New Results on Gevrey Well Posedness for the Schrödinger–Korteweg–De Vries System

Abstract

In this work, we prove that the initial value problem for the Schrödinger–Korteweg–de Vries (SKdV) system is locally well posed in Gevrey spaces for $s>-\frac{3}{4}$ and $k\ge 0$. This advancement extends recent findings regarding the well posedness of this model within Sobolev spaces and investigates the regularity properties of its solutions.

1. Introduction

In physics, nonlinear partial differential equations (PDEs) play a crucial role in describing complex real-world phenomena that linear models cannot adequately capture. Unlike linear parametric differential equations, which assume superposition and predictable behavior, nonlinear PDEs account for interactions, feedback loops, and chaotic dynamics often observed in nature. These equations are vital in various fields, including general relativity (e.g., Einstein’s field equations), fluid dynamics (e.g., turbulence), and material science (e.g., phase transitions). The nonlinear characteristics of these equations are crucial for understanding and predicting the behavior of various physical systems. They can account for phenomena such as shock waves, solitons, and pattern formation. Recent studies have focused on specific nonlinear applications, including a nonlinear time-fractional Klein–Gordon equation and a Bresse system that incorporates two fractional damping terms. Additionally, research has explored a nonlinear Liénard’s equation and a fractional differential equation related to ray tracing through a crystalline lens. This also includes a time-fractional Euler–Bernoulli beam problem with a time delay, as referenced in [1].

The regularity of solutions to nonlinear partial differential equations (PDEs) is essential for understanding their behavior and characteristics. Regularity refers to the smoothness or continuity of solutions, which significantly influences both theoretical analysis and practical applications. For example, the issue of ill posedness in Besov spaces for a three-component Novikov system is examined in [2]. The local second-order Sobolev regularity in semi-simple Lie groups for the p-Laplacian equation is discussed in [3]. Additionally, global existence and blowup for a nonlocal pseudo-parabolic equation with conical singularity and a memory term are explored in [4]. Local $C^{0,1}$-regularity for the parabolic p-Laplacian equation on the group SU(3) is established in [5]. A study on global solutions for a complex system of quadratic heat equations with a generalized kernel is presented in [6]. Moreover, the finite-time blowup of solutions for a nonlinear system of fractional differential equations is introduced. Finally, Gevrey regularity for a coupled Kadomtsev–Petviashvili II system is investigated in [7].

The periodic Cauchy problem for the Korteweg–de Vries (KdV) equation with dispersion of order $p=2k+1$, where k is a positive integer (commonly referred to as KdVp), was investigated by the authors of [8]. They demonstrated that local well posedness in time is achievable when the initial data are taken from an analytic Gevrey space of order $\sigma$. This was performed using Bourgain–Gevrey-type analytic spaces and appropriate bilinear estimates. In [9], the author investigated a class of coupled periodic Korteweg–de Vries (KdV) systems of the Majda–Biello type. The author established conditions for the exponents and specified a relationship between them to demonstrate the existence of a unique solution within analytic Gevrey spaces. In [10], the authors studied the persistence of spatial analyticity for the solutions of this Cauchy problem, given that the initial data belong to a class of analytic functions.

In [11], the authors investigated an extended KdV equation. They constructed an approximate solution in the Sobolev space $H^s$ for $s>\frac{5}{2}$, using a modified energy method. Additionally, they established the uniqueness of the solution and its continuous dependence on the initial data by employing a technique inspired by Bona and Smith. The goal of [12] is to derive and analyze a Green–Naghdi model that includes both Coriolis effects and surface tension while accounting for a non-flat bottom geometry. The authors of [13] investigated the water wave problem over uneven bottoms in a highly nonlinear regime. They developed new asymptotic models that maintain the same level of accuracy as the standard equations. These simplified models were solved explicitly, and the results were validated numerically. The Gevrey regularity of the generalized Kadomtsev–Petviashvili I equation is presented in [14].

This work examines the well posedness and regularity of the initial value problem (IVP) for the Schrdinger–Korteweg–de Vries (NLS-KDV) system:

$$\left\{\begin{array}{c}i{\partial }_t\psi +{\partial }_x^2\psi =\alpha \psi \phi +\beta {|\psi |}^2\psi ,\hfill \\ {\partial }_t\phi +{\partial }_x^3\phi +\frac{1}{2}{\partial }_x({\phi }^2)=\gamma {\partial }_x{(|\psi |}^2),\hfill \\ \psi (x,0)={\psi }_0(x),\phi (x,0)={\phi }_0(x),\hfill \end{array}\right.$$

(1)

where $\psi =\psi (x,t)$ is a complex-valued function, $\phi =\phi (x,t)$ is a real-valued function $x,t\in \mathbb{R}$, and $\alpha ,\beta ,\gamma$ are real constants. This system governs the interactions between short-wave $\psi$ and long-wave $\phi$ and arises in fluid mechanics as well as plasma physics.

This paper is organized as follows: Section 2 introduces the mathematical notations and function spaces used throughout this paper. Section 3 provides preliminary estimates that support the main results. Section 4 presents the central argument for demonstrating the local well posedness of the problem. Section 5 explores the regularity properties of the solution, completing the overall analysis. Finally, we present our conclusions in Section 6.

2. Notations and Function Spaces

In this section, it is important to recall the definition of the necessary spaces, where the analytic Gevrey spaces are defined by

$$\begin{array}{c}\hfill G^{\sigma ,\delta ,s}(\mathbb{R}):=\{g\in L^2{(\mathbb{R});\parallel g\parallel }_{G^{\sigma ,\delta ,s}(\mathbb{R})}^2=\underset{\mathbb{R}}{\int }{e}^{{2\delta |\xi |}^{\frac{1}{\sigma }}}{\langle \xi \rangle }^{2s}{|\widehat{g}(\xi )|}^{2}d\xi <\infty \},\end{array}$$

for $s\in \mathbb{R}$, $\delta >0$, and $\sigma \ge 1$, where $\langle .\rangle =1+|.|$. By convention,

$$G^{\sigma ,\delta ,s}(\mathbb{R}):=G^{\sigma ,\delta ,s}.$$

Let $k,s\in \mathbb{R}$ and $b\in (0,1)$. As in [15], denote by $X^{k,b}$ and $y^{s,b}$ the completion of $\mathcal{S}({\mathbb{R}}^2)$ with respect to norms

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X^{k,b}}={\left(\underset{{\mathbb{R}}^2}{\int }{\langle \xi \rangle }^{2k}{\langle \tau +{\xi }^2\rangle }^{2b}{|\widehat{\psi }(\tau ,\xi )|}^{2}d\tau d\xi \right)}^{\frac{1}{2}},\end{array}$$

$$\begin{array}{c}\hfill {\parallel \phi \parallel }_{Y^{s,b}}={\left(\underset{{\mathbb{R}}^2}{\int }{\langle \xi \rangle }^{2s}{\langle \tau -{\xi }^3\rangle }^{2b}{|\widehat{\phi }(\tau ,\xi )|}^{2}d\tau d\xi \right)}^{\frac{1}{2}}.\end{array}$$

At times, the analytic Gevrey–Bourgain spaces $X^{\sigma ,\delta ,k,b}$ and $Y^{\sigma ,\delta ,s,b}$ are defined for $k,s\in \mathbb{R}$, $b\in (0,1)$, $\sigma \ge 1$, and $\delta >0$:

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}={\left(\underset{{\mathbb{R}}^2}{\int }{e}^{{2\delta |\xi |}^{\frac{1}{\sigma }}}{\langle \xi \rangle }^{2k}{\langle \tau +{\xi }^2\rangle }^{2b}{|\widehat{\psi }(\tau ,\xi )|}^{2}d\tau d\xi \right)}^{\frac{1}{2}},\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,b}}={\left(\underset{{\mathbb{R}}^2}{\int }{e}^{{2\delta |\xi |}^{\frac{1}{\sigma }}}{\langle \xi \rangle }^{2s}{\langle \tau -{\xi }^3\rangle }^{2b}{|\widehat{\phi }(\tau ,\xi )|}^{2}d\tau d\xi \right)}^{\frac{1}{2}},\end{array}$$

where $\widehat{\psi }$ and $\widehat{\phi }$ are the Fourier transforms of $\psi$ and $\phi$, respectively, in both x and t variables

$$\begin{array}{c}\hfill \widehat{\psi }(\tau ,\xi )={(2\pi )}^{-1}\underset{{\mathbb{R}}^2}{\int }{e}^{-it\tau -ix\xi }\psi (x,t)dtdx,\end{array}$$

and

$$\begin{array}{c}\hfill \widehat{\phi }(\tau ,\xi )={(2\pi )}^{-1}\underset{{\mathbb{R}}^2}{\int }{e}^{-it\tau -ix\xi }\phi (x,t)dtdx.\end{array}$$

In this paper, we use C to denote generic positive constants that may vary with each occurrence.

