In this paper, we consider the Cauchy problem for the stochastic Kawahara equation:

$$ u_{t}+\alpha u_{5x}+\beta u_{3x}+\gamma u_{x}+\mu {uu}_{x}=\Phi\frac{\partial^{2}B}{\partial t\partial x}, $$

(1)

where *α*≠0, *β*, and *γ* are real numbers; *μ* is a complex number; *u* is a stochastic process defined on \((x,t)\in \mathbb {R}\times \mathbb {R_{+}}\); *Φ* is a linear operator; and *B* is a two-parameter Brownian motion on \(\mathbb {R}\times \mathbb {R_{+}}\), that is, a zero mean Gaussian process whose correlation function is given by:

$$ \mathbb{E}\left(B(x,t)B(y,s)\right)=(x\wedge y)(t\wedge s),\ \ \ t,s\geq0,\ x,y\in\mathbb{R}. $$

(2)

In general, the covariance operator *Φ* can be described by a kernel \(\mathcal {K}(x,y).\) The correlation function of the noise is then given by

$$\mathbb{E}\left(\Phi\frac{\partial^{2}B}{\partial t\partial x}(x,t)\Phi\frac{\partial^{2}B}{\partial t\partial x}(y,s)\right)=c(x,y)\delta_{t-s}, $$

where \(t,s\geq 0,\ x,y\in \mathbb {R}\), *δ* is the Dirac function and

$$c(x,y)=\int_{\mathbb{R}}\mathcal{K}(x,z)\mathcal{K}(y,z)dz. $$

Consider a fixed probability space \((\Omega,\mathcal {F},P)\) adapted to a filtration \((\mathcal {F}_{t})_{t\geq 0}\). As usual, we can rewrite the right hand side of Eq. (1) as the time derivative of a cylindrical Wiener process on \(L^{2}(\mathbb {R})\) by setting:

$$ W(t)=\frac{\partial B}{\partial x}=\sum_{i\in\mathbb{N}}\beta_{i}(t)e_{i}, $$

(3)

where \((e_{i})_{i\in \mathbb {N}}\) is an orthonormal basis of \(L^{2}(\mathbb {R})\) and \((\beta _{i})_{i\in \mathbb {N}}\) is a sequence of mutually independent real Brownian motions in \((\Omega,\mathcal {F},P)\). Let us rewrite Eq. (1) in its Itô form as follows:

$$ \left\{\begin{array}{l} du+\left(\alpha u_{5x}+\beta u_{3x}+\gamma u_{x}+\mu {uu}_{x}\right)dt=\Phi dW(t),\\ u(x,0)=u_{0}(x) \end{array}\right. $$

(4)

In order to obtain local well-posedness of Eq. (1), we mainly work on the general mild formulation of Cauchy problem (4) as below:

$$ {\begin{aligned} u(t)=U(t)u_{0}+\int_{0}^{t}U(t-s)\left(\mu {uu}_{x}\right)ds+\int_{0}^{t}U(t-s)\Phi dW(s). \end{aligned}} $$

(5)

Here, \(U(t)=\mathfrak {F}_{x}^{-1}\text {exp}\left (-it\phi (\xi)\right)\mathfrak {F}_{x}\) is the unitary group of operators related to the linearized equation:

$$ u_{t}+\alpha u_{5x}+\beta u_{3x}+\gamma u_{x}=0,\ \ \ \ (x,t)\in\mathbb{R}\times\mathbb{R_{+}}, $$

(6)

where *ϕ*(*ξ*)=*α**ξ*^{5}−*β**ξ*^{3}+*γ**ξ* is the phase function and \(\mathfrak {F}_{x}\) (or “. ̂”) is the usual Fourier transform in the *x* variable. We note that the phase function *ϕ* has non-zero singularity. This differs from the phase function of the linear Korteweg-de Vries (KdV) equation (see [1]) and causes some difficulties in the problem. To avoid these difficulties, we eliminate the singularity of the phase function *ϕ* by using the Fourier restriction operators [2]:

$${\begin{aligned} P^{N}f=\int_{|\xi|\geq N}e^{ix\xi}\hat{f}(\xi)d\xi,\ \ \ \ \ \ P_{N}f=\int_{|\xi|\leq N}e^{ix\xi}\hat{f}(\xi)d\xi,\ \ \forall N>0. \end{aligned}} $$

In the case of *Φ*≡0 (effect of the noise does not exist), Eq. (1) is reduced to the deterministic Kawahara equation:

$$ u_{t}+\alpha u_{5x}+\beta u_{3x}+\gamma u_{x}+\mu {uu}_{x}=0,\ \ \ \ (x,t)\in\mathbb{R}\times\mathbb{R_{+}}. $$

(7)

