CAUCHY PROBLEM FOR THE GENERALIZED DAVEY-STEWARTSON SYSTEMS IN BESOV SPACES AND SOME COUNTEREXAMPLES

In this paper, the Cauchy problem of the generalized ellipse-ellipse type Davey-Stewartson systems is discussed. When the dimension of space is greater than or equal to two, we get a unique global solution in Besov spaces by contraction mapping argument. Moreover, by using the F-expansion method, the exact periodic wave solutions for the generalized ellipse-ellipse type DaveyStewartson systems are discussed, some counter examples are given.


Introduction
A large amount of work are devoted to the study of Davey-Stewartson systems. The classical Davey-Stewartson systems    iu t + D 1 u = r |u| 2 u + µuv, are originally derived by Davey and Stewartson in [4] to describe quasi-monochromatic wave pockets on the surface of a shallow liquid. Here D 1 , D 2 and D 3 are partial differential operators of the second order in x, y (x, y) ∈ R 2 . D 1 = δ∂ 2 x + ∂ 2 y is either elliptic or hyperbolic, and D 2 = ∂ 2 x + m∂ 2 y (m > 0) is elliptic. Later, the case that D 2 is hyperbolic, i.e. m < 0, is derived in [5] by taking account of the effect of surface tension (or capillary). Generally, D 3 = ∂ 2 x u is a complex-valued function of (t; x, y) ∈ R + × R 2 , and v is a real-valued function of (t; x, y) ∈ R + × R 2 . And u and v are related to the amplitude and the mean velocity potential of the water wave, respectively.
As in [7], the Davey-Stewartson equations are usually classified as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic types according to the respective types of D 1 and D 2 . In recent years, Davey-Stewartson equations(1.1) have drawn much attention from many physicists and mathematicians due to the abundant physical and mathematical properties. In [3], by using the generalized Kudryashov method, Demiray and Bulut found the dark soliton solutions of DSE systems. Later, by employing the Bäcklund transformation, the Hamiltonian approach and the G-expansion method to the DSEs, new traveling solitary and kink wave solutions are obtained, see [23]. The well-posedness, decay of solutions, soliton solutions, solitary and standing waves, etc., have been quite extensively studied by many authors (see [7,9,14,15,17,25,26]).
In [10], Wang and Guo study the generalized Davey-Stewartson systems ∂xi∂xj , (a ij ) and (b ij ) are real invertible matrix. They discuss the initial value problem of the Davey-Stewartson systems for the elliptic-elliptic and hyperbolic-elliptic cases. The local and global existence, as well as uniqueness of solutions in H s are shown. Moreover, they prove that the scattering operator carries a band in H s into H s . By using the Hirota's bilinear method, the homoclinic orbits of the Davey-Stewartson Equations are obtained through the dependent variable transformation, see [30].
In [27], Wang shows that there exists s c > 0 such that the cubic (quartic) non-elliptic derivative Schrödinger equations with small data in modulation spaces Furthermore, if λ 1 |u| p1 u + λ 2 |u| p2 u in the first equation of (1.2) is replaced by a more general nonlinear function f (u), the existence of solutions is also little known. And some physical models, such as Landau-Lifshitz equation, Navier Stokes equation, can be transformed into the equation with more general nonlinear function f (u). So it is more meaningful and more difficult to study the Davey-Stewartson systems with general nonlinear term f (u). System (1.1) and (1.2) are two important cases of general nonlinear Davey-Stewartson equation. Although our study (2.1) is a special case of general nonlinear Davey-Stewartequation, it includes (1.1) and (1.2).
The purpose of this paper is to investigate the Cauchy problem of the two cases mentioned above, i.e., the generalized ellipse-ellipse and hyperbolic-ellipse type. When the dimension of space is greater than or equal to two, we get a unique global solution in Besov spaces by contraction mapping argument (see Section 2). Naturally, the reader will curious about whether these results are valid for bounded domain? To answer those interesting questions, we construct some exact periodic wave solutions for the generalized ellipse-ellipse type Davey-Stewartson systems by using the F-expansion method, and some counter examples are given (see Section 3). Furthermore, we construct some exact periodic wave solutions to show the existence of solutions for generalized ellipse-hyperbolic and hyperbolichyperbolic type Davey-Stewartson systems which is still open (see Section 4).
This paper is based on many kinds of methods which are raised, such as homogeneous balance method [28], hyperbolic function expansion method [6], nonlinear transformation method [11,13], trial function method [19], sine-cosine method [29], Jacobi elliptic function expansion method, etc. The solution gotten by these methods are mainly solitary wave solutions, shock solutions, see [6,11,13,19,28,29] and elliptic function, see [20][21][22]. To search the stability of the solutions and inspired by [18], we add perturbation in our research and discuss the evolution of the perturbation. Essentially, it is to expand the solution of nonlinear evolution equation to ε-power series and try to get the multi-order exact solutions of it. The symmetry group properties of the variable coefficient Davey-Stewartson (vcDS) systems are studied in [8]. The dromion of the Davey-Stewartson-1 equation is studied under perturbation on the large time [12]. Through the Hirota bilinear method, Ma formulate a combined fourth-order nonlinear equation while guaranteeing the existence of lump solutions of new (2+1)-dimensional nonlinear equations [16].