For a given time interval I, we define

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X_I^{\sigma ,\delta ,k,b}}:=\underset{{\psi }_{|I}^{\ast }=\psi }{inf}{\parallel {\psi }^{\ast }\parallel }_{X^{\sigma ,\delta ,k,b}},\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \phi \parallel }_{Y_I^{\sigma ,\delta ,s,b}}=\underset{{\phi }_{|I}^{\ast }=\phi }{inf}{\parallel {\phi }^{\ast }\parallel }_{Y^{\sigma ,\delta ,s,b}}.\end{array}$$

Following [15], we recall that if $b>\frac{1}{2}$, then the Sobolev Lemma implies

$$\begin{array}{c}\hfill X_I^{k,b}\hookrightarrow C(I,H^k(\mathbb{R}))\mathrm{and}{Y}_I^{s,b}\hookrightarrow C(I,H^s(\mathbb{R})).\end{array}$$

For $s,k\in \mathbb{R}$, $\sigma \ge 1$, $\delta >0$, and $b>\frac{1}{2}$, we have

$$\begin{array}{c}\hfill X_I^{\sigma ,\delta ,k,b}\hookrightarrow C(I,G^{\sigma ,\delta ,k}(\mathbb{R}))\mathrm{and}{Y}_I^{\sigma ,\delta ,s,b}\hookrightarrow C(I,G^{\sigma ,\delta ,s}(\mathbb{R})).\end{array}$$

(2)

To establish the existence of a positive time $T=T(\parallel {\psi }_0{\parallel }_{G^{\sigma ,\delta ,k}},\parallel {\phi }_0{\parallel }_{G^{\sigma ,\delta ,s}})>0$ and to demonstrate the uniqueness of a solution $(\psi (x,t),\phi (x,t))$ for the initial value problem (1), which satisfies certain conditions, we need to introduce and prove several critical lemmas in the following section.

3. Auxiliary Estimates

Let us consider the equation of the form

$$\begin{array}{c}\hfill i{\partial }_t\psi -\omega (-i{\partial }_x)\psi =F(\psi ),\end{array}$$

(3)

where $\omega$ is a measurable real-valued function and F some nonlinear function.

The Cauchy problem for (3) with initial data $\psi (0)={\psi }_0$ is rewritten as the integral equation

$$\begin{array}{c}\hfill \psi (t)=S_{\omega }(t){\psi }_0-i{\int }_0^t{S}_{\omega }(t-t^{\prime })F(\psi (t^{\prime }))d{t}^{\prime },\end{array}$$

where $S_{\omega }(t)=e^{-it\omega (-i{\partial }_x)}$ is the unitary group that solves the linear part of (3)

In this case, we will use the spaces $X^{\sigma ,\delta ,k,b}(\omega )$ and $Y^{\sigma ,\delta ,s,b}(\omega )$ for the phase functions ${\omega }_1(\xi )={\xi }^2$ and ${\omega }_2(\xi )=-{\xi }^3$. We can rewrite the system (1) in the following form:

$$\begin{array}{cc}\hfill & i{\partial }_t\psi -{\omega }_1(-i{\partial }_x)=F_1(\psi ,\phi ),\hfill \\ \hfill & i{\partial }_t\phi -{\omega }_2(-i{\partial }_x)=F_2(\psi ,\phi ),\hfill \end{array}$$

where $F_1(\psi ,\phi )=\alpha \psi \phi +\beta {|\psi |}^2\psi$ and $F_2(\psi ,\phi )=i\gamma {\partial }_x{(|\psi |}^2)-i\frac{1}{2}{\partial }_x({\phi }^2)$.

We define a cut-off function $\eta$ in $C_0^{\infty }(\mathbb{R})$ such that $\eta (t)=1$ if $|t|\le 1$ and $\eta (t)=0$ if $|t|\ge 2$, and let ${\eta }_T(t)=\eta (\frac{t}{T})$ for $0<T\le 1$.

We consider for two operators ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$ the following integral system, which is equivalent to (1):

$$\left\{\begin{array}{c}{\mathrm{\Gamma }}_1(\psi ,\phi )={\eta }_1(t)S_1(t){\psi }_0-i{\eta }_T(t){\int }_0^t{S}_1(t-t^{\prime })\{\alpha \psi \phi (t^{\prime })+{\beta |\psi |}^2\psi (t^{\prime })\}d{t}^{\prime },\hfill \\ {\mathrm{\Gamma }}_1(\psi ,\phi )={\eta }_1(t)S_2(t){\phi }_0+{\eta }_T(t){\int }_0^t{S}_2(t-t^{\prime })\{\gamma {\partial }_x{(|\psi |}^2)(t^{\prime })-\frac{1}{2}{\partial }_x({\phi }^2)(t^{\prime })\}d{t}^{\prime },\hfill \end{array}\right.$$

(4)

where

$$\begin{array}{cc}\hfill & S_1(t)=S_{{\omega }_1}(t)=e^{it{\partial }_x^2},\hfill \\ \hfill & S_2(t)=S_{{\omega }_2}(t)=e^{-t{\partial }_x^3}.\hfill \end{array}$$

For the linear estimates of (4), we have

Lemma 1 

(Lemma $2.1$ of [15]). Let $k,s\in \mathbb{R}$, $-\frac{1}{2}<b^{\prime }\le 0\le b<b^{\prime }+1$, and $T\in [0,1]$; then, for some constant $c>0$, we have

$$\begin{array}{cc}\hfill & \parallel {\eta }_1(t)S_1(t){\psi }_0{\parallel }_{X^{k,b}}\le C{\parallel {\psi }_0\parallel }_{H^k},\hfill \\ \hfill & \parallel {\eta }_1(t)S_2(t){\phi }_0{\parallel }_{Y^{s,b}}\le C{\parallel {\phi }_0\parallel }_{H^s}.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \parallel {\eta }_T(t){\int }_0^t{S}_1(t-t^{\prime })F_1(t^{\prime })d{t}^{\prime }{\parallel }_{X^{k,b}}\le C{T}^{1-b+b^{\prime }}{\parallel F_1\parallel }_{X^{k,b^{\prime }}},\hfill \\ \hfill & \parallel {\eta }_T(t){\int }_0^t{S}_1(t-t^{\prime })F_2(t^{\prime })d{t}^{\prime }{\parallel }_{Y^{s,b}}\le C{T}^{1-b+b^{\prime }}{\parallel F_2\parallel }_{Y^{s,b^{\prime }}},\hfill \end{array}$$

for $({\psi }_0,{\phi }_0)\in H^k(\mathbb{R})\times H^s(\mathbb{R})$ and $(F_1,F_2)\in X^{k,b^{\prime }}\times Y^{s,b^{\prime }}$.

Lemma 2. 

