Advancements in Gevrey Regularity for a Coupled Kadomtsev–Petviashvili II System: New Insights and Findings

Abstract

In this work, we prove that the initial value problem for a system of two Kadomtsev–Petviashvili II (KP II) equations coupled via both dispersive and nonlinear terms is locally well-posed in anisotropic Gevrey spaces $G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)\times G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)$ with $-1/3<s_1<0$ and $s_2\ge 0$. This advancement extends recent findings regarding the well-posedness of this model within anisotropic Sobolev spaces $H^{s_1,s_2}\left({\mathbb{R}}^2\right)\times H^{s_1,s_2}\left({\mathbb{R}}^2\right)$. The current strategy is based on both linear and nonlinear estimates. Additionally, to further explore the system’s temporal behavior, we establish that Gevrey regularity of order $3\rho$ (or simply Gevrey—3$\rho$ regularity in time) exists.

1. Introduction

Nonlinear partial differential equations are crucial tools in various scientific disciplines, providing a flexible framework for modeling and analyzing a range of phenomena within complex environments. The wave solutions of nonlinear equations are explored in [1]. The double sinh-Gordon equation is introduced in [2]. In [3], two nonlinear partial differential equations are studied: the double sine-Gordon equation and the Burgers equation. A coupled KdV system and its stability are presented in [4]. The Bresse system with two fractional damping terms is discussed in [5]. A nonlinear time-fractional Klein–Gordon equation is described in [6]. Additionally, in [7], a nonlinear time-fractional Euler–Bernoulli beam problem that includes time delay is addressed. A complex system of quadratic heat equations with a generalized kernel is investigated in [8]. Lastly, a generalized Kadomtsev–Petviashvili equation is stabilized in [9].

Indeed, the Kadomtsev–Petviashvili (KP) equations are fundamental models in wave dynamics, especially in scenarios where dispersion and nonlinearity interact in two-dimensional spatial environments. Among these equations, the KP-II equation is particularly relevant for describing surface waves in shallow water and phenomena in plasma physics. This paper extends the classical KP-II equations by considering a coupled system of equations that incorporates both dispersive and nonlinear interactions. Understanding the existence and regularity of solutions to these systems is crucial for analyzing the long-term behavior and stability of modeled waves. Chaotic structures and novel insights into solutions are discussed in [10] concerning the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation, along with its stability analysis. The methodology developed in [11] is applied to various traveling wave solutions of the (2+1)-dimensional extended Kadomtsev–Petviashvili equation. Two techniques are utilized in [12] to analyze new soliton-type solutions for the generalized extended (2+1)-dimensional Kadomtsev–Petviashvili equation. The Kadomtsev–Petviashvili equation, incorporating self-consistent sources, breathers, lumps, and their interactions, is examined in [13]. A new solution for the lattice Kadomtsev–Petviashvili system associated with an elliptic curve is presented in [14]. The line-soliton solutions of a coupled modified Kadomtsev–Petviashvili system in two-layer shallow water are established in [15]. Additionally, a positive multi-complexity solution for a generalized Kadomtsev–Petviashvili equation is introduced in [16]. In [17], resonant multiple solitons, N-soliton solutions, soliton molecules, and their interactions are derived for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation.

In [18], the authors studied the (2+1)-dimensional stochastic Kadomtsev–Petviashvili equation with a beta derivative. The author of [19] presented results on rogue waves for various partial differential equations, including the Kadomtsev–Petviashvili equation, the nonlinear Schrödinger equation, the Hirota equation and the Lakshmanan–Porsezian–Daniel equation. Ref. [20] analyzed the nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation, which describes the propagation of weakly dispersive waves in a fluid.

The investigation of Kadomtsev–Petviashvili system in classical Sobolev spaces is well established, with numerous solutions explored based on various exponent values. It is important to study the Kadomtsev–Petviashvili system in the anisotropic Gevrey spaces.

This work aims to study the well-posedness and regularity of the coupled KP II system in the following form:

$$\left\{\begin{array}{c}{\partial }_t\psi +{\alpha }_1{\partial }_x^3\psi +{\alpha }_2{\partial }_x^3\phi +{\gamma }_1{\partial }_x\left(\psi \phi \right)+{\gamma }_2\psi {\partial }_x\psi +{\gamma }_3\phi {\partial }_x\phi +{\partial }_x^{-1}{\partial }_y^2\psi =0,\hfill \\ {\partial }_t\phi +{\alpha }_3{\partial }_x^3\psi +{\alpha }_4{\partial }_x^3\phi +{\gamma }_4{\partial }_x\left(\psi \phi \right)+{\gamma }_5\psi {\partial }_x\psi +{\gamma }_6\phi {\partial }_x\phi +{\partial }_x^{-1}{\partial }_y^2\phi =0,\hfill \\ \psi (x,y,0)={\psi }_0(x,y),\hfill \\ \phi (x,y,0)={\phi }_0(x,y),\hfill \end{array}\right.$$

(1)

where $\psi =\psi (x,y,t)$ and $\phi =\phi (x,y,t)$ are unknown functions, and $(x,y,t)\in {\mathbb{R}}^3$ and ${\psi }_0$, ${\phi }_0$ are a given functions. ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\alpha }_4$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$, ${\gamma }_4$, ${\gamma }_5$ and ${\gamma }_6$ are real constants with ${\alpha }_1{\alpha }_2-{\alpha }_3{\alpha }_4>0$ and ${\alpha }_2{\alpha }_3>0$.

In [21], the author proved that (1) is locally well-posed in the spaces $H^{s_1,s_2}\left({\mathbb{R}}^2\right)\times H^{s_1,s_2}\left({\mathbb{R}}^2\right)$ with $-{\displaystyle \frac{1}{3}}<s_1<0$ and $s_2\ge 0$ and globally well-posed in the spaces $H^{s_1,0}\left({\mathbb{R}}^2\right)\times H^{s_1,0}\left({\mathbb{R}}^2\right)$ with $s_1>-{\displaystyle \frac{1}{14}}$.

This paper has two main objectives. First, we demonstrate that the coupled KP-II system is locally well-posed in anisotropic Gevrey spaces $G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)\times G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)$ with $-1/3<s_1<0$ and $s_2\ge 0$. Second, we establish that solutions to the system with given initial data exhibit Gevrey regularity in the time variable, specifically of the order $3\rho$. It is essential to note that for $\varrho =1$ and ${\delta }_1={\delta }_2=0$, the space $G_{s_1,s_2}^{0,0,1}$ simplifies to the anisotropic Sobolev spaces $H^{s_1,s_2}$.

Furthermore, the paper explores the time regularity of solutions, demonstrating that they exhibit Gevrey-$3\rho$ regularity with respect to time. The new bounds are much sharper than those previously available. The introduction of Gevrey-$3\rho$ regularity is crucial for gaining a deeper understanding of the temporal evolution of the system, which is essential for real-world applications where time-dependent behavior is critical.

This work is organized as follows: In the next section, we establish the foundational tools and definitions necessary for analyzing the problem, introducing key function spaces and notational conventions. Section 3 delves into deriving crucial estimates for both linear and nonlinear terms, which is essential for assessing well-posedness. In Section 4, we provide a detailed proof of the local well-posedness of the equation under consideration, demonstrating the existence, uniqueness, and continuity of solutions within appropriate function spaces. Section 5 explores the regularity of solutions, with a primary focus on Gevrey spaces and their relevance to the behavior of solutions. Finally, the Conclusion Section summarizes the main results and outlines potential directions for future research, emphasizing the implications of the findings within the broader context of the theory.

