# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 1 Introduction  
In this paper, we investigate the well-posedness of the following Cauchy problem:  
\[ \begin{cases}\partial_{t}u+\kappa(\partial_{x}u)^{2}+q(\partial_{x}u)^{3}+r( \partial_{x}u)^{4}+\delta\partial_{x}^{3}u\\ \qquad\qquad+\beta^{2}\partial_{x}^{4}u+\mu\partial_{x}^{2}u_{\epsilon}+\gamma u \partial_{x}^{2}u=0,&\quad t>0,\quad x\in\mathbb{R},\\ u(0,x)=u_{0}(x),&\quad x\in\mathbb{R},\end{cases}  \tag{1}\]  
with  
\[\gamma=0,\quad\beta\
eq 0,\quad\text{or} \tag{2}\] \[q=r=0,\quad\beta\
eq 0,\quad\kappa=2\gamma. \tag{3}\]  
Under Assumption (2), we assume on the initial datum  
\[u_{0}\in H^{2}(\mathbb{R}). \tag{4}\]  
Instead, under Assumption (3), we assume (4) or  
\[u_{0}\in H^{3}(\mathbb{R}). \tag{5}\]  
Observe that if \(q=r=\delta=\mu=\beta=\gamma=0\), Equation (1) reads  
\[\partial_{t}u+\kappa(\partial_{x}u)^{2}+\delta\partial_{x}^{3}u=0. \tag{6}\]  
Using the variable (see [1]),  
\[v=\partial_{x}u \tag{7}\]Equation (6) is equivalent to the Korteweg-de Vries equation [2]  
\[\partial_{t}v+\kappa\partial_{x}v^{2}+\delta\partial_{x}^{3}v=0, \tag{8}\]  
that has a very wide range of applications, such as magnetic fluid waves, ion sound waves, and longitudinal astigmatic waves. From a mathematical point of view, in [3; 4; 5], the Cauchy problem for (8) is studied, while in [6], the author reviewed the travelling wave solutions for (8). Moreover, in [7; 8; 9], the convergence of the solution of (8) to the unique entropy one of the Burgers equation is proven. Taking \(\kappa=r=\delta=\mu=\beta=\gamma=0\) and using the variable (7), (1) becomes  
\[\partial_{t}v+q\partial_{x}v^{3}+\delta\partial_{x}^{3}v=0, \tag{9}\]  
which is known as the modified Korteweg-de Vries equation. [10; 11; 12; 13; 14; 15] show that (9) is a non-slowly varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. In [3; 5], the Cauchy problem for (9) is studied, while in [9; 16], the convergence of the solution of (9) to the unique entropy solution of the following scalar conservation law  
\[\partial_{t}v+q\partial_{x}v^{3}=0. \tag{10}\]  
Assuming \(\kappa=1\) and \(q=r=\gamma=0\), (1) reads  
\[\partial_{t}u+(\partial_{x}u)^{2}+\delta\partial_{x}^{3}u+\beta^{2}\partial_{ x}^{4}u+\mu\partial_{x}^{2}u=0. \tag{11}\]  
Equation (11) arises in interesting physical situations, for example as a model for long waves on a viscous fluid owing down an inclined plane [17] and to derive drift waves in a plasma [18]. Equation (11) was derived also independently by [PERSON] [19; 20; 21] as a model for phase turbulence in reaction-diffusion systems and by [PERSON] [22] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. Equation (11) also describes incipient instabilities in a variety of physical and chemical systems [23; 24; 25]. Moreover, (11), which is also known as the Benney-Lin equation [26; 27], was derived by [PERSON] in the study of phase turbulence in the Belousov-Zhabotinsky reaction [28]. The dynamical properties and the existence of exact solutions for (11) have been investigated in [29; 30; 31; 32; 33; 34]. In [35; 36; 37], the control problem for (11) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [38], the problem of global exponential stabilization of (11) with periodic boundary conditions is analyzed. In [39], it is proposed a generalization of optimal control theory for (11), while in [40] the problem of global boundary control of (11) is considered. In [41], the existence of solitonic solutions for (11) is proven. In [1; 42], the well-posedness of the Cauchy problem for (11) is proven, using the energy space technique and the fixed-point method, respectively.