Let $k,s\in \mathbb{R}$, $-\frac{1}{2}<b^{\prime }\le 0\le b<b^{\prime }+1$, $T\in [0,1]$, $\delta >0$, and $\sigma \ge 1$; then, for some constant $c>0$, we have

$$\begin{array}{cc}\hfill & \parallel {\eta }_1(t)S_1(t){\psi }_0{\parallel }_{X^{\sigma ,\delta ,k,b}}\le C{\parallel {\psi }_0\parallel }_{G^{\sigma ,\delta ,k}},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \parallel {\eta }_1(t)S_2(t){\phi }_0{\parallel }_{Y^{\sigma ,\delta ,s,b}}\le C{\parallel {\phi }_0\parallel }_{G^{\sigma ,\delta ,s}},\hfill \end{array}$$

(6)

and

$$\begin{array}{cc}\hfill & \parallel {\eta }_T(t){\int }_0^t{S}_1(t-t^{\prime })F_1(t^{\prime })d{t}^{\prime }{\parallel }_{X^{\sigma ,\delta ,k,b}}\le c{T}^{1-b+b^{\prime }}{\parallel F_1\parallel }_{X^{\sigma ,\delta ,k,b^{\prime }}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \parallel {\eta }_T(t){\int }_0^t{S}_1(t-t^{\prime })F_2(t^{\prime })d{t}^{\prime }{\parallel }_{Y^{\sigma ,\delta ,s,b}}\le c{T}^{1-b+b^{\prime }}{\parallel F_2\parallel }_{Y^{\sigma ,\delta ,s,b^{\prime }}}.\hfill \end{array}$$

(8)

Proof. 

As in [8,9], the desired result is achieved by using Lemma (1) and employing the operator $A^{\delta ,\sigma }$, which is defined by

$$\begin{array}{c}\hfill \widehat{A^{\delta ,\sigma }\psi }(\xi ,\theta )=e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{\psi }(\xi )\end{array}$$

and satisfies

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}={\parallel A^{\delta ,\sigma }\psi \parallel }_{X^{k,b}},\\ \hfill {\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,b}}={\parallel A^{\delta ,\sigma }\phi \parallel }_{Y^{s,b}}\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{G^{\sigma ,\delta ,k}}={\parallel A^{\delta ,\sigma }\psi \parallel }_{H^k},\\ \hfill {\parallel \phi \parallel }_{G^{\sigma ,\delta ,s}}={\parallel A^{\delta ,\sigma }\phi \parallel }_{H^s}\end{array}$$

□

Now, we estimate the nonlinear terms needed in the proof of Theorem (1). We begin with the cubic nonlinear term.

Lemma 3. 

Let $\psi ,{\psi }^{\ast }\in X^{\sigma ,\delta ,k,b}$, with $\frac{1}{2}<b<1$, $\delta >0$, $\sigma \ge 1$, and $k\ge 0$; then, for $n\ge 0$ we have that

$$\begin{array}{cc}\hfill {\parallel |\psi |}^2{\psi \parallel }_{X^{\sigma ,\delta ,k,-n}}& \le {C\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}^3,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\parallel |\psi |}^2\psi -|{\psi }^{\ast }{|}^2{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,-n}}\le C(& {\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}^2+\parallel {\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}^2)\parallel \psi -{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}.\hfill \end{array}$$

(10)

Proof. 

In this proof, we use the estimates in Lemma $2.2$ [15] and the operator $A^{\sigma ,\delta }$. First, we note that the operator $A^{\sigma ,\delta }$ is defined by

$$\begin{array}{c}\hfill \widehat{A^{\sigma ,\delta }f}(\xi ,t)=e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{f}(\xi ,t)\mathrm{and}\widehat{A^{\sigma ,\delta }\overline{f}}(\xi ,t)=\widehat{\overline{A^{\sigma ,\delta }f}}(\xi ,t)\end{array}$$

(11)

and satisfies

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{X^{\sigma ,\delta ,k,b}({\mathbb{R}}^2)}={\parallel A^{\sigma ,\delta }f\parallel }_{X^{k,b}({\mathbb{R}}^2)}& {,\parallel f\parallel }_{Y^{\sigma ,\delta ,s,b}({\mathbb{R}}^2)}={\parallel A^{\sigma ,\delta }f\parallel }_{Y^{s,b}({\mathbb{R}}^2)},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{G^{\sigma ,\delta ,k}(\mathbb{R})}={\parallel A^{\sigma ,\delta }f\parallel }_{H^k(\mathbb{R})}& {,\parallel f\parallel }_{G^{\sigma ,\delta ,s}(\mathbb{R})}={\parallel A^{\sigma ,\delta }f\parallel }_{H^s(\mathbb{R})},\hfill \end{array}$$

(13)

where $X^{k,b}({\mathbb{R}}^2)$ and $Y^{s,b}({\mathbb{R}}^2)$ are introduced in [15].

We now examine the operator $A^{\sigma ,\delta }$ and observe that

$$\begin{array}{cc}\hfill & {\parallel |\psi |}^2{\psi \parallel }_{X^{\sigma ,\delta ,k,-n}}\hfill \\ \hfill & =\parallel A^{\sigma ,\delta }{(|\psi |}^2\psi {)\parallel }_{X^{k,-n}}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}\widehat{A^{\sigma ,\delta }{(|\psi |}^2\psi )}(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{(|\psi {|}^2\psi )}(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}{(\sqrt{2\pi })}^{-2}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{\psi }\ast \widehat{\overline{\psi }}\ast \widehat{\psi }{\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {\le \parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}{(\sqrt{2\pi })}^{-2}{\int }_{{\mathbb{R}}^4}{e}^{\delta |\xi -{\xi }_1{|}^{\frac{1}{\sigma }}}\widehat{\psi }(\xi -{\xi }_1,\tau -{\tau }_1)e^{\delta |{\xi }_1-{\xi }_2{|}^{\frac{1}{\sigma }}}\widehat{\overline{\psi }}({\xi }_1-{\xi }_2,{\tau }_1-{\tau }_2)\hfill \\ \hfill & e^{\delta |{\xi }_2{|}^{\frac{1}{\sigma }}}\widehat{\psi }({\xi }_2,{\tau }_2){{\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}d{\xi }_{1}d{\xi }_{2}d{\tau }_{1}d{\tau }_2\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}(\widehat{A^{\sigma ,\delta }\psi A^{\sigma ,\delta }\overline{\psi }{A}^{\sigma ,\delta }\psi })(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}(\widehat{|A^{\sigma ,\delta }{\psi |}^2{A}^{\sigma ,\delta }\psi })(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & =\parallel |A^{\sigma ,\delta }{\psi |}^2{A}^{\sigma ,\delta }\psi {\parallel }_{X^{k,-n}}.\hfill \end{array}$$

Note that ${\delta |\xi |}^{\frac{1}{\sigma }}\le \delta |\xi -{\xi }_1{|}^{\frac{1}{\sigma }}+\delta |{\xi }_1-{\xi }_2{|}^{\frac{1}{\sigma }}+\delta {|{\xi }_2|}^{\frac{1}{\sigma }}$.

Lemma $(2.2)$ of [15] gives

$$\begin{array}{cc}\hfill \parallel |A^{\sigma ,\delta }{\psi |}^2{A}^{\sigma ,\delta }\psi {\parallel }_{X^{k,-n}}& \le C\parallel A^{\sigma ,\delta }{\psi \parallel }_{X^{k,b}}^3\hfill \\ \hfill & ={C\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}^3.\hfill \end{array}$$

To estimate (10), we have

$$\begin{array}{cc}\hfill \widehat{A^{\sigma ,\delta }{(|\psi |}^2\psi -|{\psi }^{\ast }{|}^2{\psi }^{\ast })}& =e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{(|\psi {|}^2\psi -|{\psi }^{\ast }{|}^2{\psi }^{\ast })}\hfill \\ \hfill & =e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{{|\psi |}^2\psi }-e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{|{\psi }^{\ast }{|}^2{\psi }^{\ast }}\hfill \\ \hfill & \le \widehat{|A^{\sigma ,\delta }{\psi |}^2{A}^{\sigma ,\delta }\psi -{|A^{\sigma ,\delta }{\psi }^{\ast }|}^2{A}^{\sigma ,\delta }{\psi }^{\ast }}.\hfill \end{array}$$