2. Notation and Function Spaces

Anisotropic Gevrey spaces are given by $G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)=G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }$ with ${\delta }_1,{\delta }_2>0$; $\varrho \ge 1$ and $s_1,s_2\in \mathbb{R}$ can be defined as the completion of the Schwartz functions with respect to the norm

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}={\left({\int }_{{\mathbb{R}}^2}{e}^{2{\delta }_1{\left|\xi \right|}^{{\displaystyle \frac{1}{\varrho }}}}{e}^{2{\delta }_2{\left|\theta \right|}^{{\displaystyle \frac{1}{\varrho }}}}{\langle \xi \rangle }^{2s_1}{\langle \theta \rangle }^{2s_2}{\left|\widehat{\psi }(\xi ,\theta )\right|}^{2}d\xi d\theta \right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

where $\langle .\rangle ={(1+|.|}^2{)}^{{\displaystyle \frac{1}{2}}}$, and $\widehat{\psi }$ is the space Fourier transform of $\psi$ that is defined as

$$\begin{array}{c}\hfill \widehat{\psi }(\xi ,\theta )={\int }_{{\mathbb{R}}^2}{e}^{-i(x\xi +y\theta )}\psi (x,y)dxdy.\end{array}$$

For $s_1,s_2,b\in \mathbb{R}$ and $\beta >0$, anisotropic Bourgain spaces are given by the norm

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }}={\left({\int }_{{\mathbb{R}}^3}{\langle \xi \rangle }^{2s_1}{\langle \theta \rangle }^{2s_2}{\langle \lambda \rangle }^{2b}{\langle \eta \rangle }^{2\beta }{\left|\widehat{\psi }(\xi ,\theta ,\tau )\right|}^{2}d\xi d\theta d\tau \right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

where $\lambda =\lambda (\xi ,\theta ,\tau )=\tau -{\xi }^3+{\displaystyle \frac{{\theta }^2}{\xi }}$, $\eta =\eta (\xi ,\theta ,\tau )={\displaystyle \frac{\lambda (\xi ,\theta ,\tau )}{1+{\left|\xi \right|}^3}}$. Here, $\widehat{\psi }$ is the space time Fourier transform that is defined as

$$\begin{array}{c}\hfill \widehat{\psi }(\xi ,\theta ,\tau )={\int }_{{\mathbb{R}}^3}{e}^{-i(x\xi +y\theta +t\tau )}\psi (x,y,t)dxdydt.\end{array}$$

Furthermore, we will also need a hybrid of the analytic Gevrey and anisotropic Bourgain spaces, denoted as $X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }$ and equipped with the norm

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}={\left({\int }_{{\mathbb{R}}^3}{e}^{2{\delta }_1{\left|\xi \right|}^{{\displaystyle \frac{1}{\varrho }}}}{e}^{2{\delta }_2{\left|\theta \right|}^{{\displaystyle \frac{1}{\varrho }}}}{\langle \xi \rangle }^{2s_1}{\langle \theta \rangle }^{2s_2}{\langle \lambda \rangle }^{2b}{\langle \eta \rangle }^{2\beta }{\left|\widehat{\psi }(\xi ,\theta ,\tau )\right|}^{2}d\xi d\theta d\tau \right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

For any interval $I\subset \mathbb{R}$, we define $X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }\left(I\right)$ by the norm

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }\left(I\right)}={inf\{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\phi =\psi onI\}.\end{array}$$

Lemma 1

([9]). Let $b>{\displaystyle \frac{1}{2}}$, $s_1,s_2\in \mathbb{R}$, ${\delta }_1,{\delta }_2>0$, $\varrho \ge 1$ and $\beta >0$. Then, we have $X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }\hookrightarrow C({\mathbb{R}}_t,G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right))$, i.e., for all $\psi \in X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }$ and some constant ${\lambda }_0>0$, we have

$$\begin{array}{c}\hfill \underset{t\in [-T,T]}{sup}{\parallel \psi \left(t\right)\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)}\le {\lambda }_0{\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

(2)

3. Linear and Nonlinear Estimates

In [21], the author transforms the IVP (1) into

$$\left\{\begin{array}{c}{\partial }_t{\psi }^{*}+{\partial }_x^3{\psi }^{*}+{\gamma }_1^{*}{\partial }_x\left({\psi }^{*}{\phi }^{*}\right)+{\gamma }_2^{*}{\psi }^{*}{\partial }_x{\psi }^{*}+{\gamma }_3^{*}{\phi }^{*}{\partial }_x{\phi }^{*}+{\partial }_x^{-1}{\partial }_y^2{\psi }^{*}=0,\hfill \\ {\partial }_t{\phi }^{*}+{\partial }_x^3{\phi }^{*}+{\gamma }_4^{*}{\partial }_x\left({\psi }^{*}{\phi }^{*}\right)+{\gamma }_5^{*}{\psi }^{*}{\partial }_x{\psi }^{*}+{\gamma }_6^{*}{\phi }^{*}{\partial }_x{\phi }^{*}+{\partial }_x^{-1}{\partial }_y^2{\phi }^{*}=0,\hfill \\ {\psi }^{*}(x,y,0)={\psi }_0^{*}(x,y),\hfill \\ {\phi }^{*}(x,y,0)={\phi }_0^{*}(x,y),\hfill \end{array}\right.$$

(3)

where ${\gamma }_1^{*},{\gamma }_2^{*},{\gamma }_3^{*},{\gamma }_4^{*},{\gamma }_5^{*}$ and ${\gamma }_6^{*}$ are real values.

Note that (3) has a structure of two coupled KP II equations only in the nonlinear terms. Since the IVP (1) is equivalent to the IVP (3), it is easy to obtain results on the well-posdness of (1) from those for (3). For the sake of simplicity, from now onwards, we will drop $(\ast )$ and use the notation $\psi ,\phi ,{\psi }_0$ and ${\phi }_0$ in the system (3).

Next, we obtain the IVP as follows:

$$\left\{\begin{array}{c}{\partial }_t\psi +{\partial }_x^3\psi +{\gamma }_1{\partial }_x\left(\psi \phi \right)+{\gamma }_2\psi {\partial }_x\psi +{\gamma }_3\phi {\partial }_x\phi +{\partial }_x^{-1}{\partial }_y^2\psi =0,\hfill \\ {\partial }_t\phi +{\partial }_x^3\phi +{\gamma }_4{\partial }_x\left(\psi \phi \right)+{\gamma }_5\psi {\partial }_x\psi +{\gamma }_6\phi {\partial }_x\phi +{\partial }_x^{-1}{\partial }_y^2\phi =0,\hfill \\ \psi (x,y,0)={\psi }_0(x,y),\hfill \\ \phi (x,y,0)={\phi }_0(x,y).\hfill \end{array}\right.$$

(4)

