions for both focusing and defocusing mKdV equations with non-zero boundary conditions (NZBCs) at infinity, namely

(1)

based on the generalized Riemann-Hilbert problem, not the usual Gel’fand-Leviton-Machenko integral equation [1, 2], where , denote, respectively, the focusing and defocusing mKdV. Eq. (1) possesses the scaling symmetry with being a non-zero parameter, and can be rewritten as a Hamiltonian system under the non-zero condition

(2)

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 [15], and Schottky barrier transmission lines [16]. The well-known Miura transform [2, 12] established the relation between Eq. (1) and the KdV equation . The Hodograph transform [13, 14] was found between Eq. (1) and the Harry-Dym equation . Moreover, there exists a transformation [17] between Eq. (1) equation and the Gardner equation (also called the combined KdV-mKdV equation) [2, 18] with BCs

(3)

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 [19], and for the nonlinear Schrödinger equation by Zakharov and Shabat [20, 21], 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 [22, 23], after which the -soliton solutions and breather solutions for the focusing mKdV equation with ZBCs were found [24]. Deift and Zhou presented the long-time asymptotics of the defocusing mKdV equation with ZBCs by using the so-called steepest descent method [25]. The focusing mKdV equation with NZBCs was also studied such that the -soliton solutions were obtained for the simple-pole case with pure imaginary discrete spectra [26, 27], and the breather solutions were found for the simple-pole case with a pair of conjugate complex discrete spectra [28]. The Hamiltonian formalism of the defocusing mKdV equation with special NZBCs was given [29]. Recently, the long-time asymptotics of the simple-pole solution for the focusing mKdV equation with step-like NZBCS was studied [30].

To the best of our knowledge, there still are the following open questions on the ISTs for the defocusing mKdV equation with NZBCs and even for the focusing mKdV equation with NZBCs:

  • Though some special simple-pole solutions of the focusing mKdV equation with NZBCs were given [26, 27, 28], there still exists a natural problem whether it admits a general simple-pole solution with mixed pairs of conjugate complex discrete spectra and pure imaginary discrete spectra, i.e., -breather--soliton solutions ().

  • For the focusing mKdV equation with NZBCs, the multi-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 conjugate complex 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 [30], 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, uniformaization variable, analyticity, the symmetries and the asymptotic of Jost solutions and scattering matrix, reconstruction formula, trace formula and theta conditions.

  • There are almost no known results 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 [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], in which the inverse problems were formulated in terms of the suitable Riemann-Hilbert problems by defining uniformization variables. 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 should be pointed out that the mKdV equation differs from the NLS equation for two main reasons: i) symmetries of their Lax pairs are different; ii) the potential in the NLS equation is complex while one in the mKdV equation is real such that more complicated conditions are required for the mKdV equation. Moreover, based on the above-mentioned relations between the mKdV equation Eq. (1) and other physically important nonlinear wave equations, the obtained results can also be applied to these nonlinear wave equations.

The rest of this paper is organized as follows. In Sec. II, we present a rigorous theory of 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 results, a general simple-pole solution with pairs of conjugate complex discrete spectra and pure imaginary discrete spectra, i.e., -breather--soliton solutions, are found. In Sec. III, we derive the IST for the focusing mKdV equation with NZBCs and double poles such that a general -(breather, breather)--(bright, dark)-soliton solutions are found. In Sec. IV, 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 the conclusions and discussions in Sec. V.

Remark 1.

The complex conjugate and conjugate transpose are denoted by and , respectively. The three Pauli matrices are defined as

(4)

and with being a matrix and being a scalar variable.

It is well-known that the focusing or defocusing mKdV equation (1) is the compatibility condition, , of the ZS-AKNS scattering problem (i.e., Lax pair) [48]

(5)
(6)

where is an iso-spectral parameter, the eigenfunction is chosen as a matrix, the potential matrix is given by

(7)

and and correspond to the focusing and defocusing mKdV equations, respectively.

2 The focusing mKdV equation with NZBCs: simple poles

2.1 Direct scattering problem

The direct scattering process can determine the analyticity and the asymptotic of the scattering eigenfunctions, symmetries, and asymptotics of the scattering matrix, discrete spectrum, and residue conditions. Though defining a suitable uniformization variable [49], the two-sheeted Riemann surface for can be transformed into the standard complex -plane, on which the scattering problem is discussed conveniently.

Riemann surface and uniformization variable

Considering the following asymptotic scattering problem () of the focusing Lax pair (5) and (6) with :

(8)

with one can obtain the fundamental matrix solution of Eq. (8) as

(9)

where

(10)

To discuss the analyticity of the Jost solutions, one needs to determine the regions where Im in (cf. Eq. (10), see, e.g., Ref. [36, 45])). Since the defined is doubly branched, where the branch points are , thus one should introduce a two-sheeted Riemann surface such that is a single-valued function on its each sheet. Let with the arguments and , one obtains two single-valued branches and , respectively, on Sheet-I and Sheet-II, where the branch cut is the segment . The region where Im is the upper-half plane (UHP) on Sheet-I and the lower-half plane (LHP) on Sheet-II. The region where Im is the LHP on Sheet I and UHP on Sheet II. Besides, is real-valued on real axis and the branch cut (see Fig. 1(left)).

Remark 2.

When , i.e., , the NZBCs reduces to the ZBCs and .