For the second estimate, we have

$$\begin{array}{cc}\hfill {\parallel |\psi |}^2\psi -{|{\psi }^{\ast }|}^2{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,-n}}& =\parallel A^{\sigma ,\delta }{(|\psi |}^2\psi -|{\psi }^{\ast }{|}^2{\psi }^{\ast }{)\parallel }_{X^{k,-n}}\hfill \\ \hfill & \le \parallel |A^{\sigma ,\delta }{\psi |}^2{A}^{\sigma ,\delta }\psi -{|A^{\sigma ,\delta }{\psi }^{\ast }|}^2{A}^{\sigma ,\delta }{\psi }^{\ast }{\parallel }_{X^{k,-n}}\hfill \\ \hfill & \le C\left(\parallel A^{\sigma ,\delta }{\psi \parallel }_{X^{k,b}}^2+{\parallel A^{\sigma ,\delta }{\psi }^{\ast }\parallel }_{X^{k,b}}^2\right){\parallel A^{\sigma ,\delta }\psi -A^{\sigma ,\delta }{\psi }^{\ast }\parallel }_{X^{k,b}}\hfill \\ \hfill & =C\left({\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}^2+{\parallel {\psi }^{\ast }\parallel }_{X^{\sigma ,\delta ,k,b}}^2\right){\parallel \psi -{\psi }^{\ast }\parallel }_{X^{\sigma ,\delta ,k,b}}\hfill \end{array}$$

□

Lemma 4. 

Let $\phi ,{\phi }^{\ast }\in Y^{\sigma ,\delta ,s,b}$. Then, there exists $C>0$ such that

$$\begin{array}{cc}\hfill \parallel {\partial }_x({\phi }^2){\parallel }_{Y^{\sigma ,\delta ,s,-n}}& \le {C\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,b}}^2,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \parallel {\partial }_x({\phi }^2)-{\partial }_x({({\phi }^{\ast })}^2){\parallel }_{Y^{\sigma ,\delta ,s,-n}}\le C(& {\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,b}}+\parallel {\phi }^{\ast }{\parallel }_{Y^{\sigma ,\delta ,s,b}})\parallel \phi -{\phi }^{\ast }{\parallel }_{Y^{\sigma ,\delta ,s,b}},\hfill \end{array}$$

(15)

hold in the following:

(1) 

$1-b-min\{-\frac{2s+1}{2},\frac{4s+3}{12}\le n<\frac{1}{2}<b\le 1-n,fors\in (-\frac{3}{4},-\frac{1}{2})$;

(2) 

$\frac{5}{12}<n<\frac{1}{2}<b<\frac{7}{12},fors\in [-\frac{1}{2},0]$;

(3) 

$\frac{1}{4}\le n<\frac{1}{2}<b\le 1-n,fors\in [0,\infty ].$

Proof. 

Following [9], we obtain (14), and for (15) we use the operator $A^{\sigma ,\delta }$ and Lemma $2.3$ in [15] to obtain

$$\begin{array}{cc}\hfill \widehat{A^{\sigma ,\delta }\left({\partial }_x({\psi }^2)-{\partial }_x({({\psi }^{\ast })}^2)\right)}& =e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{\left({\partial }_x({\psi }^2)-{\partial }_x({({\psi }^{\ast })}^2)\right)}\hfill \\ \hfill & =e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{{\partial }_x({\psi }^2)}-e^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{{\partial }_x({({\psi }^{\ast })}^2)}\hfill \\ \hfill & =i\xi {(2\pi )}^{-2}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}\hat{\psi }\ast \hat{\psi }-i\xi {(2\pi )}^{-2}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{{\psi }^{\ast }}\ast \widehat{{\psi }^{\ast }}\hfill \\ \hfill & \le i\xi \widehat{{(A^{\sigma ,\delta }\psi )}^2}-i\xi \widehat{{(A^{\sigma ,\delta }{\psi }^{\ast })}^2}\hfill \\ \hfill & =\widehat{{\partial }_x{(A^{\sigma ,\delta }\psi )}^2}-\widehat{{({\partial }_x{A}^{\sigma ,\delta }{\psi }^{\ast })}^2}\hfill \\ \hfill & =\widehat{{\partial }_x{(A^{\sigma ,\delta }\psi )}^2-{({\partial }_x{A}^{\sigma ,\delta }{\psi }^{\ast })}^2}.\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill \parallel {\partial }_x{(\psi )}^2-{\partial }_x({({\psi }^{\ast })}^2){\parallel }_{Y^{\sigma ,\delta ,s,-n}}& =\parallel A^{\sigma ,\delta }{({\partial }_x{(\psi )}^2-{\partial }_x({({\psi }^{\ast })}^2)\parallel }_{Y^{s,-n}}\hfill \\ \hfill & \le \parallel {\partial }_x{(A^{\sigma ,\delta }\psi )}^2-{\partial }_x{(A^{\sigma ,\delta }{\psi }^{\ast })}^2{\parallel }_{Y^{s,-n}}\hfill \\ \hfill & \le C\left(\parallel A^{\sigma ,\delta }{\psi \parallel }_{Y^{s,b}}+{\parallel A^{\sigma ,\delta }{\psi }^{\ast }\parallel }_{Y^{s,b}}\right){\parallel A^{\sigma ,\delta }\psi -A^{\sigma ,\delta }{\psi }^{\ast }\parallel }_{Y^{s,b}}\hfill \\ \hfill & =C\left({\parallel \psi \parallel }_{Y^{\sigma ,\delta ,s,b}}+{\parallel {\psi }^{\ast }\parallel }_{Y^{\sigma ,\delta ,s,b}}\right){\parallel \psi -{\psi }^{\ast }\parallel }_{Y^{\sigma ,\delta ,s,b}}.\hfill \end{array}$$

□

Next, we prove new bilinear estimates for the interaction terms.

Lemma 5. 

For $\sigma \ge 1$ and $\delta >0$, given $k\ge 0$, $\frac{1}{6}\le n<\frac{1}{2}<b$, and $k-s\le min\{1,3n\}$, there exists a positive constant c depending only on the parameters $n,b,k$, and s such that for all $\psi ,{\psi }^{\ast }\in X^{\sigma ,\delta ,k,b}$ and $\phi ,{\phi }^{\ast }\in Y^{\sigma ,\delta ,s,b}$

$$\begin{array}{cc}\hfill {\parallel \psi \phi \parallel }_{X^{\sigma ,\delta ,k,-n}}& \le {C\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,b}}{\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,b}},\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill \parallel \psi \phi -{\psi }^{\ast }{\phi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,-n}}\le C(\parallel \psi -{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}{\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,b}}+\parallel {\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}\parallel \phi -{\phi }^{\ast }{\parallel }_{Y^{\sigma ,\delta ,s,b}}).\end{array}$$

(17)

Proof. 

By using the operator $A^{\sigma ,\delta }$ and Lemma $(3.1)$ in [15], we obtain

$$\begin{array}{cc}\hfill {\parallel \psi \phi \parallel }_{X^{\sigma ,\delta ,k,-n}({\mathbb{R}}^2)}& =\parallel A^{\sigma ,\delta }{(\psi \phi )\parallel }_{X^{k,-n}({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}\widehat{A^{\sigma ,\delta }(\psi \phi )}(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}\widehat{(\psi \phi )}(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}{(2\pi )}^{-2}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}\hat{\psi }\ast \hat{\phi }(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {\le \parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}{(2\pi )}^{-2}{e}^{{\delta |\xi |}^{\frac{1}{\sigma }}}{\int }_{{\mathbb{R}}^2}{e}^{\delta |\xi -{\xi }_1{|}^{\frac{1}{\sigma }}}\hat{\psi }(\xi -{\xi }_1,\tau -{\tau }_1)e^{\delta |{\xi }_1{|}^{\frac{1}{\sigma }}}\hat{\phi }({\xi }_1,{\tau }_1)d{\xi }_{1}d{\tau }_1{\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & {=\parallel \langle \xi \rangle }^k{\langle \tau +{\xi }^2\rangle }^{-n}\widehat{A^{\sigma ,\delta }\psi A^{\sigma ,\delta }\phi }(\xi ,\tau ){\parallel }_{L_{\xi ,\tau }^2({\mathbb{R}}^2)}\hfill \\ \hfill & =\parallel A^{\sigma ,\delta }\psi A^{\sigma ,\delta }{\phi \parallel }_{X^{k,-n}({\mathbb{R}}^2)}\hfill \\ \hfill & \le C\parallel A^{\sigma ,\delta }{\psi \parallel }_{X^{k,-n}({\mathbb{R}}^2)}{\parallel A^{\sigma ,\delta }\phi \parallel }_{Y^{s,-n}({\mathbb{R}}^{\mathbb{2}})}\hfill \\ \hfill & ={C\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,-n}({\mathbb{R}}^2)}{\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,-n}({\mathbb{R}}^{\mathbb{2}})}.\hfill \end{array}$$