Our solution concept comes from Duhamel’s formula for the IVP (4). As in [21], by applying the Fourier transform in x to (4), we obtain a differential equation and then solve it in t. Moreover, using the inverse Fourier transform, we reduce the IVP (4) to the integral system. Thus, we may write the solution as

$$\left\{\begin{array}{c}\hfill \psi \left(t\right)=S\left(t\right){\psi }_0-{\int }_0^{t}S(t-t^{\prime })F(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )d{t}^{\prime },\\ \hfill \phi \left(t\right)=S\left(t\right){\phi }_0-{\int }_0^{t}S(t-t^{\prime })G(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )d{t}^{\prime },\end{array}\right.$$

(5)

where $S\left(t\right)$ is the unitary group on $G_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }$ associated with the linear part of (4) given by

$$\begin{array}{c}\hfill \widehat{S\left(t\right){\psi }_0}(\xi ,\theta )=e^{it(i{\xi }^3-{\displaystyle \frac{{\theta }^2}{\xi }})}\widehat{{\psi }_0}(\xi ,\theta )\end{array}$$

and

$$\left\{\begin{array}{c}\hfill F(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )={\gamma }_1{\partial }_x\left(\psi \phi \right)+{\displaystyle \frac{{\gamma }_2}{2}}{\partial }_x{\psi }^2+{\displaystyle \frac{{\gamma }_3}{2}}{\partial }_x{\phi }^2,\\ \hfill G(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )={\gamma }_4{\partial }_x\left(\psi \phi \right)+{\displaystyle \frac{{\gamma }_5}{2}}{\partial }_x{\psi }^2+{\displaystyle \frac{{\gamma }_6}{2}}{\partial }_x{\phi }^2.\end{array}\right.$$

(6)

To prove the local well-posedness, let us consider the equivalent system of integral equations. Using a cut-off function satisfying $w\in C_0^{\infty }\left(\mathbb{R}\right)$, with $w=1$ in $[-1,1]$, $\mathrm{supp}w\subset [-2,2]$, define $w_{\delta }\left(t\right)=w\left({\displaystyle \frac{t}{\delta }}\right)$, for $0<\delta \le 1$.

Let us now define $({\Gamma }_1(\psi ,\phi ),{\Gamma }_2(\psi ,\phi ))$, where the maps ${\Gamma }_1$ and ${\Gamma }_2$ are given by

$$\left\{\begin{array}{c}\hfill {\Gamma }_1(\psi ,\phi )\left(t\right)=w_1\left(t\right)S\left(t\right){\psi }_0-w_1\left(t\right){\int }_0^{t}S(t-t^{\prime })F\left(t^{\prime }\right)d{t}^{\prime },\\ \hfill {\Gamma }_2(\psi ,\phi )\left(t\right)=w_1\left(t\right)S\left(t\right){\phi }_0-w_1\left(t\right){\int }_0^{t}S(t-t^{\prime })G\left(t^{\prime }\right)d{t}^{\prime }.\end{array}\right.$$

(7)

Next, we estimate the first part of $({\Gamma }_1(\psi ,\phi ),{\Gamma }_2(\psi ,\phi ))$ in the right-hand side of (7) and the integral part. Let ${\lambda }_1$ represent a general constant for the linear part, and let ${\lambda }_2$ denote a generate constant for the nonlinear part.

Lemma 2

(see [21]). Let $b>{\displaystyle \frac{1}{2}}$, $s_1,s_2\in \mathbb{R}$, and $\beta >0$. Thus, there exists a constant ${\lambda }_1>0$, such that

$$\begin{array}{c}\hfill \parallel w_1\left(t\right)S\left(t\right){\psi }_0{\parallel }_{X_{s_1,s_2,b,\beta }}\le {\lambda }_1{\parallel {\psi }_0\parallel }_{H_{s_1,s_2}},\end{array}$$

and

$$\begin{array}{c}\hfill \parallel w_1\left(t\right)S\left(t\right){\phi }_0{\parallel }_{X_{s_1,s_2,b,\beta }}\le {\lambda }_1{\parallel {\phi }_0\parallel }_{H_{s_1,s_2}},\end{array}$$

for ${\psi }_0,{\phi }_0\in H^{s_1,s_2}$.

Furthermore, for $F,G\in X_{s_1,s_2,b-1,\beta }$, there exists ${\lambda }_1>0$, such that

$$\begin{array}{c}\hfill \parallel w_1\left(t\right){\int }_0^{t}S(t-t^{\prime })F\left(t^{\prime }\right)d{t}^{\prime }{\parallel }_{X_{s_1,s_2,b,\beta }}\le {\lambda }_1{\parallel F\parallel }_{X_{s_1,s_2,b-1,\beta }},\end{array}$$

and

$$\begin{array}{c}\hfill \parallel w_1\left(t\right){\int }_0^{t}S(t-t^{\prime })G\left(t^{\prime }\right)d{t}^{\prime }{\parallel }_{X_{s_1,s_2,b,\beta }}\le {\lambda }_1{\parallel G\parallel }_{X_{s_1,s_2,b-1,\beta }}.\end{array}$$

Lemma 3.

Let $b>{\displaystyle \frac{1}{2}}$, $s_1,s_2\in \mathbb{R}$, ${\delta }_1,{\delta }_2>0$, $\varrho \ge 1$ and $\beta \ge 0$. Thus, there exists a constant ${\lambda }_1>0$, such that

$$\begin{array}{c}\hfill \parallel w_1\left(t\right)S\left(t\right){\psi }_0{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1{\parallel {\psi }_0\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(8)

and

$$\begin{array}{c}\hfill \parallel w_1\left(t\right)S\left(t\right){\phi }_0{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1{\parallel {\phi }_0\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(9)

for ${\psi }_0,{\phi }_0\in G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }$.

Moreover, for $F,G\in X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }$, there exists ${\lambda }_1>0$, such that

$$\begin{array}{c}\hfill \parallel w_1\left(t\right){\int }_0^{t}S(t-t^{\prime })F\left(t^{\prime }\right)d{t}^{\prime }{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1{\parallel F\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(10)

and

$$\begin{array}{c}\hfill \parallel w_1\left(t\right){\int }_0^{t}S(t-t^{\prime })G\left(t^{\prime }\right)d{t}^{\prime }{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1{\parallel G\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

(11)

Proof. 

As in [22,23], the desired result is achieved by using Lemma 2 and employing the operator $A^{{\delta }_1,{\delta }_2,\varrho }$, which is defined by

$$\begin{array}{c}\hfill \widehat{A^{{\delta }_1,{\delta }_2,\varrho }\psi }(\xi ,\theta )=e^{{\delta }_1{\left|\xi \right|}^{{\displaystyle \frac{1}{\varrho }}}}{e}^{{\delta }_2{\left|\theta \right|}^{{\displaystyle \frac{1}{\varrho }}}}\widehat{\psi }(\xi ,\theta )\end{array}$$

satisfies

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\sigma }}={\parallel A^{{\delta }_1,{\delta }_2,\varrho }\psi \parallel }_{X_{s_1,s_2,b,\beta }},\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}={\parallel A^{{\delta }_1,{\delta }_2,\varrho }\psi \parallel }_{H^{s_1,s_2}}.\end{array}$$

□

When studying the nonlinear aspects of the system, the bilinear form ${\partial }_x\left(\psi \phi \right)$ is critical for demonstrating the well-posedness of (4). More specifically, we obtain the following result:

Proposition 1

(see [21]). Let $s_1\in ]-{\displaystyle \frac{1}{3}},0[$, $s_2\ge 0$, $b>{\displaystyle \frac{1}{2}}$ and $\beta >{\displaystyle \frac{1}{6}}$, such that $b+\beta \le 1+s_1$, $b\le {\displaystyle \frac{1}{3}}(2+s_1)$ and $\beta \le {\displaystyle \frac{1}{3}}(1+s_1)$. Thus, the following bilinear estimate holds:

$$\begin{array}{c}\hfill \parallel {\partial }_x{\left(\psi \phi \right)\parallel }_{X_{s_1,s_2,b-1,\beta }}\le {\lambda }_2{\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }}{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }},\end{array}$$

for all $\psi ,\phi \in X_{s_1,s_2,b,\beta }$.

