Well-Posedness of a Hirota–Satsuma System Posed on a Half Line
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.
In 1981, Hirota and Satsuma  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  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  proved that the system is locally well-posed in , for , when , and globally well-posed in for , when . Furthermore, in 2007, Panthee and Silva  verified that the system is locally well-posed in , for , when , and globally well-posed in , for , when . Moreover, in 1994, Feng  demonstrated that the system is locally well-posed, for , and in 2022, Zhao and Lv  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 . 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 .
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 .
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 , 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 .
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 .
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 .
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 .
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 .
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 , we havewhere and are given byandSimilarly, as abovementioned, we haveWe set the ball of aswhere and