Abstract
We first give a Bäcklund transformation from the KdV equation to a new nonlinear evolution equation. We then derive two Bäcklund transformations with two pseudopotentials, one of which is from the KdV equation to the new equation and the other from the new equation to itself. As applications, by applying our Bäcklund transformations to known solutions, we construct some novel solutions to the new equation.
1. Introduction
In 1895 two Dutch mathematicians Korteweg and de Vries derived a nonlinear wave equation which is now called KdV equation and adopts the canonical form [1]Since the KdV equation (1) possesses the solitary wave solutionwhere and are constants, it provides a theoretical confirmation of the existence of the solitary wave observed in 1834 by the Scottish engineer John Scott Russell on the Union canal. In 1965 Zabusky and Kruskal [2] discovered that the interaction of two solitary wave solutions is elastic, and therefore they called this kind of solutions solitons. In 1967 Gardner et al. [3] related the solution of the Cauchy initial value problem for the KdV equation (1) to the inverse scattering problem for a one-dimensional linear Schrödinger equation and derived the analytical expressions of -soliton solutions. The -soliton solutions to the KdV equation (1) can also be derived from the Darboux [4] and Bäcklund transformations [5], which have been extended and applied to a large variety of nonlinear evolution equations (see [6–13]).
It is well known that the KdV equation (1) is connected to the potential KdV equationvia . By use of this connection, Wahlquist and Estabrook [14] obtained the following Bäcklund transformation for the KdV equation (1), as well as for the potential KdV equation (3).
Theorem 1 (see [14]). If is a solution of the potential KdV equation (3), then the systemon is integrable, where is an arbitrary constant. Moreover, also satisfies (3). So the integrable system (4) defines a Bäcklund transformation for the potential KdV equation (3), and it also gives a Bäcklund transformation for the KdV equation (1) which is defined by
Note that (4) is a Lax pair of the potential KdV equation (3). In this paper, we first give another Lax pair of (3) and show that it defines a Bäcklund transformation from the KdV equation (1) to a new nonlinear evolution equation (7) (see Theorem 2). Then by combining the Bäcklund transformations given in Theorems 1 and 2, we derive two Bäcklund transformations defined by two pseudopotentials (see Theorem 8): one is from (1) to (7) (see (22)) and the other from (7) to itself (see (23)). As applications, by applying our Bäcklund transformations to known solutions of (1), (3), or (7), we construct some novel solutions to (7) (see Examples 6, 7, 11, and 12).
2. Bäcklund Transformation with One Pseudopotential
In this section, we prove the following result.
Theorem 2. If is a solution of (3), then the system on is integrable. Moreover, satisfies the nonlinear evolution equation
Proof. From (3) and (6) we haveTherefore ; that is, (6) is integrable.
From the first equation of (6) we haveSubstituting the above equations into the second equation of (6) yields that satisfiesand thereforeSo satisfies (7).
To our knowledge, (7) has not been reported in literature previously.
Remark 3. If is a solution of (7), thenis a solution of (3).
Remark 4. The integrable system (6) gives a Bäcklund transformation from the potential KdV equation (3) to (7)which is defined by the pseudopotential [15].
Remark 5. The integrable system (6) also defines a Bäcklund transformation with the pseudopotential :which takes a solution of the KdV equation (1) to a solution of (7).
Substituting a known solution of (3) into the integrable system (6), one can get a solution of (7). In the next two examples, from stationary and kink solutions of (3), we generate solutions of (7).
Example 6. Substituting the stationary solution of the potential KdV equation (3)into the integrable system (6) yields the following rational solution of (6),and therefore of (7)The profiles of the solution (17) at are shown in Figure 1. Note that, at , the solution curve has three pieces; as approaches , the piece located in the middle becomes smaller and smaller and finally disappears.
Example 7. Substituting the kink solution (see Figure 2) of the potential KdV equation (3)(where is an arbitrary nonzero constant) into the integrable system (6) yields the following solution of (6),and therefore of (7)Figure 3 illustrates the wave form of the solution (20) for .
3. Bäcklund Transformations with Two Pseudopotentials
By combining Theorems 1 and 2, we have the following result. The proof is obvious (see, for example, [16]).
Theorem 8. If is a solution of (3), then the system on and is integrable, where is an arbitrary constant. Moreover, satisfies (7).
Remark 9. The integrable system (21) gives a Bäcklund transformation from the KdV equation (1) to (7) as follows:which is defined by two pseudopotentials and [15].
Remark 10. The integrable system (21) also defines a Bäcklund transformation with two pseudopotentials and for (7) as follows:
Applying the Bäcklund transformation (22) or (23) to a known solution of (1) or of (7), it is usually not easy to get explicit formulas for and when solving the integrable system (21). But in this case, one can use an ordinary differential equation solver (see, for example, [13, 16]) to obtain numerical approximations of and . In the next two examples, we use this method.
Example 11. Applying the Bäcklund transformation (22) to the one-soliton solution (see Figure 4)of the KdV equation (1), where is an arbitrary nonzero constant, we generate solution of (7). Figure 5 consists of the wave forms of for with initial values and at (first row) and at (second row), which shows the evolution with time of this solution.
Example 12. Applying the Bäcklund transformation (23) to the rational solution (17) of (7), we generate new solution of (7). Figure 6 shows the profiles of for with initial values and at (first row), at (second row), and at (third row), which describes the evolution with time of this solution.
Competing Interests
The author declares that they have no competing interests.
Acknowledgments
This work is supported by the National Natural Science Foundation of China through 11571299 and the Natural Science Foundation of Jiangsu Province through BK20151304. The author would like to thank Dr. Lai Zhang from Umeå University for his help of plotting the figures.