For (17) (see reference [15]), we have

$$\begin{array}{cc}\hfill \parallel \psi \phi -{\psi }^{\ast }{\phi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,-n}({\mathbb{R}}^2)}& =\parallel A^{\sigma ,\delta }(\psi \phi -{\psi }^{\ast }{\phi }^{\ast }){\parallel }_{X^{k,-n}({\mathbb{R}}^2)}\hfill \\ \hfill & =\parallel A^{\sigma ,\delta }(\psi \phi )-A^{\sigma ,\delta }({\psi }^{\ast }{\phi }^{\ast }){\parallel }_{X^{k,-n}({\mathbb{R}}^2)}\hfill \\ \hfill & =\parallel A^{\sigma ,\delta }\psi A^{\sigma ,\delta }\phi )-A^{\sigma ,\delta }{\psi }^{\ast }{A}^{\sigma ,\delta }{\phi }^{\ast }{)\parallel }_{X^{k,-n}({\mathbb{R}}^2)}\hfill \\ \hfill & \le C(\parallel A^{\sigma ,\delta }\psi -A^{\sigma ,\delta }{\psi }^{\ast }{\parallel }_{X^{k,b}}\parallel A^{\sigma ,\delta }\phi {\parallel }_{Y^{s,b}}+\parallel A^{\sigma ,\delta }{\psi }^{\ast }{\parallel }_{X^{k,b}}\parallel A^{\sigma ,\delta }\phi -A^{\sigma ,\delta }{\phi }^{\ast }{\parallel }_{Y^{s,b}})\hfill \\ \hfill & =C(\parallel \psi -{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}\parallel \phi {\parallel }_{Y^{\sigma ,\delta ,s,b}}+\parallel {\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}\parallel \phi -{\phi }^{\ast }{\parallel }_{Y^{\sigma ,\delta ,s,b}}).\hfill \end{array}$$

□

Lemma 6. 

Let $\sigma \ge 1$, $\delta >0$, $k\ge 0$, $b>\frac{1}{2}$, and $0\le n\le b$; there exists a positive constant C depending only on $n,b,k$, and s such that for ${\psi }_i,{\psi }_i^{\ast }\in X^{\sigma ,\delta ,k,b}$, $i=1,2$,

$$\begin{array}{c}\hfill \parallel {\partial }_x({\psi }_1\overline{{\psi }_2}){\parallel }_{Y^{\sigma ,\delta ,s,-n}}\le C\parallel {\psi }_1{\parallel }_{X^{\sigma ,\delta ,k,b}}{\parallel {\psi }_2\parallel }_{X^{\sigma ,\delta ,k,b}},\end{array}$$

(18)

$$\begin{array}{c}\hfill \parallel {\partial }_x(|{\psi }_1{|}^2)-{\partial }_x(|{\psi }_1^{\ast }{|}^2{)\parallel }_{Y^{\sigma ,\delta ,s,-n}}\le C((\parallel {\psi }_1{\parallel }_{X^{\sigma ,\delta ,k,b}}+\parallel {\psi }_1^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}})\parallel {\psi }_1-{\psi }_1^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,b}}),\end{array}$$

(19)

hold in the following cases:

1/ 

$s-2k\le min\{3n-2b-\frac{1}{2},-\frac{1}{2}\}$ for $k\in [0,\frac{1}{2}]$;

2/ 

$s-k\le 3n-b-\frac{1}{2}$ for $k\in (\frac{1}{2},\infty )$ and $b\in (\frac{1}{2},k]$.

Proof. 

Using a similar approach to the previous proof, we apply Lemma $3.2$ in [15] and the operator $A^{\delta ,\sigma }$ to establish Lemma 6.  □

We will now present the theorem establishing the solution’s uniqueness. The proof is derived using the lemmas presented above in this section.

Theorem 1. 

Let $k\ge 0$, $s>-\frac{3}{4}$, $\delta >0$, and $\sigma \ge 1$. Then, for any $({\psi }_0,{\phi }_0)\in G^{\sigma ,\delta ,k}(\mathbb{R})\times G^{\sigma ,\delta ,s}(\mathbb{R})$, provided the following:

(i) 

$k-1⩽s⩽2k-\frac{1}{2}fork\in [0,\frac{1}{2}]$;

(ii) 

$k-1⩽s<k+\frac{1}{2}fork\in (\frac{1}{2},\infty )$.

there exist a positive time $T=T(\parallel {\psi }_0{\parallel }_{G^{\sigma ,\delta ,k}},\parallel {\phi }_0{\parallel }_{G^{\sigma ,\delta ,s}})>0$ and a unique solution $(\psi (x,t),\phi (x,t))$ of the initial value problem (1), satisfying

$$\begin{array}{c}\hfill \psi \in C([0,T],G^{\sigma ,\delta ,k}(\mathbb{R}))and\phi \in C([0,T],G^{\sigma ,\delta ,s}(\mathbb{R})).\end{array}$$

(20)

4. Local Well Posedness and Proof

We are now ready to estimate all terms in (4) by using the estimates in the above lemmas. We consider the following function space where we seek our solution:

$$\begin{array}{c}\hfill {\omega }_{\theta }=\{(\psi ,\phi )\in X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }\times Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }{;\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}\le r_1{and\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}\le r_2\},\end{array}$$

where $0<\theta \ll$ 1 and $r_i$, $i=1,2$, will be chosen below.

${\omega }_{\theta }$ is a complete metric space with the norm

$$\begin{array}{c}\hfill {\parallel (\psi ,\phi )\parallel }_{{\omega }_{\theta }}={\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}+{\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}.\end{array}$$

Let $n=\frac{1}{2}-2\theta$, $b^{\prime }=-n$, and $b=\frac{1}{2}+\theta$, with $0<\theta \le {\theta }^{\prime }(k,s)$ and ${\theta }^{\prime }(k,s)$, satisfy the following conditions (see [15]):

$${\theta }^{\prime }(k,s)=\left\{\begin{array}{cc}min\{\frac{1}{12},-\frac{(2s+1)}{2},\frac{(4s+3)}{12}\},\hfill & \mathrm{if}(k,s)\in [0,\frac{1}{2}]\times (-\frac{3}{4},-\frac{1}{2}),\hfill \\ \frac{1}{24}-,\hfill & \mathrm{if}(k,s)\in [0,\frac{1}{2}]\times [-\frac{1}{2},\infty ),\hfill \\ min\{\frac{1}{24},\frac{2k-1}{2},\frac{2k-2s+1}{14}\},\hfill & \mathrm{if}(k,s)\in (\frac{1}{2},\infty )\times (-\frac{1}{2},\frac{2k+1}{2}).\hfill \end{array}\right.$$