Proposition 2.

Let $s_1\in ]-{\displaystyle \frac{1}{3}},0[$, $s_2\ge 0$, $b>{\displaystyle \frac{1}{2}}$$,{\delta }_1,{\delta }_2>0$, $\varrho \ge 1$ and $\beta >{\displaystyle \frac{1}{6}}$, such that $b+\beta \le 1+s_1$, $b\le {\displaystyle \frac{1}{3}}(2+s_1)$ and $\beta \le {\displaystyle \frac{1}{3}}(1+s_1)$. Thus, the following bilinear estimate is established:

$$\begin{array}{c}\hfill \parallel {\partial }_x{\left(\psi \phi \right)\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_2{\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\sigma }}{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(12)

for all $\psi ,\phi \in X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }$.

Proof. 

In a similar manner to [22,23], we apply Proposition 1 and the operator $A^{{\delta }_1,{\delta }_2,\varrho }$ to demonstrate Proposition 2. □

4. Local Well-Posedness and Proof

To prove the main theorem regarding the well-posedness of System (4), we need to establish two important lemmas. To this end, let us define the following relevant spaces:

$$\begin{array}{c}\hfill M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }:=X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }\times X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho },\end{array}$$

and

$$\begin{array}{c}\hfill N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }:=G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\times G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho },\end{array}$$

with norm

$$\begin{array}{c}\hfill {\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}={\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel (\psi ,\phi )\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}={\parallel \psi \parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \phi \parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

The following lemmas will be used to prove Theorem 1. In the subsequent lemma, we will define the ball $X(0,R)$ and show that the map ${\Gamma }_1\times {\Gamma }_2$ is a contraction from $X(0,R)$ to $X(0,R)$.

Lemma 4.

Let $s_1\in ]-{\displaystyle \frac{1}{3}},0[$, $s_2\ge 0$, ${\delta }_1,{\delta }_2>0$, $\varrho \ge 1$, $b>{\displaystyle \frac{1}{2}}$ and $\beta >{\displaystyle \frac{1}{6}}$, such that $b+\beta \le 1+s_1$, $b\le {\displaystyle \frac{1}{3}}(2+s_1)$ and $\beta \le {\displaystyle \frac{1}{3}}(1+s_1)$. Then, for all $({\psi }_0,{\phi }_0)\in N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }$, the map ${\Gamma }_1\times {\Gamma }_2:X(0,R)\to X(0,R)$ is a contraction ($X(0,R)$), where

$$\begin{array}{c}\hfill X(0,R):=\{(\psi ,\phi )\in M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho };\parallel (\psi ,\phi ){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le R\},\end{array}$$

and

$$R:=2{\lambda }_1{\parallel ({\psi }_0,{\phi }_0)\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}.$$

Proof. 

In this proof, we define ${\Gamma }_i(\psi ,\phi )={\Gamma }_i$ and ${\Gamma }_i({\psi }^{*},{\phi }^{*})={\Gamma }_i^{*}$ for $i=\{1,2\}$. Using Lemma 3 and Proposition 2, we obtain

$$\begin{array}{cc}\hfill \parallel {\Gamma }_1{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}& \le \parallel w_1(t)S(t){\psi }_0{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel w_1(t){\int }_0^{t}S(t-t^{\prime })F(t^{\prime })d{t}^{\prime }\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & \le {\lambda }_1{\parallel {\psi }_0\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1\left(|{\gamma }_1|\parallel {\partial }_x{(\psi \phi )\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\displaystyle \frac{|{\gamma }_2|}{2}}\parallel {\partial }_x{\psi }^2{\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\displaystyle \frac{|{\gamma }_3|}{2}}{\parallel {\partial }_x{\phi }^2\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\right)\hfill \\ & \le {\lambda }_1{\parallel {\psi }_0\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1{\lambda }_2{\lambda }^{\prime }\left({2\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}^2+{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}^2\right)\hfill \\ & ={\lambda }_1{\parallel {\psi }_0\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1{\lambda }_2{\lambda }^{\prime }{\left({\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\right)}^2.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \parallel {\Gamma }_1{(\psi ,\phi )\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}& \le {\lambda }_1\parallel {\psi }_0{\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1{\lambda }_2{\lambda }^{\prime }{\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}^2,\hfill \end{array}$$

(13)

where

$${\lambda }^{\prime }=max\{{\displaystyle \frac{|{\gamma }_1|}{2}},{\displaystyle \frac{|{\gamma }_2|}{2}},{\displaystyle \frac{|{\gamma }_3|}{2}}\}.$$

Similarly, for $\parallel {\Gamma }_2{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}$, we obtain

$$\begin{array}{cc}\hfill \parallel {\Gamma }_2{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}& \le \parallel w_1(t)S(t){\phi }_0{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel w_1(t){\int }_0^{t}S(t-t^{\prime })G(t^{\prime })d{t}^{\prime }\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & \le {\lambda }_1\parallel {\phi }_0{\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1{\lambda }_2{\lambda }^{\prime \prime }{\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}^2,\hfill \end{array}$$

(14)

where

$${\lambda }^{″}:=max\{{\displaystyle \frac{|{\gamma }_4|}{2}},{\displaystyle \frac{|{\gamma }_5|}{2}},{\displaystyle \frac{|{\gamma }_6|}{2}}\}.$$

From (13) and (14), we obtain

$$\begin{array}{cc}\hfill \parallel ({\Gamma }_1,{\Gamma }_2){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}& \le {\lambda }_1\parallel ({\psi }_0,{\phi }_0){\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{\prime \prime }){\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}^2.\hfill \end{array}$$

If we choose $R>0$, such that

$$4{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{″})R<1,$$

(15)

we obtain

$$\begin{array}{cc}\hfill \parallel ({\Gamma }_1,{\Gamma }_2){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}& \le {\lambda }_1\parallel ({\psi }_0,{\phi }_0){\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{″}){\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}^2\hfill \\ & \le {\displaystyle \frac{R}{2}}+{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{″})R^2\hfill \\ \hfill & \le R.\hfill \end{array}$$

(16)

Assumption (15) gives

$$2{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{″})R<1.$$

Therefore,

$$\parallel ({\Gamma }_1,{\Gamma }_2){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\in X(0,R).$$

