Construction of soliton equations using special polynomials

Abstract

A simple, algorithmic approach is proposed for the construction of the most general family of equations of a given scaling weight, possessing, at least, the same single-soliton solution as a given, lower scaling weight equation. The construction exploits special polynomials–differential polynomials in the solution, u, of an evolution equation, which vanish identically when u is a single-soliton solution. Applying the approach to different types of evolution equations yields new results concerning the most general families of evolution equations in a given scaling weight, which possess solitary wave solutions. The same method can be applied in the identification of families of evolution equations of mixed scaling weight (and, in general, of any structure), which admit single-soliton solutions of a desired form.

Introduction

The literature devoted to the search for nonlinear evolution equations possessing solitary wave solutions is very rich. One important question has been the identification of families of equations admitting single- and multi-soliton solutions (in particular, integrable equations) within larger classes of differential equations, because, often, in physical applications, a general class of equations arises, rather than a single equation, e.g., in the realm of perturbations. One research direction has dealt with the search for equations, which emanate in higher scaling weights from a given equation and admit solitary wave solutions. Of these, the most popular example is the hierarchy of the symmetries of the Korteweg-de Vries (KdV) equation. In the latter case, the scaling weight is defined by assigning (∂_t,  ∂_x, u) the weights (3, 1, 2), respectively. With these assignments, the KdV equation,

$$u_t=u_3+6{\mathit{uu}}_1\text{,}\left(u_k\equiv {\partial }_x^{k}u\right)\text{,}$$

has scaling weight 5. Its single-soliton solution is given by

$$u(t\text{,}x)=\frac{A}{\cosh {\left[k(x+\mathit{vt})\right]}^2}\text{,}$$

with A = 2k², and v = 4k².

Consider the most general scaling-weight 7 equation:

$$u_t=u_5+a_1{\mathit{uu}}_3+a_2{u}_1{u}_2+a_3{u}^2{u}_1$$

(u₅ is not multiplied by a constant, as such a coefficient may be absorbed by re-scaling of t and a re-definition of a_i, 1  ⩽ i ⩽ 3). A particular integrable version of Eq. (3) is obtained when the differential polynomial on its right-hand side is the scaling-weight-7 symmetry of the KdV equation:

$$u_t=S_3^{\text{KdV}}[u]\text{.}$$

$S_3^{\text{KdV}}$ is given by [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]:

$$S_3^{\text{KdV}}[u]=u_5+10{\mathit{uu}}_3+20u_1{u}_2+30u^2{u}_1\text{.}$$

Eq. (4) admits the same soliton solutions of Eq. (1), modulo proper adaptation of soliton velocities. For example, in Eq. (2) one now has v =  (4k²)².

Consider now the most general scaling-weight 9 equation:

$$u_t=S[u]=a_1{u}^3{u}_1+a_2{u}^2{u}_3+a_3{\mathit{uu}}_1{u}_2+a_4(u_1{)}^3+a_5{\mathit{uu}}_5+a_6{u}_1{u}_4+a_7{u}_2{u}_3+u_7\text{.}$$

Again, a particular integrable version of Eq. (6) is

$$u_t=S_4^{\text{KdV}}[u]\text{.}$$

where $S_4^{\text{KdV}}$ is the scaling-weight-9 symmetry of the KdV equation:

$$S_4^{\text{KdV}}[u]=u_7+14{\mathit{uu}}_5+42u_1{u}_4+70u_2{u}_3+70u^2{u}_3+280{\mathit{uu}}_1{u}_2+70(u_1{)}^3+140u^3{u}_1\text{.}$$

Eq. (7) has the same soliton solutions as Eq. (1), modulo proper adaptation of soliton velocities.