Cauchy problem
In this section we study the Cauchy problem for the generalized Davey-Stewartson systems 1≤i,j≤n bij ξiξj F ψ, where F is Fourier transformation.
Then (2.1) is equivalent to the following form Here f (u) ∈ C k (k ∈ Z), is a nonlinear function and for any given p, For brevity, in the following, we denote  Now, we state the main result as follows.
Theorem 2.1. Let n ≥ 2, 4 n ≤ p, q < ∞, max(s(q), s(p)) ≤ s < ∞, [s] ≤ p. If u 0 ∈ H s and there exist a δ > 0 such that u 0 H s < δ, then there exists a unique solution u of the Cauchy problem(2.3) satisfying Throughout this paper, we use a variety of function spaces, Lebesgue spaces L r , Bessel potential spaces H s,r , Besov spaces B s r,2 . The definition of L r and H s,r is as usual, and an equivalent definition of the norm onḂ s r,2 is that u Ḃs where [s] denotes the largest integer less than or equal to s, For some additional basic results on Besov spaces, one can refer to [1,24]. In the following, C stands for a constant that may be different numbers in different places. For any r ∈ [1, ∞], r denotes the duality number of r, i.e. 1 r + 1 r = 1.

Main lemmas
The main tools used here are time spaces L p − L p estimates for solutions of linear Schrödinger equations in Lebesgue space. These estimates are usually named generalized Strichartz inequalities. The method of the proof of the main result is a contraction mapping argument. Let us recall that some estimates for linear Schrödinger equations in Lebesgue-Besov spaces which are established by Cazenave and Weissler in [2]. Lemma 2.1. For all s ∈ R, r, q ∈ [2, α(n)). S(t) is semi-group of operator and i∂/∂t + A is generating operator of S(t), then we have (i) If u 0 ∈Ḣ s , then S(t)u 0 ∈ L γ(r) (0, ∞;Ḃ s r,2 ), and there exists a constant C > 0, such that S(t)u 0 L γ(r) (0,∞;Ḃ s r,2 ) ≤ C u 0 Ḣs ; (2.9)
Then, there exists a constant C > 0 only depending on r , n, s, such that uv Ḃs By a similar method in [9], one can have the following result.

The proof of Theorem 2.1
The Cauchy problem of (2.1) is essentially equivalent to the following integral equation  And for any u, v ∈ D, the metric d(u, v) is d(u, v) = u − v L p+2 (0,∞;L r(p) )∩L q+3 (0,∞;L r(q) ) . (2.43) Considering the mapping and . (2.48) So, for any u ∈ D, Thus, J is a contraction mapping on (D, d), and has a unique fixed point u ∈ D. From Lemma 1, we deduce that there exists a unique solution u of the Cauchy problem (2.1) satisfying This finishes the proof of the Theorem.

Explicit periodic wave solutions and some counter examples
The F-expansion method is the generalization of Jacobi elliptic function expansion method. In this section we mainly consider the general Davey-Stewartson systems where u is a complex-valued function, r, µ, b are real constants.
(ii) When n = 2, λ = (1 + 4m 2 ), the Lam equation is The corresponding Lam function is defined by The corresponding Lam function is defined by The corresponding lam function is defined by

Degenerate solution
When the m → 1, snξ → tanh ξ, the zeroth-order solution of DSI degenerates into where and k j , α j are constants, ξ 0 , η 0 are arbitrary constants, and (Σ l j=1 kj 2 −Σ n j=l+1 kj 2 ) r > 0. This is solitary wave solution that we frequently see, and we call it shock wave solution.
Similarly by cnξ → sec h, dnξ → sec hξ, when m → 1, the first-order solution of DSI is to degenerate into This is a bell shaped solitary wave solution, pulse shock wave solution.
The second-order solution of DSI is to degenerate into

The more exact solution of DSI
One can get more solutions of Davey-Stewartson equations: , one can get the zeroth-order solution of DSI which is The first-order solution is The second-order solution is (4.23), one can get the zeroth-order solution of DSI which is r > 0, v 0 = c − 2ldn 2 ξ, η = Σ n j=1 α j k j + [(Σ n j=1 k j 2 )(2 − m 2 ) − Σ n j=1 α j 2 − 2c]t + η 0 . (4.46) The first-order solution is snξcnξdnξ. (4.47) The second-order solution is

Conclusion
In this paper, we prove that the Cauchy problem of generalized Davey-Stewartson systems has a unique solution in C(0, ∞; H s )∩L p+2 (0, ∞; B s r(p),2 )∩L q+3 (0, ∞; B s r(q),2 ). What's more interesting, we construct some explicit period wave solution of the generalized Davey-Stewartson by F-expansion method, as well as some multi-order exact solutions.
From the discussion above, it can be seen that (i) One can get many zeroth-order solutions of nonlinear evolution equations by using F-expansion or Jacobi elliptic function expansion, which only related to the correlation chart of P, Q, R and the solution of P F 4 + QF 2 + R.
(ii) The form of the first-order equation is the same as that of the Lam equation. So one can get the first-order solution by solving the Lam equation. The form of the second-order equation is the same as the inhomogeneous Lam equation, and one can obtain the second-order solution by the particular solution of the inhomogeneous term.
(iii) One can obtain the degenerate solution by discussing the limit cases of the multi-order exact solutions. The method is valid to get the multi-order exact solutions of some other nonlinear evolution equations. At the same time, one can get many kinds of solitary wave solutions.
(iv) By the contraction mapping theorem, one can deduce that there exists a unique solution of the Cauchy problem of generalized Davey-Stewartson systems.