Now, we need to show that ${\Gamma }_1\times {\Gamma }_2$ is a contraction. To this end, we assume that $(\psi ,\phi ),({\psi }^{*},{\phi }^{*})\in X(0,R)$. Therefore,

$$\begin{array}{cc}\hfill \parallel {\Gamma }_1-{\Gamma }_1^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}& \le \parallel {\int }_0^{t}S(t-t^{\prime })F(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )d{t}^{\prime }\hfill \\ & -{\int }_0^{t}S(t-t^{\prime })F({\psi }^{*},{\phi }^{*},{\partial }_x{\psi }^{*},{\partial }_x{\phi }^{*})d{t}^{\prime }{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & \le \parallel {\int }_0^{t}S(t-t^{\prime })\left[F(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )-F({\psi }^{*},{\phi }^{*},{\partial }_x{\psi }^{*},{\partial }_x{\phi }^{*})\right]{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & \le {\lambda }_1{\parallel F(\psi ,\phi ,{\partial }_x\psi ,{\partial }_x\phi )-F({\psi }^{*},{\phi }^{*},{\partial }_x{\psi }^{*},{\partial }_x{\phi }^{*})\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & \le {\lambda }_1{\displaystyle (}|{\gamma }_1|\parallel {\partial }_x\left[\psi (\phi -{\phi }^{*})\right]{\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+|{\gamma }_1|\parallel {\partial }_x\left[{\phi }^{*}(\psi -{\psi }^{*})\right]{\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & +{\displaystyle \frac{|{\gamma }_2|}{2}}\parallel {\partial }_x\left[(\psi +{\psi }^{*})(\psi -{\psi }^{*})\right]{\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\displaystyle \frac{|{\gamma }_3|}{2}}{\parallel {\partial }_x\left[(\phi +{\phi }^{*})(\phi -{\phi }^{*})\right]\parallel }_{X_{s_1,s_2,b-1,\beta }^{{\delta }_1,{\delta }_2,\varrho }})\hfill \\ & \le {\lambda }_1{\lambda }_2{\displaystyle (}|{\gamma }_1{|\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\parallel \phi -{\phi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+|{\gamma }_1|\parallel {\phi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}{\parallel \psi -{\psi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & +{\displaystyle \frac{|{\gamma }_2|}{2}}\parallel \psi +{\psi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\parallel \psi -{\psi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\displaystyle \frac{|{\gamma }_3|}{2}}\parallel \phi +{\phi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}{\parallel \phi -{\phi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }})\hfill \\ & \le {\lambda }_1{\lambda }_2{\lambda }^{\prime }\left({2\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \phi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel {\phi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\right){\parallel \phi -{\phi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & +\left(2\parallel {\phi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \psi \parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel {\psi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\right){\parallel \psi -{\psi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & \le {\lambda }_1{\lambda }_2{\lambda }^{\prime }{\displaystyle (}{(2\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+2\parallel ({\psi }^{*},{\phi }^{*}){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }})\parallel \phi -{\phi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\hfill \\ & +{(2\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+2\parallel ({\psi }^{*},{\phi }^{*}){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }})\parallel \psi -{\psi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }})\hfill \\ & \le 2{\lambda }_1{\lambda }_2{\lambda }^{\prime }\left({\parallel (\psi ,\phi )\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel ({\psi }^{*},{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\right)\hfill \\ & \left(\parallel \psi -{\psi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}+{\parallel \phi -{\phi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel {\Gamma }_1-{\Gamma }_1^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le 4{\lambda }_1{\lambda }_2{\lambda }^{\prime }R{\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(17)

where ${\lambda }^{\prime }=max\{{\displaystyle \frac{|{\gamma }_1|}{2}},{\displaystyle \frac{|{\gamma }_2|}{2}},{\displaystyle \frac{|{\gamma }_3|}{2}}\}$.

In the same way, we obtain

$$\begin{array}{c}\hfill \parallel {\Gamma }_2-{\Gamma }_2^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le 4{\lambda }_1{\lambda }_2{\lambda }^{\prime \prime }R{\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(18)

where

$${\lambda }^{″}:=max\{{\displaystyle \frac{|{\gamma }_4|}{2}},{\displaystyle \frac{|{\gamma }_5|}{2}},{\displaystyle \frac{|{\gamma }_6|}{2}}\}.$$

From (17) and (18), we have

$$\begin{array}{c}\hfill \parallel ({\Gamma }_1-{\Gamma }_1^{*}),{\Gamma }_2-{\Gamma }_2^{*}{)\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le 4{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{\prime \prime })R{\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

(19)

Assumption (15) confirms that the map ${\Gamma }_1\times {\Gamma }_2$ is a contraction.

Finally, we conclude for $({\psi }_0,{\phi }_0)\in X(0,R)$, via (16), that ${\Gamma }_1\times {\Gamma }_2$ maps $X(0,R)$ into $X(0,R)$ and by (19) ${\Gamma }_1\times {\Gamma }_2$ is a contraction. Using the Banach fixed point theorem, there is exactly one fixed point because of ${\Gamma }_1\times {\Gamma }_2\in X(0,R)$. □

Now, we will demonstrate that the solution $(\psi ,\phi )$ depends continuously on the initial conditions $({\psi }_0,{\phi }_0)$.

Lemma 5.

Let $s_1\in ]-{\displaystyle \frac{1}{3}},0[$, $s_2\ge 0$, ${\delta }_1,{\delta }_2>0$, $\varrho \ge 1$, $b>{\displaystyle \frac{1}{2}}$ and $\beta >{\displaystyle \frac{1}{6}}$. Then, for all $({\psi }_0,{\phi }_0)$$,({\psi }_0^{*},{\phi }_0^{*})\in N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }$, we have

$$\begin{array}{c}\hfill \parallel (\psi -{\psi }^{*},\phi -{\phi }^{*}){\parallel }_{(C{([0,T],G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right))}^2}\le 4{\lambda }_0{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{″}){\parallel ({\psi }_0-{\psi }_0^{*},{\phi }_0-{\phi }_0^{*})\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

(20)

Proof. 

Using Lemma 1 in this proof, if $(\psi ,\phi )$ and $({\psi }^{*},{\phi }^{*})$ are two solutions to (4) corresponding to initial data $({\psi }_0,{\phi }_0)$ and $({\psi }_0^{*},{\phi }_0^{*})$, we have

$$\begin{array}{c}\hfill \parallel \psi -{\psi }^{*}{\parallel }_{C([0,T],G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)}\le {\lambda }_0{\parallel \psi -{\psi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\end{array}$$

(21)

and

$$\begin{array}{c}\hfill \parallel \phi -{\phi }^{*}{\parallel }_{C([0,T],G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)}\le {\lambda }_0{\parallel \phi -{\phi }^{*}\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

(22)

By taking $(\psi ,\phi ),({\psi }^{*},{\phi }^{*})\in X(0,R)$ and $4{\lambda }_1{\lambda }_2({\lambda }^{\prime }+{\lambda }^{″})R\le C<1$, we have

$$\begin{array}{c}\hfill \parallel \psi -{\psi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1\parallel {\psi }_0-{\psi }_0^{*}{\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+4{\lambda }_1{\lambda }_2{\lambda }^{\prime }R{\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }},\end{array}$$

(23)

and

$$\begin{array}{c}\hfill \parallel \phi -{\phi }^{*}{\parallel }_{X_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1\parallel {\phi }_0-{\phi }_0^{*}{\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+4{\lambda }_1{\lambda }_2{\lambda }^{\prime \prime }R{\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

(24)

Moreover, from (23) and (24), we have

$$\begin{array}{c}\hfill \parallel (\psi -{\psi }^{*},\phi -{\phi }^{*}){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\lambda }_1\parallel ({\psi }_0-{\psi }_0^{*},{\phi }_0-{\phi }_0^{*}{\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}+C{\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*})\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

Thus,

$$\begin{array}{c}\hfill \parallel (\psi -{\psi }^{*},\phi -{\phi }^{*}){\parallel }_{M_{s_1,s_2,b,\beta }^{{\delta }_1,{\delta }_2,\varrho }}\le {\displaystyle \frac{{\lambda }_1}{(1-C)}}\parallel ({\psi }_0-{\psi }_0^{*},{\phi }_0-{\phi }_0^{*}{\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

In the end, from Lemma (1), we have

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}\parallel (\psi -{\psi }^{*},\phi -{\phi }^{*}){\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}\le {\displaystyle \frac{{\lambda }_0{\lambda }_1}{(1-C)}}\parallel ({\psi }_0-{\psi }_0^{*},{\phi }_0-{\phi }_0^{*}{\parallel }_{N_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }}.\end{array}$$

□

We are now ready to present the theorem regarding the well-posedness of System (4).

