Abstract
We study the hierarchy commonly defined as an infinite sequence of partial differential equations which begins with the Korteweg–de Vries equation and its modified version. An important feature of the hierarchy is its highly nonlinear property. In this regard, obtaining solutions for the members of the hierarchy poses a great problem. In this paper, we propose a method to allow for the construction of solutions to the full hierarchy. Our approach involves a recursion operator in the conservation law of the hierarchy. The efficiency of the method is demonstrated by selected examples. In certain cases, we obtain snoidal solutions.
1. Introduction
The Korteweg–de Vries (KdV) equation,attributed to the Dutch mathematicians Diederik Korteweg and Gustav de Vries (1895), is a mathematical model of waves on shallow water surfaces whose origin has a long history. To this day, the KdV equation is still considered as one of the most important nonlinear partial differential equations as it has a wide variety of applications [1]. It was originally derived to model the propagation of weakly dispersive nonlinear water waves and serve as a model equation for any physical system in consideration [2]. In most contexts, is the function that denotes the elongation of the wave at space and time [3].
KdV equation (1) naturally extends to an infinite sequence of integrable nonlinear partial differential equations of solitonic characters [4] and can be considered as the initiator of the KdV hierarchy, denoted bywhere denotes the total derivative and denotes the integral with respect to . The second member of the hierarchy is given when in (2), viz.
Note that, for higher order , the equations become increasingly nonlinear and higher order in derivatives.
If the nonlinear term in equation (1) is replaced by , then the most important case other than when , is when . This yields the so-called modified KdV (mKdV) equation, given by [5, 6]which may also be connected to the mKdV hierarchy:
Here, the case of gives the second member of the mKdV hierarchy:
Another famous hierarchy is the Burgers’ hierarchy and its solutions [7, 8].
Completely integrable nonlinear equations, such as those of the KdV hierarchy, are endowed with many special mathematical properties. They are of interest due to their infinite conservation laws [9] and symmetries [10], bi- or tri-Hamiltonian structures, their Painlevé property [11], Lax pairs [12], etc. The solving of such equations is deeply connected to the inverse scattering transform [13, 14] and Hirota’s direct method [15]. The aforementioned importance of the KdV hierarchy has motivated this study.
The mKdV equations are related to the KdV equation through the Miura transformation, which maps the solutions of the KdV equations to the solutions of the mKdV equations [16].
The KdV hierarchies are an example of higher-order water wave models, which are of great significance, and they play crucial roles especially to study physical systems and the necessary material properties needed to manipulate waves in a desired manner [17].
In this paper, we propose a method to solve the full KdV hierarchy and extend the method to include the mKdV hierarchy. This is a novel proposal as, to the best of our knowledge, no such endeavour has appeared in the literature. The entire hierarchy is highly nonlinear and of higher order in derivatives, thereby posing an extremely challenging problem to solve. We formulate our method based on a transformation of variables (derived from the point symmetries of the hierarchy), effectively reducing the partial differential hierarchy into an ordinary differential hierarchy. The latter is connected to a transformed conservation law of the hierarchy, and the knowledge of this transformed conservation law forms a general approach to solving the hierarchy for all .
This paper is organised as follows. In Section 2, we briefly present some theoretical considerations, and Section 3 discusses some general properties of the hierarchies. Section 4 contains the main results and a description of our method, and Section 5 elaborates on some applications of our method. Section 6 provides some alternate solutions.
2. Preliminaries
The procedure for determining point symmetries for an arbitrary system of equations is well known [18]. Consider unknown functions which depend on independent variables , i.e., and , with indices and . Letbe a system of nonlinear differential equations, where represents the derivative of with respect to . We consider the following symmetry:given bywhere is extended to all derivatives appearing in the equation through an appropriate prolongation. A current is conserved if it satisfiesalong the solutions of the given equation. Equation (10) is called a local conservation law.
Suppose that is a symmetry of system (7) and is a conserved vector of (7). Then, if and satisfythe symmetry is said to be associated with [19]. Equation (11) is closely related to a Noether theorem, but in [19], it was proved that this result holds without the existence of Lagrangian. The transformation for the KdV equation enables the construction of a Lagrangian density [20] so that Noether’s theorem may be applied for its conservation laws. However, we have opted to study the KdV equation in the absence of Lagrangian.