As aforesaid in [3–5], Eq. (7) is a fifth-order shallow water wave equation. It arises in study of the water waves with surface tension, in which the Bond number takes on the critical value, where the Bond number represents a dimensionless magnitude of surface tension in the shallow water regime. If we consider a realistic situation, in which a non-constant pressure affects on the surface of the fluid or the bottom of the layer is not flat, it is meaningful to add a forcing term to Eq. (7). This term can be given by the gradient of the exterior pressure or of the function whose graph defines the bottom [6, 7]. This paper focuses on the case when the forcing term is of additive white noise type. This leads us to study the stochastic fifth-order shallow water wave Eq. (1). By means of white noise functional analysis, the analytical white noise functional solutions for the nonlinear stochastic partial differential equations (SPDEs) can be investigated. This subject is attracting more and more attention [8–15].

It is well known that the Cauchy problem (4) is locally well-posed for data in \(H^{s}(\mathbb {R}),\ s\in \mathbb {R}\), if for any finite time *T*, there exists a locally continuous mapping that transfers \(u_{0}\in H^{s}(\mathbb {R})\) to a unique solution \(u\in C\left ([0,T];H^{s}(\mathbb {R})\right)\). If the solution mapping exists for all time, we say that the Cauchy problem (4) is globally well-posed [16].

In [17], Huo obtained a local well-posedness result in \(H^{s}(\mathbb {R})(s>-11/8)\) for the Kawahara equation. Moreover, Jia and Huo [18] proved the local well-posedness of the Kawahara and modified Kawahara equations for data in \(H^{s}(\mathbb {R})\) with *s*>−7/4 and *s*≥−1/4 respectively. The first well-posedness result for the Kaup-Kupershmidt equations was presented by Tao and Cui [19]. They proved that their Cauchy problems are locally well-posed in \(H^{s}(\mathbb {R})\) for *s*>5/4 and *s*>301/108, respectively. Thereafter, Zhao and Gu [20] lowered the regularity of the initial data space to *s*>9/8 and improved the preceding result in [19]. Also, using a Fourier restriction method, a local well-posedness result for the Kaup-Kupershmidt equations was established in [18] for data in \(H^{s}(\mathbb {R})\) with *s*>0 and *s*>−1/4, respectively.

If *α*=*γ*=0, the model (7) is minified to the famous KdV equation:

$$ u_{t}+\beta u_{3x}+\mu {uu}_{x}=0,\ \ \ \ (x,t)\in\mathbb{R}\times\mathbb{R_{+}}. $$

(8)

The well-posedness of Eq. (8) was studied by Kenig, Ponce, and Vega [21]. They proved that its Cauchy problem is locally well-posed in \(H^{s}(\mathbb {R})\) for *s*>−3/4. Also, Ponce [1] discussed the general fifth-order shallow water wave equation:

$$ {\begin{aligned} u_{t}+u_{x}+c_{1}u u_{x}+c_{2}u_{3x}+c_{3} u_{x} u_{xx}+c_{4}u u_{3x}+c_{5} u_{5x}=0 \ \ \ (x,t)\in\mathbb{R}\times\mathbb{R_{+}} \end{aligned}} $$

(9)

and gave a global well-posedness result of its Cauchy problem for data in \(H^{4}(\mathbb {R})\). The well-posedness of the SPDEs has been the subject of a large amount of work. de Bouard and Debussche [22] considered the stochastic KdV equation forced by a random term of white noise type. They proved existence and uniqueness of solutions in \(H^{1}(\mathbb {R})\) and existence of martingale solutions in \(L^{2}(\mathbb {R})\) in the case of additive and multiplicative noise, respectively. Since that time, many researchers paid more attention to investigate the Cauchy problems for some SPDEs and have obtained a number of local and global well-posedness results [23–25].

The goal of this paper is to investigate the Cauchy problem of the stochastic Kawahara Eq. (1), where the random force is of additive white noise type. By employing a Fourier restriction method, a Banach fixed point theorem, and some basic inequalities, we show that Eq. (1) is locally well-posed for data in \(H^{s}(\mathbb {R}),\ s>-7/4\). Also, we give global existence for \(L^{2}(\mathbb {R})\) solutions. An outline of this paper is as follows. The “Main results” section contains precise statement of our new results and some important function spaces. In the section “The stochastic convolution estimate”, we give an estimation of the stochastic convolution term via a Fourier restriction method and some basic inequalities. In the section “Local well-posedness: proof of Theorem 1”, we use the stochastic estimation proved in the section “The stochastic convolution estimate” and the Banach fixed point theorem to obtain a local well-posedness result of Eq. (1). In the section “Global well-posedness: proof of Theorem 2”, we extend our technique and show global well-posedness result of Eq. (1). The “Summary and discussion” section is devoted to the summary and discussion.