Theorem 1.

Let $s_1\in ]-{\displaystyle \frac{1}{3}},0[$, $s_2\ge 0$, ${\delta }_1,{\delta }_2>0$, $\varrho \ge 1$ and $({\psi }_0,{\phi }_0)\in G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\times G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }$. If $b>{\displaystyle \frac{1}{2}}$ and $\beta >{\displaystyle \frac{1}{6}}$ satisfy the hypothesis $b+\beta \le 1+s_1$, $b\le {\displaystyle \frac{1}{3}}(2+s_1)$ and $\beta \le {\displaystyle \frac{1}{3}}(1+s_1)$, then there exists $T=T(\parallel {\psi }_0{\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }},\parallel {\phi }_0{\parallel }_{G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }})$ and a unique solution $(\psi ,\phi )$ of (4) on the time interval, as follows:

$$\begin{array}{c}\hfill (\psi ,\phi )\in C([0,T],G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right))\times C([0,T],G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)).\end{array}$$

Proof. 

Lemmas 4 and 5 provide the desired result. □

5. Gevrey’s Regularity

In this section, we will demonstrate in the following theorem that the solution to the coupled system of KP-II (4) equations exhibits Gevrey regularity with respect to the time variable t. Specifically, we aim to establish the estimates given in Equations (25) and (26), as outlined in Proposition 3. To prove these estimates, we will employ mathematical induction on l. Additionally, when estimating nonlinear terms, we will utilize Leibniz’s formula.

Theorem 2.

Let $-{\displaystyle \frac{1}{3}}<s_1<0$, $s_2\ge 0$, ${\gamma }_1,{\gamma }_2>0$, $\varrho \ge 1$ and $(\psi ,\phi )\in C([0,T],G_{s_1,s_2}^{{\gamma }_1,{\gamma }_2,\varrho }\left({\mathbb{R}}^2\right))\times C([0,T],G_{s_1,s_2}^{{\gamma }_1,{\gamma }_2,\varrho }\left({\mathbb{R}}^2\right))$ be the solution of (4). Then, $(\psi ,\phi )\in G^{3\varrho }\left([0,T]\right)\times G^{3\varrho }\left([0,T]\right)$ in the time variable t.

For the proof of the Theorem 2, it is enough to prove the following results:

Proposition 3.

Let $n,m,l\in \{0,1,2,\cdots \}$, for some $L>0$. Thus, we have

$$\begin{array}{c}\hfill |{\partial }_t^l{\partial }_x^m{\partial }_y^n\psi (x,t)|\le L^{l+m+n+1}{((m+n+3l)!)}^{\varrho }{M}^l,\end{array}$$

(25)

and

$$\begin{array}{c}\hfill |{\partial }_t^l{\partial }_x^m{\partial }_y^n\phi (x,t)|\le L^{l+m+n+1}{((m+n+3l)!)}^{\varrho }{M}^l,\end{array}$$

(26)

where

$$M:=L^2+max({\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}},{\displaystyle \frac{|{\gamma }_4|L}{2^{\varrho }}})+max({\displaystyle \frac{|{\gamma }_2|L}{2^{\varrho }}},{\displaystyle \frac{|{\gamma }_5|L}{2^{\varrho }}})+max({\displaystyle \frac{|{\gamma }_3|L}{2^{\varrho }}},{\displaystyle \frac{|{\gamma }_6|L}{2^{\varrho }}})+{\displaystyle \frac{1}{6^{\varrho }}}.$$

Proof. 

To prove Proposition 3, we will use induction on l. For the case when $l=0$, Inequalities (25) and (26) follow from the result below (see, [22,23]).

$$\begin{array}{c}\hfill |{\partial }_x^m{\partial }_y^n\psi (x,t)|\le L^{m+n+1}{(m!)}^{\varrho }{(n!)}^{\varrho }\le L^{m+n+1}{((m+n)!)}^{\varrho },\end{array}$$

(27)

and

$$\begin{array}{c}\hfill |{\partial }_x^m{\partial }_y^n\phi (x,t)|\le L^{m+n+1}{(m!)}^{\varrho }{(n!)}^{\varrho }\le L^{m+n+1}{((m+n)!)}^{\varrho },\end{array}$$

(28)

Thus, $(\psi ,\phi )$ is a $G^{\varrho }\times G^{\varrho })$ in $x,y$ for for all $s_1>-{\displaystyle \frac{1}{3}}$ and $s_2\ge 0$.

For $l=1$, we have

$$\begin{array}{cc}\hfill |{\partial }_t{\partial }_x^m{\partial }_y^n\psi (x,t)|& \le |{\partial }_x^{m+3}{\partial }_y^n\psi |+|{\gamma }_1\left|\right|{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|+|{\gamma }_2\left|\right|{\partial }_x^m{\partial }_y^n\left(\psi {\partial }_x\psi \right)|\hfill \\ \hfill & +|{\gamma }_3\left|\right|{\partial }_x^m{\partial }_y^n\left(\phi {\partial }_x\phi \right)|+|{\partial }_x^{m-1}{\partial }_y^{n+2}\psi |\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill |{\partial }_t{\partial }_x^m{\partial }_y^n\phi (x,t)|& \le |{\partial }_x^{m+3}{\partial }_y^n\phi |+|{\gamma }_4\left|\right|{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|+|{\gamma }_5\left|\right|{\partial }_x^m{\partial }_y^n\left(\psi {\partial }_x\psi \right)|\hfill \\ \hfill & +|{\gamma }_6\left|\right|{\partial }_x^m{\partial }_y^n\left(\phi {\partial }_x\phi \right)|+|{\partial }_x^{m-1}{\partial }_y^{n+2}\phi |\hfill \end{array}$$

(30)

The terms in (29) can be estimated as

$$\begin{array}{c}\hfill |{\partial }_x^{m+3}{\partial }_y^n\psi |\le L^{m+n+1+1}{((m+n+3·1)!)}^{\varrho }{L}^2.\end{array}$$

(31)

For the second term in (29), we have

$$\begin{array}{c}\hfill |{\gamma }_1\left|\right|{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|\le |{\gamma }_1|\left|\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_x^{p+1}{\partial }_y^q\phi \right)\right|.\end{array}$$

For $p\le m$ and $q\le n$, we have the following inequality:

