Well-Posedness of a Hirota–Satsuma System Posed on a Half Line

Abstract

This paper studies the initial boundary value problem of the Hirota–Satsuma system posed on the half line. . For , we demonstrate that the abovementioned system is locally well-posed in by utilizing several analytic boundary forcing operators.

1. Introduction

In 1981, Hirota and Satsuma [1] introducedwhere , ; , are the real functions modeling the interactions of two long waves with different dispersion relations. Nowadays, (1) is called the Hirota–Satsuma system. Additionally, Hirota and Ohta in [2] derived the Hirota–Satsuma system as a reduction of a special hierarchy of coupled bilinear equations.

The initial value problem for the Hirota–Satsuma system on the whole line and periodic domain has been extensively studied. In 2005, Angulo [3] proved that the system is locally well-posed in , for , when , and globally well-posed in for , when . Furthermore, in 2007, Panthee and Silva [4] verified that the system is locally well-posed in , for , when , and globally well-posed in , for , when . Moreover, in 1994, Feng [5] demonstrated that the system is locally well-posed, for , and in 2022, Zhao and Lv [6] confirmed that the system is locally well-posed in , for , when .

Without losing generalization, for the remainder of this paper, we assume that and .

This study will investigate the initial boundary value problem for the Hirota–Satsuma system posed on the half line as follows:

The main result regarding the IBVP (2) can be stated as follows:

Theorem 1. Let . Then, there existsand a local solution of IBVP (2), in the distributional sense, such that

Moreover, the mapping is locally Lipschitz-continuous from to .

2. Presenting the Solution

2.1. Solution to the Initial Value Problem

We define the linear unitary groupwhich associated to the linear equation asso that

Thus, if we setthen solves the linear problem

2.2. Solution to the Boundary Value Problem

First, we introduce the Duhamel boundary forcing operator suggested by Colliander and Kenig [7]. Its definition iswhich is defined for all and signifies the Airy function

From this definition, we observe that

Next, we introduce the generalization of operator used in Holmer [8].

For and with , we havewhere . Then, by using (12), we obtain

For any , , we can address IBVP (2) by replacing in (12) with for a suitable .

The proofs of the following Lemmas exhibited in this section are presented in [8].

Lemma 1. (Spatial continuity and decay properties for ). If and fix . Then, we haveFor is continuous in for all . For and satisfies the following decay bounds:

Lemma 2. (Values of at ). For and , we have

Now, by adopting [8], we obtain the solution to the boundary value problem by using the class. Specifically, let , , if we setfrom Lemmas 1 and 2,

Moreover, if is given and we setthen solves the linear problem

2.3. Solution to the Initial-Boundary Value Problem

From the Lemmas presented and consideringwherethen solves the linear problem

2.4. Nonhomogeneous Versions

We introduce the Duhamel nonhomogeneous solution operator associated with the KdV equation asand it follows that

2.5. Solution to the Hirota–Satsuma System

Using (9), (21), (24), and (26), we obtain the solution of IBVP (2) aswhere

3. Main Estimates

In this section, we introduce the space trace estimates, the time trace estimates, and the Bourgain regularities.

Let and ; we introduce the classical Bourgain spaces associated with as the completion of the Schwartz space under the normwhere .

To obtain our results, we define the following auxiliary modified Bourgain spaces. Let and be the completion of with respect to the normsand

The following Lemma is an important estimate presented [9].

Lemma 3. Let or , and . Then, we have

3.1. Space Trace Estimates

Lemma 4. Let .(a)If , then we have(b)If and , then we have(c)If , then we have

Proof. See reference [8].

3.2. Time Trace Estimates

Lemma 5. Let .(a)For all , we have(b)For all , we have(c)For all , we have

Proof. See reference [8].

3.3. Bourgain Regularities

Lemma 6. Let .(a)For all , if , we have(b)For all , and , we have(c)For all and , we have

Proof. See references [8, 10].

3.4. Bilinear Estimates

The following Lemma was proposed by Zhao and its proof can be found in [6].

Lemma 7. If , then , we haveand

4. Proof of the Main Theorem

Before proving the main theorem, some helpful properties of the Sobolev spaces are outlined in the results that follow. The proof of them can be found in [7].

Lemma 8. Let . If , then there is a constant such that

Lemma 9. Let . If with ; then, we have and there is a constant such that

Proof of Theorem 1. Let , and be the extensions of , and such that , , and . Let such that Lemma 7 is valid.

Step 1. Solution mapping and spaces.
Take , . By Section 2.5 we define the operator , which is given byandwhere and are given byandWe consider in the Banach space , wherewith norm

Step 2. Estimates of boundary values.
For this purpose, by using Lemmas 4–6 (b), we show that and .
By hypothesizing , Lemmas 8 and 5 (a) implyNow, Lemmas 3, 5 (c), 7, and 8 implywhere is adequately small.
If , then, and Lemma 8 shows that . If , then due to the compatibility condition, and Lemma 9 shows that . Thus, equations (52) and (53) reveal that .
By combining the abovementioned, we obtainSimilarly, from Lemmas 3, 5 (a), 5 (c), 7, and 8, we haveIt follows thatSimilarly, as , we obtainIndeed, the abovementioned equations show that .
Thus, we obtain

Step 3. Contraction of the solution mapping.
From Lemmas 3, 4, 7, and equation (54), we haveFrom Lemmas 3, 5, 7, and equation (54), we haveFrom Lemmas 3, 6, 7, and equation (54), we haveby combining the abovementioned, we obtainFrom Lemmas 3–7, and equation (58), we haveInspired from [11], we havewhere and are given byandSimilarly, as abovementioned, we haveWe set the ball of aswhere and .
By restricting on the ball , by (62) and (63) and (67), we obtainThen, we take to be suitably small, such thatandThus, defines a contraction in , and we obtain a fixed point in in . Therefore, the IBVP (2) is solved in the sense of distributions by

5. Conclusions

According to [4, 6], the asymmetry of the nonlinear component makes the asymmetrical product space better suited for the Hirota–Satsuma system than the symmetric product space . Thus, this paper studies the IBVP of the Hirota–Satsuma system in the asymmetrical product space. Inspired by [8, 11], we propose the following directions for further consideration:

Remark 1. The skill used in this paper is also applicable to the IBVP of the Hirota–Satsuma system posed on the left half line :

Remark 2. In addition to discussing the left half line, we may consider other boundary conditions on the right half line. More specifically, let denote the differential operator associated with the KdV equationwhere . Sincewith any of the following boundary values:For any one of the boundary values a, b, c, and d, operator generates an operator group by the Phillips–Lumper theorem. We will investigate those questions in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

An earlier version of this article [12] was presented as a preprint on the Research Square website.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was financially supported by the Natural Science Foundation of Zhejiang Province (grant no. LY18A010024) and National Natural Science Foundation of China (grant no. 12075208).