Abstract
The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and construct n-th Bäcklund transformation for this nKdV equation in the form of the determinants.
1. Introduction
It is well known that infinitely many symmetries or flows appear for finding evolution equations, and a general method was proposed by Olver, which was applied to the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV), Burgers, and sine-Gordon equations [1, 2]. Then, Olver’s concept of a recursion operator for symmetries of an evolution equation was extended to negative powers of the operator, and some negative-order Korteweg–de Vries (nKdV) equations were derived, including the following nKdV equation [3–12]:
Several years later, Lou reobtained the negative KdV hierarchy from the conformal invariance of its Schwartz form [13]. This new method used for the KdV equation can be extended to get more symmetries from known ones for arbitrary partial differential equations without using a recursion operator. Also, the theory can be used as a new criterion to verify whether a model is integrable or not. Based on the regular KdV system, Qiao originally studied nKdV equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions through bilinear Bäcklund transformations [14–17]. By using the simplified form of Hirota’s direct method, Wazwaz developed the nKdV equation and negative-order Kadomtsev–Petvishvili (nKP) equation in 2 + 1 dimensions and found multiple soliton solutions with free coefficients [18]. However, Kudryashov showed that existence of multisoliton solutions for the nonlinear evolution equation is the consequence of complete integrability [19, 20]. According to Theorem 1 in Reference [14], the nKdV equation (1) admits a Lax pair with the parameter λ as follows:and also possesses a Lax pair without the parameter as follows:
This paper is organized as follows: In Section 2, with the aid of the Lax pair, the residual symmetry of the nKdV equation (1) is derived, and this nonlocal symmetry is localized by introducing four auxiliary variables. Subsequently, we can obtain the finite symmetry transformation by solving the initial value problem. In Section 3, through localizing the linear superposition of multiple residual symmetries and constructing the infinite transformation for the nKdV equation, multiple residual symmetries and n-th Bäcklund transformation are obtained. A direct result shows that one can derive special soliton solutions from some seed solutions. A brief summary is given in Section 4.
2. Nonlocal Symmetry and Finite Transformation of the nKdV Equation
First, under the infinitesimal transformations and with the infinitesimal parameter ε, the symmetries and of the nKdV equation (1) can be expressed as a solution of their linear equations:which means that equation (1) is form invariant. At the same time, we can obtain the linear form of equation (2) under the infinitesimal transformation as follows:That is, the symmetric equations of the Lax pair (2) are
Second, supposing the symmetries and with the auxiliary variable ψ asthe symmetry can be derived from equations (1), (6a)-(6b), and (7) as follows:where the auxiliary variable ψ satisfies and the corresponding symmetric equation is where f is a auxiliary function and its infinitesimal transformation is The direct result is
So, equation (1) can be expressed by the auxiliary variable ψ through equation (2) as
The consistent condition of the auxiliary function f is derived from equation (10) as
Indeed, equation (11) has a typical Schwarzian form of the nKdV equation defined as follows:which is invariant under the Möbius transformation
The corresponding symmetric equation of equation (11) is
Since the nonlocal symmetry could not be used to construct the explicit solutions of a partial differential equation (PDE) directly, we need to transform these components to local ones. In this part, we will seek an enlarged system which possesses a Lie point symmetry for the nonlocal symmetry. For this purpose, we further introduce four auxiliary variables , and q, which need to obey the rule
The related symmetries areunder the infinitesimal transformations
It can be verified that equations (7)–(9), (14), and (16) have the following solution:
This is a local Lie point symmetry of the prolonged equations (1), (2), and (15) with .
Correspondingly, the initial value problem can be written as follows:
By solving this initial value problem, one can obtain the following finite transformation theorem.
Theorem 1. If is a solution of the extended system (1), (2), and (15), so iswhere ε is an arbitrary group parameter.
3. Bäcklund Transformation Related to Multiple Residual Symmetries
Considering the intrusion of the spectral parameter λ in the nonlocal symmetry of equation (7), we can derive infinitely many residual symmetries of the fields u and , that is,where are spectral functions of the Lax pair in equation (2) with different spectral parameters .
Just as the case of , to find the finite transformation of equation (20), we have to introduce a suitable prolonged system such that the symmetry can be localized to a Lie point symmetry. The corresponding finite transformation can be summarized as the following theorem.
Theorem 2. If is a solution of the enlarged systemthen the symmetry (20) is localized to a Lie point symmetry as follows:
Proof. The enlarged system (21a)–(21e) has the following linearized form:We first consider the special case, i.e., for any fixed while in equations (22a)–(22f). In this case, we obtain the localized symmetry for , and from equation (17) as follows:For , we eliminate through equation (21b) by taking and , respectively. Then, we haveSubstituting into equation (23b) with and vanishing through equation (25), we haveIt can be easily verified that equation (26) has the following solution:The symmetry for , and can be easily obtained from equation (23e) with :After taking the linear combination of the above results for all , Theorem 2 is proved.
When a nonlocal symmetry is localized to a Lie point symmetry, searching for its finite transformation is inevitable according to Lie’s first principle. For the Lie point symmetry (22a)–(22f), its initial value problem hasThen, one can get the following n-th Bäcklund theorem for the enlarged system (21a)–(21e) by solving (29a)–(29g).
Theorem 3. If is a solution of the prolonged nKdV equations (21a)–(21e), so is wherewhere , and are three determinants of the matrices , and respectively, which are defined as follows:
From Theorem 3, we can obtain an infinite number of new solutions from a suitable seed solution of the nKdV equation (1). Especially, one can obtain recursive soliton solutions of this system from the known one. For example, taking the seed solution for equation (1), it is not difficult to verify that equations (21a)–(21e) possess the following solution:
The corresponding first three multiple wave solutions for equation (1) arefor the line soliton solution:
For illustrating more details, the parameters are set as follows: Figure 1 displays the bell-like bright and dark solitons for the above condition of equations (34a)–(34b). Figure 1(a) shows a line dark soliton for with the amplitude 1, while Figure 1(b) shows a bright one for with the amplitude 2. Similarly, Figure 2 shows the collision of two-resonant solitons in equation (36), and Figure 3 shows interactions of three-resonant solitons expressed in equation (37).
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4. Summary
The nonlocal symmetry of the nKdV equation is obtained with the aid of its Lax pair. After introducing four auxiliary variables , h, p, and q, an enlarged system which possesses a Lie point symmetry for the nonlocal symmetry is taken. By applying Lie’s first theorem for the localized point symmetries, we obtain the corresponding finite transformation. Furthermore, we can localize the linear superposition of multiple residual symmetries and construct the infinite transformation for the nKdV equation. From Theorem 3, the n-th Bäcklund transformation can be expressed in a compact way of determinants. According to this conclusion, one can derive special soliton solutions from some seed solutions.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11775104). WC thanks the financial support from the High Level Talents Projects of Lishui City (Grant no. 2017RC16).