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\le \left({\displaystyle \genfrac{}{}{0pt}{}{m+n}{p+q}}\right).\end{array}$$

(32)

By (32), we have

$$\begin{array}{cc}\hfill |{\gamma }_1||{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|& \le |{\gamma }_1|\left|\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m+n}{p+q}}\right)\left({\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_x^{p+1}{\partial }_y^q\phi \right)\right|\hfill \\ \hfill & \le |{\gamma }_1|\sum _{p=0}^m\sum _{q=0}^n{\displaystyle \frac{{((m+n)!)}^{\varrho }}{{((p+q)!)}^{\varrho }{((m+n-p-q)!)}^{\varrho }}}\hfill \\ \hfill & L^{m-p+n-q+1}{((m-p+n-q)!)}^{\varrho }{L}^{p+1+q+1}{((p+1+q)!)}^{\varrho }\hfill \\ \hfill & \le |{\gamma }_1|L^{m+n+3}{((m+n)!)}^{\varrho }\sum _{p=0}^m\sum _{q=0}^n{(p+1+q)}^{\varrho }.\hfill \end{array}$$

However,

$$\begin{array}{c}\hfill \sum _{p=0}^m\sum _{q=0}^n(p+1+q)={\displaystyle \frac{(m+1)(n+1)(m+n+2)}{2}}.\end{array}$$

Thus,

$$\begin{array}{cc}\hfill |{\gamma }_1\left|\right|{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|& \le |{\gamma }_1|L^{m+n+3}{((m+n)!)}^{\varrho }{\displaystyle \frac{{(m+1)}^{\varrho }{(n+1)}^{\varrho }{(m+n+2)}^{\varrho }}{2^{\varrho }}}\hfill \\ \hfill & \le L^{m+n+1+1}{((m+n)!)}^{\varrho }{(m+n+1)}^{\varrho }{(m+n+2)}^{\varrho }{(m+n+3)}^{\varrho }{\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}}\hfill \\ \hfill & \le L^{m+n+1+1}{((m+n+3.1)!)}^{\varrho }{\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}}.\hfill \end{array}$$

(33)

For the third and the fourth terms in (29), we have

$$\begin{array}{c}\hfill |{\gamma }_2\left|\right|{\partial }_x^m{\partial }_y^n\left(\psi {\partial }_x\psi \right)|\le L^{m+n+1+1}{((m+n+3.1)!)}^{\varrho }{\displaystyle \frac{|{\gamma }_2|L}{2^{\varrho }}},\end{array}$$

(34)

and

$$\begin{array}{c}\hfill |{\gamma }_3\left|\right|{\partial }_x^m{\partial }_y^n\left(\phi {\partial }_x\phi \right)|\le L^{m+n+1+1}{((m+n+3.1)!)}^{\varrho }{\displaystyle \frac{|{\gamma }_3|L}{2^{\varrho }}}.\end{array}$$

(35)

For the fifth term in (29), we have

$$\begin{array}{cc}\hfill |{\partial }_x^{m-1}{\partial }_y^{n+2}\psi |& \le L^{m-1+n+2+1}{((m-1+n+2)!)}^{\varrho }\hfill \\ \hfill & \le L^{m+n+1+1}{((m+n+3·1)!)}^{\varrho }{\displaystyle \frac{1}{{(m+n+2)}^{\varrho }{(m+n+3)}^{\varrho }}}\hfill \\ \hfill & \le L^{m+n+1+1}{((m+n+3·1)!)}^{\varrho }{\displaystyle \frac{1}{6^{\varrho }}},\hfill \end{array}$$

(36)

From (31) and (33)–(36), we obtain

$$\begin{array}{c}\hfill |{\partial }_t{\partial }_x^{m-1}{\partial }_y^{n+2}\psi |\le L^{m+n+1+1}{((m+n+3·1)!)}^{\varrho }{M}^1\mathrm{for} \mathrm{all}x,y\in \mathbb{R},t\in [0,T].\end{array}$$

The bounds for each term in (30) can be derived in a similar manner as those in (29). To obtain these bounds, simply replace ${\gamma }_1,{\gamma }_2$ and ${\gamma }_3$ with ${\gamma }_4,{\gamma }_5$, and ${\gamma }_6$, respectively, in the bounds found for Equation (30). Thus, we obtain

$$\begin{array}{c}\hfill |{\partial }_t{\partial }_x^m{\partial }_y^n\phi |\le L^{m+n+1+1}{((m+n+3·1)!)}^{\varrho }{M}^1\mathrm{for} \mathrm{all}x,y\in \mathbb{R},t\in [0,T].\end{array}$$

We assume that (25) and (26) hold for $l\ge 1$, where $m,n\in \{0,1,2,\cdots \}$. Then, we prove it for $l+1$ and $m,n\in \{0,1,2,\cdots \}$. Thus, we obtain

$$\begin{array}{cc}\hfill |{\partial }_t^{l+1}{\partial }_x^m{\partial }_y^n\psi (x,t)|& \le |{\partial }_t^l{\partial }_x^{m+3}{\partial }_y^n\psi |+|{\gamma }_1\left|\right|{\partial }_t^l{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|+|{\gamma }_2\left|\right|{\partial }_t^l{\partial }_x^m{\partial }_y^n\left(\psi {\partial }_x\psi \right)|\hfill \\ \hfill & +|{\gamma }_3\left|\right|{\partial }_t^l{\partial }_x^m{\partial }_y^n\left(\phi {\partial }_x\phi \right)|+|{\partial }_t^l{\partial }_x^{m-1}{\partial }_y^{n+2}\psi |\hfill \end{array}$$

(37)

and

$$\begin{array}{cc}\hfill |{\partial }_t^{l+1}{\partial }_x^m{\partial }_y^n\phi (x,t)|& \le |{\partial }_t^l{\partial }_x^{m+3}{\partial }_y^n\phi |+|{\gamma }_4\left|\right|{\partial }_t^l{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|+|{\gamma }_5\left|\right|{\partial }_t^l{\partial }_x^m{\partial }_y^n\left(\psi {\partial }_x\psi \right)|\hfill \\ \hfill & +|{\gamma }_6\left|\right|{\partial }_t^l{\partial }_x^m{\partial }_y^n\left(\phi {\partial }_x\phi \right)|+|{\partial }_t^l{\partial }_x^{m-1}{\partial }_y^{n+2}\phi |.\hfill \end{array}$$

(38)

The terms in (37) can be estimated as

$$\begin{array}{cc}\hfill |{\partial }_t^l{\partial }_x^{m+3}{\partial }_y^n\psi |& \le L^{(l+1)+m+n+1}{((m+n+3+3·l)!)}^{\varrho }{L}^2{M}^l\hfill \\ \hfill & =L^{(l+1)+m+n+1}{((m+n+3·(l+1))!)}^{\varrho }{L}^2{M}^l.\hfill \end{array}$$

(39)

For the second term in (37), we have

$$\begin{array}{cc}\hfill |{\gamma }_1||{\partial }_t^l{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|& \le |{\gamma }_1|\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_t^l{\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_x^{p+1}{\partial }_y^q\phi \right)\hfill \\ \hfill & +|{\gamma }_1|\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_t^l{\partial }_x^{p+1}{\partial }_y^q\phi \right)\hfill \\ \hfill & +|{\gamma }_1|\sum _{k=0}^{l-1}\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_t^{l-k}{\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_t^k{\partial }_x^{p+1}{\partial }_y^q\phi \right).\hfill \end{array}$$