We have

$$\begin{array}{cc}\hfill \parallel {\mathrm{\Gamma }}_1{(\psi ,\phi )\parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}& \le C_0{\parallel {\psi }_0\parallel }_{G^{\sigma ,\delta ,k}}+C_1{T}^{\theta }\left[{\parallel \psi \phi \parallel }_{X^{\sigma ,\delta ,k,-\frac{1}{2}+2\theta }}+{\parallel |\psi |}^2{\psi \parallel }_{X^{\sigma ,\delta ,k,-\frac{1}{2}+2\theta }}\right],\hfill \\ \hfill & \le C_0{\parallel {\psi }_0\parallel }_{G^{\sigma ,\delta ,k}}+C_1{T}^{\theta }\left[{\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}{\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}+{\parallel |\psi \parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}^3\right],\hfill \\ \hfill & \le C_0{\parallel {\psi }_0\parallel }_{G^{\sigma ,\delta ,k}}+C_1{T}^{\theta }\left[r_1{r}_2+r_1^3\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel {\mathrm{\Gamma }}_2{(\psi ,\phi )\parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}& \le C_0{\parallel {\phi }_0\parallel }_{G^{\sigma ,\delta ,s}}+C_2{T}^{\theta }\left[\parallel {\partial }_x({\phi }^2){\parallel }_{Y^{\sigma ,\delta ,s,-\frac{1}{2}+2\theta }}+\parallel {\partial }_x{(|\psi |}^2{)\parallel }_{Y^{\sigma ,\delta ,s,-\frac{1}{2}+2\theta }}\right]\hfill \\ \hfill & \le C_0{\parallel {\phi }_0\parallel }_{G^{\sigma ,\delta ,s}}+C_2{T}^{\theta }\left[{\parallel \phi \parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}^2+{\parallel \psi \parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}^2\right]\hfill \\ \hfill & \le C_0{\parallel {\phi }_0\parallel }_{G^{\sigma ,\delta ,s}}+C_2{T}^{\theta }\left[r_1^2+r_2^2\right].\hfill \end{array}$$

Now, taking $r_1=2C_0{\parallel \psi \parallel }_{G^{\sigma ,\delta ,k}}$ and $r_2=2C_0{\parallel \phi \parallel }_{G^{\sigma ,\delta ,s}}$, we have that

$$\begin{array}{c}\hfill \parallel {\mathrm{\Gamma }}_1{(\psi ,\phi )\parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}\le \frac{r_1}{2}+C_1{T}^{\theta }[r_1^3+r_1{r}_2],\end{array}$$

(21)

so

$$T^{\theta }\le \frac{1}{2C_1[r_1^2+r_2]}.$$

and

$$\begin{array}{c}\hfill \parallel {\mathrm{\Gamma }}_2{(\psi ,\phi )\parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}\le \frac{r_2}{2}+C_2{T}^{\theta }[r_1^2+r_2^2].\end{array}$$

(22)

Also,

$$T^{\theta }\le \frac{r_2}{C_2[r_1^2+r_2^2]}.$$

Hence, $({\mathrm{\Gamma }}_1(\psi ,\phi ),{\mathrm{\Gamma }}_2(\psi ,\phi ))\in {\omega }_{\theta }$ for

$$\begin{array}{c}\hfill T^{\theta }\le \frac{1}{2}min\left\{\frac{1}{C_1[r_1^2+r_2]},\frac{r_2}{C_2[r_1^2+r_2^2]}\right\}\end{array}$$

(23)

Similarly, we have

$$\begin{array}{c}\hfill \parallel {\mathrm{\Gamma }}_1(\psi ,\phi )-{\mathrm{\Gamma }}_1({\psi }^{\ast },{\phi }^{\ast }){\parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}\le c_3{T}^{\theta }[r_1^2+r_1+r_2][\parallel \psi -{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}+\parallel \phi -{\phi }^{\ast }{\parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}]\end{array}$$

$$\begin{array}{c}\hfill \parallel {\mathrm{\Gamma }}_2(\psi ,\phi )-{\mathrm{\Gamma }}_2({\psi }^{\ast },{\phi }^{\ast }){\parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}\le c_4{T}^{\theta }[r_1+r_2][\parallel \psi -{\psi }^{\ast }{\parallel }_{X^{\sigma ,\delta ,k,\frac{1}{2}+\theta }}+\parallel \phi -{\phi }^{\ast }{\parallel }_{Y^{\sigma ,\delta ,s,\frac{1}{2}+\theta }}].\end{array}$$

Hence,

$$\begin{array}{c}\hfill \parallel ({\mathrm{\Gamma }}_1(\psi ,\phi ),{\mathrm{\Gamma }}_2(\psi ,\phi ))-({\mathrm{\Gamma }}_1({\psi }^{\ast },{\phi }^{\ast }),{\mathrm{\Gamma }}_2({\psi }^{\ast },{\phi }^{\ast })){\parallel }_{{\omega }_{\theta }}\le \frac{1}{2}{\parallel (\psi ,\phi )-({\psi }^{\ast },{\phi }^{\ast })\parallel }_{{\omega }_{\theta }}\end{array}$$

for

$$\begin{array}{c}\hfill T^{\theta }\le \frac{1}{4}min\left\{\frac{1}{c_3[r_1^2+r_1+r_2]},\frac{1}{c_4[r_1+r_2]}\right\}\end{array}$$

(24)

Therefore, the map ${\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2:{\omega }_{\theta }\to {\omega }_{\theta }$ is a contraction mapping, and we obtain a unique fixed point that solves the equation for any T that satisfies (23) and (24).

5. Regularity of Solution

In this section, we are going to prove that the solution $(\psi ,\phi )$ to (1) has Gevrey regularity in the time variable where $\psi$ is a complex-valued function and $\phi$ is a real-valued function; more precisely, we will prove the following result.

Theorem 2. 

Let $k⩾0$, $s>-\frac{3}{2}$, $\delta >0$, $\sigma ⩾1$, and $(\psi ,\phi )\in C([0,T],G^{\sigma ,\delta ,k}(\mathbb{R}))\times C([0,T],G^{\sigma ,\delta ,s}(\mathbb{R}))$ be the solution of (1); then, $(\psi ,\phi )\in G^{2\sigma }([0,T])\times G^{3\sigma }([0,T])$ in the time variable t.

For the proof of this theorem, it is enough to prove the following results.

Proposition 1. 

Let $m,l\in \{0,1,2,...\}$; we have

$$\begin{array}{c}\hfill |{\partial }_t^l{\partial }_x^m\psi (x,t)|\le L^{l+m+1}{\left((m+2l)!\right)}^{\sigma }{M}^l\end{array}$$

(25)

and

$$\begin{array}{c}\hfill |{\partial }_t^l{\partial }_x^m\phi (x,t)|\le L^{l+m+1}{\left((m+3l)!\right)}^{\sigma }{N}^l\end{array}$$

(26)

where $M=L^1+\frac{|\alpha |}{2^{\sigma }}+\frac{|\beta |L^1}{2^{\sigma }}$ and $N=L^2+\frac{L^1}{2^{\sigma +1}}+\frac{|\gamma |L^1}{2^{\sigma }}$

Proof. 

For the proof of the proposition, we use proof by induction on l for $l=0$ and $m\in \{0,1,2,\cdots \}$. The inequalities in Equations (25) and (26) are derived from the following result:

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

and

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

which allows us to conclude that $(\psi ,\phi )\in G^{\sigma }\times G^{\sigma }$ in x for all $k⩾0$ and $s>-\frac{3}{2}$. For proof of this result, see [8].

