Cauchy Problem of the Generalized Zakharov Type System in R

In this paper, we consider the initial value problem for a two-dimensional generalized Zakharov system with quantum effects. We prove the existence and uniqueness of global smooth solutions to the initial value problem in the Sobolev space through making a priori integral estimates and the Galerkin method.


Introduction
In the recent years, special interest has been devoted to quantum corrections to the Zakharov equations for Langmuir waves in a plasma []. By use of a quantum fluid approach, the following modified Zakharov equations are obtained: where H is the dimensionless quantum parameter given by the ratio of the ion plasmon and electron thermal energies. For H = , this system was derived by Zakharov in [] to model a Langmuir wave in plasma. The Zakharov system attracted many scientists' wide interest and attention [-].
In this paper, we deal with the following generalized Zakharov system: where (E, n) : (x, t) ∈ R  × R and the initial data are taken to be E| t= = E  (x), n| t= = n  (x), n t | t= = n  (x).
To study a smooth solution of the generalized Zakharov system, we transform it into the following form: with initial data Now we state the main results of the paper.
Then there exists a unique global smooth solution of the initial value problem ()-().
The obtained results may be useful for better understanding the nonlinear coupling between the ion-acoustic waves and the Langmuir waves in a two-dimensional space.

A priori estimates
Proof Taking the inner product of () and E, then taking the imaginary part, we have Hence, we get We thus get Lemma ..
where C is a positive constant,  ≤ j m ≤ α ≤ , where Proof Take the inner products of () and -E t . Since Letting and noticing (), we obtain Proof By Hölder's inequality, Young's inequality and Lemma ., it follows that From Lemma . we get Take the inner products of Eq. () and ϕ. It follows that Hence, from Eq. () we get Using Gronwall's inequality, we obtain that We thus get Lemma ..
Proof Differentiating () with respect to t, then taking the inner products of the resulting equation and E t , we have By Lemma ., it follows that Differentiating Eq. () with respect to t, then taking the inner products of the resulting equation and n t , we have Noting that From Eq. () and () we get By Gronwall's inequality, it follows that Take the inner products of Eq. () and ϕ t . It follows that Take the inner products of Eq. () and E. It follows that From Eq. () we get From () we obtain From () we obtain We thus get Lemma ..
Lemma . Suppose that f  , f  ∈ H s ( ), ⊆ R n . Then we have where the constant C s is independent of f  and f  .
Proof Lemma . is true when m =  (Lemma .). Suppose that Lemma . is true when m = k, (k ≥ ). Take the inner products of () and (-) k+ k+ ϕ. It follows that By using Gronwall's inequality, we have

From () and () we get
Differentiating () with respect to t, then taking the inner products of the resulting equation and (-) k+ k+ E t , we obtain Since By using Gronwall's inequality, we get This means Lemma . is true when m = k + . Thus Lemma . is proved completely.

Existence and uniqueness of solution
Now, with these lemmas, we are able to prove Theorem .. First we obtain the existence and uniqueness of the global generalized solution of problem ()-(). Now, one can estimate the following theorem.
Then there exists a global smooth solution of the initial value problem ()-().
where the constants C are independent of l and D. By compact argument, some subsequence of (E l , n l , ϕ l ), also labeled by l, has a weak limit (E, n, ϕ). More precisely By the continuation extension principle, from the conditions of the theorem and a priori estimates in Section , we can get the existence of a global generalized solution for problem ()-(). By Lemma . and the Sobolev imbedding theorem, Theorem . is proved.
Next, we prove the uniqueness of a solution for problem ()-().
Then the global solution of the initial value problem ()-() is unique.
Proof Suppose that there are two solutions E  , n  , ϕ  and E  , n  , ϕ  . Let From ()-() we get with initial data Take the inner product of () and E. Since Take the inner product of () and ϕ. Since Hence from () and () we get By using Gronwall's inequality and noticing (), we arrive at Theorem . is proved. This completes the proof of Theorem ..

Results and discussion
One can regard ()-() as the Langmuir turbulence parameterized by H and study the asymptotic behavior of systems ()-() when H goes to zero.

Conclusions
By a priori integral estimates and the Galerkin method, we have the following conclusion.
Suppose that E  (x) ∈ H l+ (R  ), n  (x) ∈ H l+ (R  ), n  (x) ∈ H l (R  ), l ≥ . Then there exists a unique global smooth solution of the initial value problem ()-().