(40)

For the first term in (40), we have

$$\begin{array}{cc}\hfill & |{\gamma }_1||\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_t^l{\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_x^{p+1}{\partial }_y^q\phi \right)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{3}}{L}^{(l+1)+m+n+1}{(m+n+3·(l+1))!)}^{\varrho }{\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}}{M}^l.\hfill \end{array}$$

(41)

For the second term in (40), we have

$$\begin{array}{cc}\hfill & |{\gamma }_1|\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_t^l{\partial }_x^{p+1}{\partial }_y^q\phi \right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{3}}{L}^{(l+1)+m+n+1}{(m+n+3·(l+1))!)}^{\varrho }{\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}}{M}^l.\hfill \end{array}$$

(42)

For the third term in (40), we have the following inequality:

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\le \left({\displaystyle \genfrac{}{}{0pt}{}{l+m+n}{k+p+q}}\right).\end{array}$$

(43)

Hence,

$$\begin{array}{cc}\hfill & |{\gamma }_1\left|\right|\sum _{k=0}^{l-1}\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{q}}\right)\left({\partial }_t^{l-k}{\partial }_x^{m-p}{\partial }_y^{n-q}\psi \right)\left({\partial }_t^k{\partial }_x^{p+1}{\partial }_y^q\phi \right)|\hfill \\ \hfill & \le |{\gamma }_1|\sum _{k=0}^{l-1}\sum _{p=0}^m\sum _{q=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{l+m+n}{k+p+q}}\right)L^{l-k+m-p+n-q+1}{((m-p+n-q+3·(l-k))!)}^{\varrho }{M}^{l-k}\hfill \\ \hfill & L^{k+p+1+q+1}{((p+1+q+3·k)!)}^{\varrho }{M}^k\hfill \\ \hfill & \le {\displaystyle \frac{1}{3}}{L}^{(l+1)+m+n+1}{((m+n+3·(l+1))!)}^{\varrho }{\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}}{M}^l.\hfill \end{array}$$

(44)

From (41), (42) and (44), we obtain

$$\begin{array}{c}\hfill |{\gamma }_1||{\partial }_t^l{\partial }_x^{m+1}{\partial }_y^n\left(\psi \phi \right)|\le L^{(l+1)+m+n+1}{((m+n+3·(l+1))!)}^{\varrho }{\displaystyle \frac{|{\gamma }_1|L}{2^{\varrho }}}{M}^l.\end{array}$$

(45)

For the third and the fourth terms in (37), we have

$$\begin{array}{c}\hfill |{\gamma }_2||{\partial }_t^l{\partial }_x^m{\partial }_y^n\left(\psi {\partial }_x\psi \right)|\le L^{(l+1)+m+n+1}{((m+n+3·(l+1))!)}^{\varrho }{\displaystyle \frac{|{\gamma }_2|L}{2^{\varrho }}}{M}^l,\end{array}$$

(46)

and

$$\begin{array}{c}\hfill |{\gamma }_3||{\partial }_t^l{\partial }_x^m{\partial }_y^n\left(\phi {\partial }_x\phi \right)|\le L^{(l+1)+m+n+1}{((m+n+3·(l+1))!)}^{\varrho }{\displaystyle \frac{|{\gamma }_3|L}{2^{\varrho }}}{M}^l.\end{array}$$

(47)

For the last term in (37), we have

$$\begin{array}{c}\hfill |{\partial }_t^l{\partial }_x^{m-1}{\partial }_y^{n+2}\left(\phi {\partial }_x\phi \right)|\le L^{(l+1)+m+n+1}{((m+n+3·(1+l))!)}^{\varrho }{\displaystyle \frac{1}{6^{\varrho }}}{M}^l.\end{array}$$

(48)

Finally, from (39) and (45)–(48), we obtain

$$\begin{array}{c}\hfill |{\partial }_t^{l+1}{\partial }_x^m{\partial }_y^n\psi |\le L^{(l+1)+m+n+1}{((m+n+3·(1+l))!)}^{\varrho }{M}^{l+1}\mathrm{for} \mathrm{all}x,y\in \mathbb{R},t\in [0,T].\end{array}$$

The same works for

$$\begin{array}{c}\hfill |{\partial }_t^{l+1}{\partial }_x^m{\partial }_y^n\phi |\le L^{(l+1)+m+n+1}{((m+n+3·(1+l))!)}^{\varrho }{M}^{l+1}\mathrm{for} \mathrm{all}x,y\in \mathbb{R},t\in [0,T].\end{array}$$

Consequently, $(\psi ,\phi )\in G^{3\varrho }\times G^{3\varrho }$ in the time variable t. □

6. Conclusions

We have demonstrated that the initial value problem for a system of two Kadomtsev–Petviashvili (KP II) equations, which are coupled through both dispersive and nonlinear terms, is locally well-posed in anisotropic Gevrey spaces $G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)\times G_{s_1,s_2}^{{\delta }_1,{\delta }_2,\varrho }\left({\mathbb{R}}^2\right)$. Here, the parameters satisfy $-{\displaystyle \frac{1}{3}}<s_1<0$ and $s_2\ge 0$. This advancement builds on recent findings regarding the well-posedness of this model within anisotropic Sobolev spaces $H^{s_1,s_2}\left({\mathbb{R}}^2\right)\times H^{s_1,s_2}\left({\mathbb{R}}^2\right)$. Our approach relies on both linear and nonlinear estimates. Furthermore, to better understand the system’s temporal behavior, we establish that the Gevrey regularity of order $3\rho$ is maintained.

In our upcoming work, we aim to examine the local well-posedness of the coupled system of periodic Kawahara equations with initial data in Gevrey spaces $G^{s,\delta ,\varrho }$ with $s>0$, $\delta >0$ and $\varrho \ge 1$ placed on the circle $\mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}$, as follows:

$$\left\{\begin{array}{c}{\partial }_t\psi +\alpha {\partial }_x^5\psi +\beta {\partial }_x^3\psi +\gamma {\partial }_x\psi +{\displaystyle \frac{\mu }{2}}\psi {\partial }_x\psi +{\displaystyle \frac{\mu }{2}}\phi {\partial }_x\phi =0,\hfill \\ {\partial }_t\phi +\alpha {\partial }_x^5\phi +\beta {\partial }_x^3\phi +\gamma {\partial }_x\phi +{\displaystyle \frac{\mu }{2}}\psi {\partial }_x\phi +{\displaystyle \frac{\mu }{2}}\phi {\partial }_x\psi =0,\hfill \\ \psi (x,0)={\psi }_0\left(x\right),\phi (x,0)={\phi }_0\left(x\right).\hfill \end{array}\right.$$

(49)

where $\psi =\psi (x,t)$ and $\phi =\phi (x,t)$ are unknown functions, $x\in \mathbb{T}$ and $t\in \mathbb{R}$, while ${\psi }_0$ and ${\phi }_0$ are given functions.

Here, the parameters $\alpha \ne 0$, $\beta$ and $\gamma$ are real numbers, and $\mu$ is a complex number. We will also concentrate on a new strategy for analyzing the Gevrey regularity of the solution over time t.