For $l=1$ and $m\in \{0,1,2,\cdots \}$, we have

$$\begin{array}{c}\hfill |{\partial }_t{\partial }_x^m\psi |\le |{\partial }_x^{m+2}\psi |+|\alpha ||{\partial }_x^m\psi \phi |+|\beta ||{\partial }_x^m{|\psi |}^2\psi |\end{array}$$

(27)

The terms of (27) can be estimated as

$$\begin{array}{cc}\hfill |{\partial }_x^{m+2}\psi |& ⩽L^{m+2+1}{\left((m+2)!\right)}^{\sigma }\hfill \\ \hfill & ⩽L^{m+1+1}{\left((m+2.1)!\right)}^{\sigma }{L}^1.\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill |\alpha ||{\partial }_x^m\psi \phi |& ⩽|\alpha |\sum _{k=0}^m\left(\genfrac{}{}{0pt}{}{m}{k}\right)|{\partial }_x^{m-k}\psi ||{\partial }_x^k\phi |\hfill \\ \hfill & ⩽|\alpha |\sum _{k=0}^m\frac{{(m!)}^{\sigma }}{{(k!)}^{\sigma }{((m-k)!)}^{\sigma }}{L}^{m-k+1}{\left((m-k)!\right)}^{\sigma }{L}^{k+1}{\left((k+1)!\right)}^{\sigma }\hfill \\ \hfill & ={|\alpha |(m!)}^{\sigma }\sum _{k=0}^m{(k+1)}^{\sigma }{L}^{m+2}\hfill \end{array}$$

At this stage, we use the fact that

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

So,

$$\begin{array}{cc}\hfill |\alpha ||{\partial }_x^m\psi \phi |& ⩽|\alpha |L^{m+2}{(m!)}^{\sigma }\frac{{(m+1)}^{\sigma }{(m+2)}^{\sigma }}{2^{\sigma }}\hfill \\ \hfill & =L^{m+1+1}{\left((m+2.1)!\right)}^{\sigma }\frac{|\alpha |}{2^{\sigma }}.\hfill \end{array}$$

(29)

In the previous term, by applying Leibniz’s rule twice, we obtain the following result:

$$\begin{array}{cc}\hfill |\beta ||{\partial }_x^m{|\psi |}^2\psi |& ⩽|\beta |\sum _{k=0}^m\sum _{k_1=0}^k\left(\genfrac{}{}{0pt}{}{m}{k}\right)\left(\genfrac{}{}{0pt}{}{k}{k_1}\right)\hfill \\ \hfill & \left|{\partial }_x^{m-k}\psi \right||{\partial }_x^{k-k_1}|\psi ||\left|{\partial }_x^{k_1}|\psi |\right|\hfill \\ \hfill & ⩽|\beta |\sum _{k=0}^m\sum _{k_1=0}^k\frac{{(m!)}^{\sigma }}{{(k!)}^{\sigma }{((m-k)!)}^{\sigma }}\frac{{(k!)}^{\sigma }}{{(k_1!)}^{\sigma }{((k-k_1)!)}^{\sigma }}\hfill \\ \hfill & L^{m-k+1}{\left((m-k)!\right)}^{\sigma }{L}^{k-k_1+1}{\left((k-k_1)!\right)}^{\sigma }{L}^{k_1+1}{\left((k_1)!\right)}^{\sigma }.\hfill \end{array}$$

Finally,

$$\begin{array}{cc}\hfill |\beta ||{\partial }_x^m{|\psi |}^2\psi |& ⩽L^{m+1+1}{\left((m+2.1)!\right)}^{\sigma }\frac{|\beta |L^1}{2^{\sigma }}.\hfill \end{array}$$

(30)

From (28)–(30) we obtain

$$\begin{array}{c}\hfill |{\partial }_t{\partial }_x^m\psi |⩽L^{m+1+1}{\left((m+2.1)!\right)}^{\sigma }{M}^1,\end{array}$$

(31)

where

$$M:=L^1+\frac{|\alpha |}{2^{\sigma }}+\frac{|\beta |L^1}{2^{\sigma }}.$$

□

We assume that (25) is correct for l, and then we prove that for $(l+1)$ we obtain

$$\begin{array}{c}\hfill |{\partial }_t^{l+1}{\partial }_x^m\psi |⩽|{\partial }_t^l{\partial }_x^{m+2}\psi |+|\alpha ||{\partial }_t^l{\partial }_x^m\psi \phi |+|\beta ||{\partial }_t^l{\partial }_x^m{|\psi |}^2\psi |.\end{array}$$

(32)

These terms are estimated as follows:

$$\begin{array}{cc}\hfill |{\partial }_t^l{\partial }_x^{m+2}\psi |& ⩽L^{m+1+l+1}{\left((m+2+2.l)!\right)}^{\sigma }{M}^l\hfill \\ \hfill & =L^{m+(l+1)+1}{\left((m+2.(l+1))!\right)}^{\sigma }{M}^l{L}^1,\hfill \end{array}$$

(33)

and

$$\begin{array}{cc}\hfill |\alpha ||{\partial }_t^l{\partial }_x^m\psi \phi |& ⩽|\alpha |\sum _{p=0}^l\sum _{k=0}^m\left(\genfrac{}{}{0pt}{}{l}{p}\right)\left(\genfrac{}{}{0pt}{}{m}{k}\right)|{\partial }_t^{l-p}{\partial }_x^{m-k}\psi ||{\partial }_t^p{\partial }_x^k\phi |\hfill \\ \hfill & ⩽|\alpha |\sum _{p=0}^l\sum _{k=0}^m\left(\genfrac{}{}{0pt}{}{l}{p}\right)\left(\genfrac{}{}{0pt}{}{m}{k}\right)L^{l-p+m-k+1}{\left((m-k+2.(l-p))!\right)}^{\sigma }{M}^{l-p}\hfill \\ & L^{p+k+1}{\left((k+2.p)!\right)}^{\sigma }{M}^p\hfill \\ \hfill & ⩽L^{m+(l+1)+1}{\left((m+2(l+1))!\right)}^{\sigma }{M}^l\frac{|\alpha |}{2^{\sigma }}.\hfill \end{array}$$

(34)

The same applies to

$$\begin{array}{cc}\hfill |\beta ||{\partial }_t^l{\partial }_x^m{|\psi |}^2\psi |& ⩽|\beta |\sum _{k=0}^m\sum _{k_1=0}^k\sum _{p=0}^l\sum _{p_1=0}^l\left(\genfrac{}{}{0pt}{}{m}{k}\right)\left(\genfrac{}{}{0pt}{}{k}{k_1}\right)\left(\genfrac{}{}{0pt}{}{l}{p}\right)\left(\genfrac{}{}{0pt}{}{p}{p_1}\right)\hfill \\ \hfill & |{\partial }_t^{l-p}{\partial }_x^{m-k}|\psi |||{\partial }_t^{p-p_1}{\partial }_x^{k-k_1}|\psi |||{\partial }_t^{p_1}{\partial }_x^{k_1}\psi |\hfill \\ \hfill & ⩽|\beta |\sum _{k=0}^m\sum _{k_1=0}^k\sum _{p=0}^l\sum _{p_1=0}^l\left(\genfrac{}{}{0pt}{}{m}{k}\right)\left(\genfrac{}{}{0pt}{}{k}{k_1}\right)\left(\genfrac{}{}{0pt}{}{l}{p}\right)\left(\genfrac{}{}{0pt}{}{p}{p_1}\right)\hfill \\ \hfill & L^{l-p+m-k+1}{\left((m-k+2(l-p))!\right)}^{\sigma }{M}^{l-p}\hfill \\ \hfill & L^{p-p_1+k-k_1+1}{\left((k-k_1+2(p-p_1))!\right)}^{\sigma }{M}^{p-p_1}\hfill \\ \hfill & L^{p_1+k_1+1}{\left((k_1+2(p_1))!\right)}^{\sigma }{M}^{p_1}.\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{cc}\hfill |\beta ||{\partial }_t^l{\partial }_x^m{|\psi |}^2\psi |& ⩽L^{(l+1)+m+1}{\left((m+2(l+1))!\right)}^{\sigma }{M}^l\frac{|\beta |L^1}{2^{\sigma }}\hfill \end{array}$$

(35)

Finally, by using (33)–(35) we arrive at

$$\begin{array}{c}\hfill |{\partial }_t^{l+1}{\partial }_x^m\psi |⩽L^{m+(l+1)+1}{\left((m+2(l+1))!\right)}^{\sigma }{M}^{l+1}.\end{array}$$

Now, we prove (26); for $l=1$ and $m\in \{0,1,2,\cdots \}$, we have

$$\begin{array}{c}\hfill |{\partial }_t{\partial }_x^m\phi |⩽|{\partial }_x^{m+3}\phi |+\frac{1}{2}|{\partial }_x^{m+1}({\phi }^2)|+|\gamma ||{\partial }_x^{m+1}{(|\psi |}^2)|\end{array}$$