3. Generalised Properties of the KdV and mKdV Hierarchy
A standard calculation of the symmetries of equation (1), using condition (9), reveals that it has the following four Lie point symmetries:
The second member of the hierarchy, with , or equation (3) has the following three symmetries:
If one repeats the Lie symmetry method for higher members, it is easy to see that hierarchy (2) possesses the Lie point symmetriesfor with Lie bracket relations , given in Table 1.
A similar investigation of mKdV hierarchy (5) gives that has the following three symmetries:and has the symmetries
As before, if one repeats the Lie symmetry method for higher members, it is easy to see that hierarchy (5) possesses the Lie point symmetriesfor with Lie bracket relations in Table 2.
As for the conservation laws of the above hierarchies, we notice several interesting properties. There exist many ways to compute conservation laws, and we opt for the multiplier approach [21].
Below are the cases of the KdV hierarchy when and , where the conservation laws are where . Equation (1) has the following three conservation laws:and finally,
These conservation laws can also be found via Noether’s theorem if the problem is reformulated to possess Lagrangian. The calculations are straightforward but tedious. Reports of these quantities or their equivalent appear in [20, 22], with more in [23].
Equation (3), i.e., , has the following two conservation laws:
As for the mKdV equation, equation (4) has the following two conservation laws:
Similarly, equation (6), , has the conservation laws
We now establish a result that is the foundation of our approach and that is the conservation law of each hierarchy. As we shall show, such a conservation law can be manipulated to solve the entire hierarchy for all values of . To begin, we establish the conserved vector of hierarchy (2) by the following theorem.
Theorem 1. The KdV hierarchy possesses the conserved vector along the solutions of equation (2), i.e., a component of the conserved vector admits a recursion operator.
Proof. Suppose the conservation law iswhere is the conserved density and is the conserved flux. Then, from equation (2), we havealong the solutions of equation (2), and the result follows.
Hence, the conserved density for every member of the hierarchy is , while the conserved flux is . The fluxes for the first few members of hierarchy (18) for and (21) for are confirmed by the above theorem.
Similarly, we can prove a result for the mKdV hierarchy.
Theorem 2. The mKdV hierarchy possesses the conserved vector along the solutions of equation (5), i.e., a component of the conserved vector admits a recursion operator.
One can easily check that the conserved vectors (24) for and (26) for arise from this theorem.
In the next section, we give a method to find solutions of the entire hierarchy.
4. A Method to Solve the Full Hierarchy-Type I Solutions
To proceed, we require a symmetry generator to be associated with a conserved vector of a given equation. Based on the previous section, (14), and (17), we notice that the symmetries and are possessed by the respective hierarchies for all . In particular, we observe that the point symmetry , when applied to condition (11), isand for the second symmetry,
Therefore, both symmetries satisfy the association condition, and we conclude that they are associated with every conservation law of Theorem 1, i.e., with any of the conservation law of the KdV hierarchy member. A similar result holds for the mKdV hierarchy, and here, we conclude that the same symmetries are associated with every conservation law of Theorem 2.
Next, we recall the fundamental theorem on double reduction [24, 25], which states that there exist functions such that
The transformed conserved quantity may be expressed aswhere and are similarity variables connected to an associated symmetry .
Since and are associated with the conserved vector , we consider the linear combination ( is a constant) to obtain the similarity transformationfor the KdV hierarchy, and similarly,for the mKdV hierarchy.
Therefore, we may establish the following results for .
Theorem 3. The conserved quantity of KdV hierarchy equation (2) can be reduced towhere .
Proof. Application of (32) gives usand in the new variables, by transformation (33), equation (36) transforms toAs examples, corresponding to of (18) is given byfor , and for (21) in the case of is
Theorem 4. The conserved quantity of mKdV hierarchy equation (5) can be reduced towhere , and the proof is similar to that of Theorem 3.
Also, to this end, examples of for the mKdV hierarchy include of (24) given byfor , and for of (26) is given byfor .
That is, the above results can be used to find for any value of , for both KdV and mKdV hierarchies. Based on equation (31), we have that , is a constant. Therefore, we have reduced the entire partial differential KdV and mKdV hierarchies to ordinary differential hierarchies. These ordinary differential hierarchies may then be solved for any .
5. Type I Solutions
In this section, we illustrate the applicability of the above method and theory in establishing solutions to members of the KdV and mKdV hierarchy. The solutions obtainable via our method in Section 4 will be referred to as type I solutions. Below, we set for simplicity.