Figure 1: Fousing mKdV equation with NZBCs. Left: the first sheet of the Riemann surface, showing the discrete spectrums, the region where (grey) and the region where (white). Right: the complex -plane, showing the discrete spectrums [zeros of (blue) in grey region and those of (red) in white region], the region where (grey), the region where (white) and the orientation of the contours for the Riemann-Hilbert problem.

Before we continue to study the properties of the Jost solutions and scattering datas, it is convenient to introduce a uniformization variable defined by the conformal mapping [49]:

(11)

whose inverse mapping is derived as

(12)

The mapping relation between the two-sheeted Riemann -surface (Fig. 1(left)) and complex -plane (Fig. 1(right)) is observed as follows:

  • On the Sheet-I of the Riemann surface, as , while on the Sheet-II of the Riemann surface, as ;

  • The Sheet-I and Sheet-II, excluding the branch cut, are mapped onto the exterior and interior of the circle of radius , respectively;

  • The branch cut is mapped into the circle of radius . In particular, the segment of the Sheet-I (Sheet-II) is mapped onto the part in the first (second) quadrant of complex -plane, and the segment of Sheet-I (Sheet-II) is mapped onto the part in the third (fourth) quadrant of complex -plane;

  • The real axis is mapped onto the real axis. In particular, of the Sheet-I (Sheet-II) is mapped onto () and of the Sheet-I (Sheet-II) is mapped onto ();

  • The region for Im (Im ) of the Riemann surface is mapped onto the grey (white) domain in the complex -plane. In particular, the UHP of the Sheet-I (Sheet-II) is mapped onto the grey (white) domain of the UHP in the complex -plane, and the LHP of the Sheet-I (Sheet-II) is mapped onto the white (grey) domain of the LHP in the complex -plane.

For convenience, we denote the grey and white domains in Fig. 1 (right) by

(13)

respectively. In the following, we will consider our problem on the complex -plane instead of -plane. With the help of the inverse mapping (12), one can rewrite the fundamental matrix solution of the asymptotic scattering problem as , where

(14)
Remark 3.

As , , whereas , in which its inverse matrix exists, i.e.,

(15)

Moreover, we find that , and

(16)

which allows one to define the Jost solutions as simultaneous solutions of both parts of the Lax pair (5) and (6).

Properties of Jost solutions

As usual, the continuous spectrum is the set of all values of satisfying [36]. Let be the circle of radius (see Fig. 1 (right)). Then, the continuous spectrum is denoted by . We will seek for the simultaneous solutions of the Lax pair (5, 6), i.e., the so-called Jost solutions, such that

(17)

The modified Jost solutions is introduced though dividing by the asymptotic exponential oscillations

(18)

such that

(19)

Then the Jost integral equation can be obtained from Eqs. (5) by the constant variation approach

(20)

where .

Lemma 1.

Given a series and a function on an interval , where and are matrix-valued functions. If converges uniformly on the interval and , , then and

Proposition 1.

Suppose , then the Jost integral equation (20) has unique solutions defined by Eq. (18) in . Moreover, the columns and can be extended analytically to and continuously to , and and can be extended analytically to and continuously to , where is the -th column of .

Proof.

As , . Let , which satisfy (cf. Eq. (20))

For the first column of , one has

(21)

where . As usual, a Neumann series for is introduced as

(22)

with ’s defined by

For , we have

For , we restrict to the domain , where the domain is defined by . Combining with the fact that as , one can infer that there exists a positive constant such that for . Therefore, for . Furthermore, one can obtain the recursion inequality:

(23)

In terms of Eq. (23) one can find the estimate for each term of the series as

(24)

which implies that the series converges uniformly in the domain . Thus is continuous in and analytic in . Besides, the arbitrariness of infers that is continuous in and analytic in . As , Eq. (24) also can infer that

(25)

which derive that converges uniformly in . With Lemma 1, one obtains

(26)

Eq. (26) with the arbitrary of illustrates the existence of the solution for Eq. (21). Using the following inequality,

(27)

one gives the asymptotic of :

To illustrate the uniqueness of the solution for integral equation (21), we need to prove that the homogeneous integral equation has only zero solution. Now we write the homogeneous integral equation as

It is easy to infer that

(28)

Let . Then one can obtain the following inequality for :

that is,

By the monotonicity, one obtains

which derives . Combining with in Eq. (28), one can infer . Then the uniqueness is proved. The existence, uniqueness, continuity, and analyticity of can also follow from . Similarly, one can also verify the corresponding properties of using the above-mentioned approach and here we omit them. ∎

Corollary 1.

Suppose , then Eq. (5) has unique solutions defined by Eq. (17) in . Moreover, and can be extended analytically to and continuously to , and and can be extended analytically to and continuously to , where is the -th column of .

Proof.

The proof can be deduced from Proposition 1 by Eq. (18). ∎

Proposition 2.

Suppose , then the Jost integral equation (20) has unique solutions defined by Eq. (18) in . Besides, and can be extended analytically to and continuously to while and can be extended analytically to and continuously to .

Proof.

It follows from Proposition 1 that we here only need to verify the existence, uniqueness and continuity of at .

satisfies the following integral equation:

where is the argument of . Similarly, we introduce a Neumann series

with

As , one has

which can be used to get the recursion inequality:

In the condition of , one has