Well-posedness of initial value problem of Hirota-Satsuma system in low regularity Sobolev space

1.   Introduction

Hirota-Satsuma system coupled by two KdV equations:

(where $\alpha\in {\mathbb{R}}$, $\beta\in {\mathbb{R}}$; $u = u(x, t)$, $v = v(x, t)$ are real functions) was introduced by Hirota, Satsuma in ^[1] to describe the interactions of two long waves with different dispersion relations, also was derived by Hirota R, Ohta Y in ^[2] as a reduction of a special hierarchy of coupled bilinear equations.

The main progress of soliton solutions of Hirota-Satsuma system is as follows: In 2000, Tam and Ma in ^[3] considered some particular special expansions in the direct method to derive the one- and three-cKdV soliton solutions with a profile different in form to the classical solitons. In 2003, Hu and Liu in ^[4] derived generalized M-solitons solutions of the Grammian type by means of a Darboux transformation. In 2020, Prado and Cisneros-Ake in ^[5] carried out a systematic analysis of multi-soliton solution based on the direct method to fully describe its N+M interacting multisoliton solutions holding a typical hyperbolic profile (For more detail, see ^[6]).

The progress of well-posedness of Hirota-Satsuma system can be summarized as: In 1994, Feng proved in ^[7] that Hirota-Satsuma system posed on the whole line is locally well-posed $H^s({\mathbb{R}})\times H^{s}({\mathbb{R}})$, if $s > 2$. In 2005, Angulo showed in ^[8] that Hirota-Satsuma system posed on periodic domain is locally well-posed in $H_{periodic}^s(0, L)\times H_{periodic}^{s}(0, L)$, for $s\ge 0$, when $\alpha = -1$ and globally well-posed in $H_{periodic}^s(0, L)\times H_{periodic}^{s}(0, L)$ for $s\ge 1$, when $\alpha\neq-1$, $0$. In 2007, Panthee, Silva verified in ^[9] that Hirota-Satsuma system posed on periodic domain is locally well-posed in $H_{periodic}^s(0, L)\times H_{periodic}^{1+s}(0, L)$, for $s\ge -\frac12$ and global well-posed in $H_{periodic}^s(0, L)\times H_{periodic}^{s+1}(0, L)$ for $s\ge -\frac3 {14}$ when $\alpha = -1$.

In this paper, we will study the initial value problem of Hirota-Satsuma system:

As shown in ^[9] that asymmetrical product space $H^s({\mathbb{R}})\times H^{s+1}({\mathbb{R}})$ is more suitable to Hirota-Satsuma system than the symmetrical product space $H^s({\mathbb{R}})\times H^{s}({\mathbb{R}})$ since the asymmetry of nonlinear term $uv_x$.

Definition 1.1. Let $s\in {\mathbb{R}}$, $b\in {\mathbb{R}}$, Bourgain space $X_{s, b}$ associated with $\partial_t\pm\alpha\partial^3_x$ is defined to be the closure of the Schwartz space $S(R^2)$ under the norm:

where $\langle\cdot\rangle = (1+|\cdot|)$, ${\mathcal{F}}u = \widehat{u} (\xi, \tau)$ denote as Fourier transformation of $u$ with respect to $t$ and $x$.

Obviously, when $s_1 \le s_2$, $b_1 \le b_2$, $\|u\|_{X_{s_1, b_1}} \le \|u\|_{X_{s_2, b_2}}$.

The main result is:

Theorem 1.2. Let $s > -\frac 18$. Then for any initial data $(\phi, \psi)\in H^s({\mathbb{R}})\times H^{1+s}({\mathbb{R}})$, there exists $T = T(\|(\phi, \psi)\|_{H^s\times H^{1+s}})$, such that there is unique solution of initial value problem (1.1) on [0, T).

Conservative of mass

conservative of energy

and local well-posedness (Theorem 1.2) imply that: For $\alpha = -1$ and $\beta > 0$, initial value problem (1.1) is globally well-posed in $H^s({\mathbb{R}})\times H^{1+s}({\mathbb{R}})$ if $s\ge 0$.

The following sections are arranged as follows: Bilinear estimate will be established in Section 2 which is the core of the Fourier restriction norm method; Locally well-posedness will be proved in Section 3 by Banch's fixed point theorem; We give some remarks in Section 4 to point out some simple facts about the Hirota-Satsuma system.

