Doubly and Triply Periodic Waves Solutions for the KdV Equation *

Based on the arbitrary constant solution, a series of explicit doubly periodic solutions and triply periodic solutions for the Korteweg-de Vries (KdV) equation are first constructed with the aid of the Darboux transformation method.


Introduction
The famous KdV equation is a shallow water wave equation early derived by Korteweg de and Vries, its first application was discovered in the study of collision-free hydro-magnetic waves in 1960.Subsequently, it has arisen in a number of physical contexts, such as stratified internal waves, ionacoustic waves, plasma physics, lattice dynamics and so on.Following the further studies of these physical problems, its exact solutions have attracted much attention and have been extensively studied [1][2][3][4][5][6][7].However, in contrast to solitary wave solutions, the analytic periodic solutions represent only a small subclass of its known solutions, and multi-periodic solutions are scarce.It is always useful to seek more and various multi-periodic solutions for recovering interactions among some simple periodic waves in a nonlinear medium.We know that the Darboux transformation method is the main method to construct exact multi-soliton solutions, and this method is scarcely used for solving multiperiodic solutions [8][9][10].In the paper, not only explicit doubly periodic solutions are available, but also a group of explicit triply periodic solutions is obtained by means of the Darboux transformation method.

Doubly Periodic Solutions
According to [11], the linear system is the Lax pair for Equation (1), with the Darboux matrix where , are the spectral parameters.The monograph [11] further points out, if is a known solution to Equation (1), then becomes new solution generated from , with , , , , , , , , , , , , , is the fundamental solution matrix to the lax pair on .
i Only solving the fundamental solution matrix of the lax pair corresponding to constant solution 0 , it is possible to construct multi-periodic solutions to the KdV Equation (1).Substituting into the system (2) yields , then we can assert that both the system (6) and the following linear system 0 0 1 have exactly the same solutions.Under the condition for 0 u   , by the eigenvalue method, we obtain the complex-valued fundamental solution matrix to the above system e e , e e ia ia ia ia ia ia where Because the real and imaginary parts of a complex-valued solution are also solutions, we thus take as the fundamental solution matrix to the the system (6), where , , From (5), we have in the above formula, choosing respectively, with (4), the periodic wave solutions Now we construct the doubly periodic solutions generated from , thanks to (4), we see that we first give 1  , then substitute 0  and 1  into (11).
Again according to [11], we can obtain the fundamental solution matrix to the lax pair associated with the known periodic wave solution in the following manner where 1, 0 . After combining (5) and ( 12), choosing     , we get Substituting ( 9) and ( 13) into (11), we have new doubly periodic solution Again substituting ( 10) and ( 13) into (11), we obtain another new doubly periodic solution Similarly, choosing 1 1 0, 1 which implies the doubly periodic solutions and Specially, although 2 3 u  is a doubly periodic solution, its structure is very similar to a given two-soliton solution in [1].

Triply Periodic Solutions
As shown in [11], the fundamental solution matrix to the lax pair associated with the doubly periodic wave solution can be given by substituting ( 12) into (19), in exactly the same manner as in Section 2, we get Owing to ( 4) and ( 11), we have Here, we set tan and  into (20), we obtain triply periodic solution Similarly, we have