(36)

The terms of (36) can be estimated as

$$\begin{array}{cc}\hfill |{\partial }_x^{m+3}\phi |& ⩽L^{m+3+1}{\left((m+3)!\right)}^{\sigma }\hfill \\ \hfill & =L^{m+1+1}{\left((m+3.1)!\right)}^{\sigma }{L}^2.\hfill \end{array}$$

(37)

Regarding the second term in (36), we derive the following:

$$\begin{array}{cc}\hfill \frac{1}{2}|{\partial }_x^{m+1}({\phi }^2)|& ⩽\frac{1}{2}\sum _{k=0}^{m+1}\left(\genfrac{}{}{0pt}{}{m+1}{k}\right)|{\partial }_x^{m+1-k}\phi ||{\partial }_x^k\phi |\hfill \\ \hfill & ⩽\frac{1}{2}\sum _{k=0}^{m+1}\frac{{((m+1)!)}^{\sigma }}{{((m+1-k)!)}^{\sigma }{(k!)}^{\sigma }}{L}^{m+1-k+1}{\left((m+1-k)!\right)}^{\sigma }{L}^{k+1}{\left(k!\right)}^{\sigma }\hfill \\ \hfill & ⩽\frac{1}{2}{((m+1)!)}^{\sigma }\sum _{k=0}^{m+1}{(k+1)}^{\sigma }{L}^{m+3}\hfill \\ \hfill & ⩽\frac{1}{2}{((m+1)!)}^{\sigma }\frac{{(m+2)}^{\sigma }{(m+3)}^{\sigma }}{2^{\sigma }}{L}^{m+3}\hfill \\ \hfill & =L^{m+1+1}{((m+3.1)!)}^{\sigma }\frac{L^1}{2^{\sigma +1}}.\hfill \end{array}$$

(38)

A similar calculation for the third term of (36) yields

$$\begin{array}{cc}\hfill |\gamma ||{\partial }_x^{m+1}{(|\psi |}^2)|& ⩽L^{m+1+1}{((m+3.1)!)}^{\sigma }\frac{|\gamma |L^1}{2^{\sigma }}.\hfill \end{array}$$

(39)

From (37)–(39), we obtain

$$\begin{array}{c}\hfill |{\partial }_t{\partial }_x^m\phi |⩽L^{m+1+1}{((m+3.1)!)}^{\sigma }{N}^1,\end{array}$$

(40)

where $N=L^2+\frac{L^1}{2^{\sigma +1}}+\frac{|\gamma |L^1}{2^{\sigma }}$.

We assume that (26) is correct for l, and then we prove that for $(l+1)$ we have

$$\begin{array}{c}\hfill |{\partial }_t^{l+1}{\partial }_x^m\phi |⩽|{\partial }_t^l{\partial }_x^{m+3}\phi |+\frac{1}{2}|{\partial }_t^l{\partial }_x^{m+1}({\phi }^2)|+|\gamma ||{\partial }_t^l{\partial }_x^{m+1}{(|\psi |}^2)|.\end{array}$$

(41)

These terms are estimated as follows:

$$\begin{array}{cc}\hfill |{\partial }_t^l{\partial }_x^{m+3}\phi |& ⩽L^{m+3+l+1}{\left((m+3+3.l)!\right)}^{\sigma }\hfill \\ \hfill & =L^{m+(l+1)+1}{\left((m+3.(l+1))!\right)}^{\sigma }{N}^l{L}^2.\hfill \end{array}$$

(42)

For the second term of (41), we have

$$\begin{array}{cc}\hfill \frac{1}{2}|{\partial }_t^l{\partial }_x^{m+1}({\phi }^2)|& ⩽\frac{1}{2}\sum _{p=0}^l\sum _{k=0}^{m+1}\left(\genfrac{}{}{0pt}{}{l}{p}\right)\left(\genfrac{}{}{0pt}{}{m+1}{k}\right)|{\partial }_t^{l-p}{\partial }_x^{m+1-k}\phi ||{\partial }_t{\partial }_x^k\phi |\hfill \\ \hfill & ⩽\frac{1}{2}\sum _{p=0}^l\sum _{k=0}^{m+1}\left(\genfrac{}{}{0pt}{}{l+m+1}{p+k}\right)\hfill \\ \hfill & L^{l-p+m+1-k+1}{\left((m+1-k+3.(l-p))!\right)}^{\sigma }{L}^{p+k+1}{\left((k+3.p)!\right)}^{\sigma }\hfill \\ \hfill & ⩽L^{m+(l+1)+1}{\left((m+3.(l+1))!\right)}^{\sigma }{N}^l\frac{L^1}{2^{\sigma +1}}.\hfill \end{array}$$

(43)

Using the same calculus for the third term of Equation (41), we find that

$$\begin{array}{cc}\hfill |\gamma ||{\partial }_t^l{\partial }_x^{m+1}{(|\psi |}^2)|& ⩽L^{m+(l+1)+1}{\left((m+3.(l+1))!\right)}^{\sigma }{N}^l\frac{|\gamma |L^1}{2^{\sigma }}.\hfill \end{array}$$

(44)

Finally, by combining (42)–(44), we deduce that

$$\begin{array}{c}\hfill |{\partial }_t^{l+1}{\partial }_x^m\phi |⩽L^{m+(l+1)+1}{\left((m+3.(l+1))!\right)}^{\sigma }{N}^{l+1}.\end{array}$$

Consequently, $(\psi ,\phi )\in G^{2\sigma }([0,T])\times G^{3\sigma }([0,T])$.

6. Conclusions

We have demonstrated that the initial value problem for the Schrödinger–Korteweg–de Vries (SKdV) system is locally well posed in Gevrey spaces, specifically $G^{\sigma ,\delta ,k}(\mathbb{R})\times G^{\sigma ,\delta ,s}(\mathbb{R})$. The parameters must satisfy $k\ge 0$ and $s>-\frac{3}{4}$. This advancement builds on recent findings regarding the well posedness of this model within Sobolev spaces, namely $H^k(\mathbb{R})\times H^s(\mathbb{R})$. Additionally, to further understand the system’s temporal behavior, we establish the Gevrey regularity properties of its solutions. Also, we focus on the local existence of solutions to gain an initial understanding of their behavior over a short period. This approach helps to establish a basic framework in Gevrey spaces. To explore global existence and persistence, further conditions or estimates are needed, which go beyond the scope of our initial study. We plan to address these aspects in future research, once we have a clearer understanding of the system’s long-term dynamics.

An interesting direction for future work involves extending the current results to two-dimensional (or higher-dimensional) systems, where fundamental challenges arise. Specifically, there are several key areas to address:

• Developing estimates for the cubic nonlinear terms (9) and (10);
• Establishing bilinear estimates for the interaction terms presented in (16) and (17);
• Enhancing the proof of solution regularity by refining the estimates (34) and (35).

We will also investigate the local well posedness of a coupled system of generalized Kadomtsev–Petviashvili II equations relating to the Cauchy problem defined as follows:

$$\begin{array}{cc}\hfill & {\partial }_t\psi -{|D_x|}^{\alpha }{\partial }_x\psi +{\partial }_x^{-1}{\partial }_{yy}\psi +\psi {\partial }_x\psi =0,\hfill \\ \hfill & \psi (x,y,0)={\psi }_0,\hfill \end{array}$$

where $\psi (x,y,t)=\psi$ is an unknown function, $(x,y)\in {\mathbb{R}}^2$, $t\in \mathbb{R}$, and $\alpha \ge 4$, while ${\psi }_0$ is a given function. Here, $|D_x{|}^{\alpha }$ is the Fourier multiplier given by $\widehat{|D_x{|}^{\alpha }\psi }(\xi )={|\xi |}^{\alpha }\hat{\psi }(\xi )$ and and ${\partial }_x^{-1}$ is defined by its Fourier multiplier $-i{\xi }^{-1}$. We will also concentrate on a new strategy for analyzing the Gevrey regularity of the solution over time t.