In the following, without lose of generalization, we assume that $\alpha = -1$, $\beta = 1$.

2.   Bilinear estimates

2.1.   Some lemmas

$D^s$ denote the s-order derivative defined by:

Lemma 2.1. Denote $\widehat{F}_\rho(\xi, \tau) = \frac{f(\xi, \tau)}{(1+|\tau-\xi^3|)^\rho}$, then

(1) If $\rho > \frac 12$, then

where $\chi\in C_0^\infty$ satisfying: When $|\xi|\le 1$, $\chi(\xi) = 1$; when $|\xi| > 2$ then $\chi(\xi) = 0$.

(2) If $\rho > \frac 38$, $0\le \theta\le \frac 18$, then

(3) If $\rho > \frac 5{12}$, then

(4) If $\rho > \frac \theta 2$, where $\theta \in [0, 1]$, then

(5) If $\rho > \frac 13$, then

Proof. (2.1)–(2.5) are Lemmas 2.3–2.7 in ^[10].

Lemma 2.2. Assume $f$, $f_1$, $f_2$ are Schwartz functions, then

where $\int_\ast d\delta = \int\limits_{\xi = \xi_1+\xi_2, \; \tau = \tau_1+\tau_2}d\xi_1 d\xi_2 d\tau_1 d\tau_2$.

Let $Z$ be Abelian addition group with invariable measure $d\xi$. For integer $k\ge 2$, we denote $\Gamma_k(Z)$ as the hyperplane:

Define $[k, Z]$-multiplier as function $m: \Gamma_k(Z)\mapsto C$. If $m$ is $[k, Z]$-multiplier, define $\|m\|_{[k, Z]}$ the norm of $[k, Z]$-multiplier as the infimum of $C$ such that

Lemma 2.3. If $m(\xi)$ and $M(\xi)$both are $[k, Z]$-multipliers, and $\forall\; \xi\in\Gamma_k(Z)$, $|m(\xi)|\le |M(\xi)|$, then

Proof. See ^[11] for the detail.

2.2.   Bilinear estimates

Proposition 2.4. If $s\ge -\frac 18$, $\frac 12 < b < \frac 9{16}$, then $\forall \; b' > \frac12$, we have

and

Proof. It is enough to prove (2.6). Since the proof of (2.7) is just a minor modification of that of (2.6). Besides, it is enough to show the case of $s\le 0$. Since when $s > 0$, we have:

This inequality and the results of $s = 0$ implies the result of $s > 0$.

By Plancherel Theorem, in order to prove (2.6), it is enough to prove

where $\bar{f}\in L^2({\mathbb{R}}^2)$ and $\bar{f}\ge 0$;

By the definition of $[k, Z]$-multiplier, if

then (2.6) holds.

By symmetry, it is enough to consider $|\xi_1|\le|\xi_2|$. Let $r = -s$, then $\frac18 > r\ge 0$.

Denote $\widehat{F}_\rho(\xi, \tau) = \frac{\bar{f}(\xi, \tau)}{(1+|\tau-\xi^3|)^\rho}$, $\widehat{F}_\rho^j(\xi, \tau) = \frac{f_j(\xi, \tau)}{(1+|\tau-\xi^3|)^\rho}$, $j = 1, 2$.

Case 1. $|\xi|\le 2$.

Subcase 1.1. $|\xi_1|\le 1$. We have $|\xi_2| = |\xi-\xi_1|\le|\xi| +|\xi_1|\le 3$, thus,

We applied (2.2) of Lemma 2.1 and Lemma 2.2 here.

Subcase 1.2. $|\xi_1|\ge 1$. By symmetrical assumption, $|\xi_2|\ge1$. For $r\le \frac 18$, we have

We applied (2.2) of 2.1 and Lemma 2.2 here.

Case 2. $|\xi|\ge 2$.

Case 2.1. $|\xi_1|\le 1$. We have $|\xi_2| = |\xi-\xi_1| \ge|\xi|-|\xi_1|\ge 1$, thus

Here (2.1) and (2.4) of Lemma 2.1 and Lemma 2.2 are used. Besides, $b < \frac 9{16}$ is also required.

Case 2.2. $|\xi_1|\ge 1$. By symmetrical assumption, $1\le |\xi_1|\le |\xi_2|$.