5.1. The KdV Hierarchy
Let us consider in Theorem 3 to get the reduced conserved component (38); that is, we solve
We find that this equation has an implicit solution
Suppose we choose the free parameters to be ; then, the integral in (44) is evaluated to bewhere EllipticF is the incomplete elliptic integral of the first kind, and the solution to (43) becomesor in the original independent variables, by reversing transformation (33),where JacobiSN is an inverse of elliptic integrals and doubly periodic elliptic functions. These solutions appear graphically in Figure 1.
Next, we consider the second member of the hierarchy, , in Theorem 3 to get the reduced conserved component (39). That is, we solve
In this case, we find two solutions of the second member of the hierarchy, viz.or secondly,
5.2. The mKdV Hierarchy
This time, let us consider in Theorem 4 to get the reduced conserved component (41); that is, we solve
The solution of (51) is of two cases, namely,and secondly,
The latter may be expressed in original variables as
This solution has 2D and 3D plots in Figure 2.
6. Type II Solutions
As seen above, both hierarchies admit conservation laws, such as (19) or (22), independent of Theorems 1 and 2. Now, we cannot transcribe these conservation laws to theorems with a recursion operator as was done in Theorems 1 and 2. Nonetheless, a function may still be obtained in such cases, using the same formula (32) and transformation (33) or (35). This will lead to other solutions, which we call type II.
For example, in the KdV hierarchy, for of (19) is given byfor , and for of (22) is given byfor .
As for the mKdV hierarchy, for , we have for of (23) which is given byand for of (25) is given byfor . Below, we explore some solutions that arise out of these functions.
6.1. Type II Solution to the KdV Hierarchy
A type II solution corresponding to solving (55) yields an implicit solution
Here, the above integral is equal to
Suppose we let ; then, the explicit solution to (59) isor in original variables,
The progression of these solutions appears in Figure 3, and they are visibly periodic in nature.
6.2. Type II Solution to the mKdV Hierarchy
A type II solution corresponding to solving (57) has an implicit solution
The integral is evaluated to be
Suppose we let ; then, the solution to (63) isorwhose graphical representation appears in Figure 4.
7. Conclusions
In the study of differential equations, equations that are highly nonlinear and that possess higher-order derivatives are almost impossible to solve. We have proposed a scheme to overcome this problem and aid the solution of, in particular, the KdV and mKdV infinite hierarchy.
The well-known (solitary wave) solution of the KdV equation involves the hyperbolic secant function [26], but Korteweg and de Vries were interested in cnoidal solutions, expressible in terms of Jacobi’s elliptic CN functions [27]. Given the mathematical relations between the Jacobi SN and CN solutions, , sech, and tanh functions, our above solutions for KdV may be related to the known ones, but in that case, the recovery of known or related solutions validates our approach. As for our solutions for KdV , we find no connection to any known results. It is possible to find many more solutions.
In the analysis of our solutions, we divided our solutions to be of two types: type I and type II. Type I is the most interesting solution as it is derived from a recursion operator within the conservation law of the KdV and mKdV infinite hierarchy. In both solution types, the knowledge of association between symmetry and conserved components was exploited and formed the basis of our approach to reduce the order of the partial differential hierarchy to an ordinary differential hierarchy. Consequently, our method has many significant uses and can be extended to solve other infinite hierarchies. Specifically, it may be applied to any hierarchy in possession of a recursion operator, for example, the Kaup–Kupershmidt hierarchy. Furthermore, it would be interesting to attempt a study of systems of nonlinear equations with known recursion operators, such as the Hirota–Satsuma system or the nonlinear Schrödinger system of real equations.
An advantage of our approach is that it can easily be implemented into computer algebra programs such as Maple or Mathematica. A disadvantage is that, at higher-order members of the hierarchy, one may struggle to solve the reduced conservation laws, simply because the computations are too involved and computer algebra programs may run out of memory to complete the necessary calculations.
In the known literature, there are numerous methods to solve members of the KdV hierarchy, for example, the generalized Kudryashov method [28], the double Laplace transform [29, 30], the differential transform method [31], the tanh-expansion method [32], the exp-function method [33], and the -expansion method [34]. Our approach involves a recursion operator and conservation law to aid the analysis of the nonlinear partial differential hierarchy. To the best of our knowledge, this is the first time that a study has conceived an approach for dealing with the entire KdV hierarchy.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work received financial support from the National Research Foundation of South Africa (118 047).