Since $(\tau_1-\xi_1^3)+(\tau_2-\xi_2^3)-(\tau-\xi^3) = 3\xi\xi_1\xi_2$, at lease one of the following 3 cases will occur:

By this fact, we divide Case 2.2 into 3 different subcases as follows:

Case 2.2.1. When (a) occurs. If $r+b-1\le \frac 18$ and $r\ge b > \frac 12$, then

Here (2.2) of Lemma 2.1 and Lemma 2.2 are used.

The above results implies that if $r+b-1\le \frac 18$ and $r\ge b > \frac 12$, then

By Lemma 2.3, when $r\le \frac 18$, (2.8) still holds. Indeed, since $\xi = \xi_1+\xi_2$, we have $\langle\xi\rangle\le\langle \xi_1\rangle \langle\xi_2\rangle$. If $r_1\le r_2$, then

Case 2.2.2. When (b) occurs. If $r+b'\ge 1$, $0 < r-b'\le \frac 1{16}$, we have

where (2.2) of Lemma 2.1 and Lemma 2.2 is used here. Besides, it is required that $b < \frac 58$.

When $r\le \frac 18$, the results is implied by Lemma 2.3.

Case 2.2.3. When (c) occurs. The proof is similar to Case 2.2.2, we omit the detail.

3.   Proof of the main theorem

Take $\theta \in C^\infty_0({\mathbb{R}})$ such that: When $t\in [-\frac{1}2, \frac 12]$, $\theta\equiv 1$ and supp $\theta \subseteq (-1, \; 1)$. Denote $\theta_\delta(t) = \theta(\frac t\delta)$. Let $U(t)$ $(t\in {\mathbb{R}})$ denote fundamental solution operator of the Airy equation: $v_t\pm v_{xxx} = 0$:

Lemma 3.1. Let $s\in R$, $\frac 12 < b < b'\le 1$, $0 < \delta\le 1$, then

Proof. See ^[10].

In the following, we will give the

Proof of Theorem 1.1:

Proof. For $s\ge -\frac 18$, let $(\phi, \psi)\in H^s\times H^{1+s}$ and $\|(\phi, \psi)\|_{H^s\times H^{1+s}}\equiv\|\phi\|_{H^s}+\|\psi\|_{H^{1+s}} = r$. Define

then $B_r$ is Banach space, whose norm is

For $(u, v)\in B_r$, define the mapping

We will prove that $\Phi\times \Psi_{(\phi, \psi)}[u, v]$ map $B_r$ into $B_r$.

By (3.1)–(3.3) in Lemma 3.1 and bi-linear estimate (2.7), there exists $b$, $b'$ satisfying $\frac12 < b < b'\le \frac 9{16}$ such that

Similarly, by (3.1)–(3.3) of Lemma 3.1 and bilinear estimate (2.6), we have

Thus, by the estimates (3.4) and (3.5), we have

Thus, when taking $\delta < [(2C)^2r]^{\frac1{b-b'}}$, $\Phi\times \Psi_{(\phi, \psi)}[u, v]$ mapping $B_r$ into $B_r$.

Similar to (3.4) and (3.5), for $\delta$ determined above, we have

Thus, $\Phi\times \Psi_{(\phi, \psi)}[u, v]$ is contract mapping.

Finally, by Banach theorem, $\forall\; t$ $(0 < t\le 1)$, in the ball $B_r$, the mapping $\Phi\times \Psi_{(\phi, \psi)}[u, v]$ have unique fixed point $(u, v)$ satisfying

4.   Conclusions

Remark 4.1. Although, the main result in this paper covered the results of ^[7], it must be not the sharp results when compare it with ^[9].

Remark 4.2. When compare it with ^[9], we conjecture that the initial value problem of Hirota-Satsuma system maybe locally well-posed in $H^s({\mathbb{R}})\times H^{s+1}({\mathbb{R}})$, for any $s > -\frac34$. We'll investigate this question in the future.

Remark 4.3. We are interested in well-posedness of initial boundary value problem of the Hirota-Satsum system, especially well-posedness with low regularity datum. We'll show the results in elsewhere.

Acknowledgments

This work is financially supported by the Natural Science Foundation of Zhejiang Province (No. LY18A010024, No. Y19A050005) and National Natural Science Foundation of China (No. 12075208).

Conflict of interest

The authors declare that they have no conflicts of interest.