However, the construction described above cannot provide a solution for the broader task of identifying all higher scaling-weight equations possessing solitary-wave solutions in the form of KdV solitons. For example, in scaling weight 7, even under the restriction to integrable equations that have soliton solutions of the same structure as those of the KdV equation, an evolution equation, which is not included in the hierarchy of symmetries of the KdV equation, exists-the integrable Sawada–Kotera (SK) equation [25]:

$$u_t=S_2^{\text{SK}}[u]=u_5+15{\mathit{uu}}_3+15u_1{u}_2+45u^2{u}_1\text{,}\left(u_k\equiv {\partial }_x^{k}u\right)\text{.}$$

Caudrey et al. [26] identified the SK equation as a member of a new hierarchy of independent evolution equations. The hierarchy emanates from the equation:

$$u_t+u_1+{\partial }_x{A}^{(n)}=0\text{,}{A}^{(n)}\equiv A^{(n)}(u\text{,}{u}_2\text{,}{u}_4\text{,}\dots \text{,}{u}_{2n})\text{,}$$

where A⁽ⁿ⁾ are differential polynomials in u. The members of the hierarchy are independent evolution equations of different scaling weights. The first two members in the hierarchy are the KdV (scaling weight 5) and SK (scaling weight 7) equations, and only those were found in [26] to have N-soliton solutions. In the next scaling weights (9 and 11), two new members of the hierarchy of Eq. (10) were found. However, it was only possible to show that they have 1-and 2-soliton solutions. For example, after re-scaling so as to comply with our notation, their scaling weight 9 equation reads:

$$u_t=S_2^{\text{CDG}}[u]=u_7+28{\mathit{uu}}_5+28u_1{u}_4+70u_2{u}_3+210u^2{u}_3+420{\mathit{uu}}_1{u}_2+420u^3{u}_1\text{.}$$

Modulo adaptation of soliton velocity to the scaling weight, Eqs. (9), (11) have the same single-soliton solution, Eq. (2), as the KdV equation. Their multiple-(N-) soliton solutions (N ⩾ 2 for Eq. (9), and N = 2 for Eq. (11)) can be obtained by the Hirota method [27]. They have the same structure as the multiple-solitons solutions of the KdV equation, except for the coefficients in Hirota’s form.

Other lines of research have led to higher-scaling-weight equations with soliton solutions that are not of the KdV type. Of this kind are solutions of the (scaling-weight-7) Kaup–Kupershmidt (KK) equation [28], [29], [30], [31],

$$u_t=S_2^{\text{KK}}[u]=u_5+30{\mathit{uu}}_3+75u_1{u}_2+180u^2{u}_1\text{.}$$

Eq. (12) is integrable, and has N-soliton solutions for any N ⩾ 1. Its single-soliton solution is:

$$u(t\text{,}x)=2k^2\frac{\left\{1+2\cosh \left(2k(x+16k^{4}t)\right)\right\}}{{\left\{2+\cosh \left(2k(x+16k^{4}t)\right)\right\}}^2}\text{.}$$

Although Eq. (13) differs in structure from the KdV-type soliton, Eq. (2), multiple-soliton solutions can still be obtained through the Hirota method [32].

Another kind of solitary wave solutions has been identified in [33] on equal footing with the KdV and KK solitons, but of a more general form. Expressing the solution, u, in terms of the “potential”, $p={\partial }_x^{-1}u$, one requires that the solution be a solitary wave, and imposes the condition that the equation u(p) = 0 has two roots, which corresponds to solutions that vanish at infinity. This procedure, applied to the higher-order KdV equations of a mixed scaling weight (5 + 7), leads to the “generalized KK” (GKK) solitons, the structure of which is given by

$$u(t\text{,}x)=4k^2\frac{\left(Q\cosh [2k(\xi +\delta )]+1\right)}{{\left(\cosh [2k(\xi +\delta )]+Q\right)}^2}(\xi =x+\mathit{vt})\text{.}$$

The remarkable feature of this form is that it contains the KK-single soliton solution of Eq. (13) and the KdV-soliton of Eq. (2) as special cases: The former corresponds to Q = 2 and the latter-to either Q = 1 (with wave number k) or Q = 0 (with wave number 2k). It has been found in [33] that Eq. (14) may represent solutions of several families of higher-order KdV equations of a mixed scaling weight (5 + 7) with single-soliton solution shapes that are different from the standard single-hump solution. Two-and three-GKK soliton solutions have been also constructed in [33]. A particular case of solutions belonging to one of the families can be obtained from the solution found in [34] for a specific wave number. Solutions of the form (14) with two discrete values of Q have been identified in [35] (and further studied in [36]) as solutions of an equation, which is not of the KdV type. Rather, it is a bidirectional extension of the KK equation of sixth-order in space and second order in time.

Through yet a different approach, the evaluation of functional derivatives of conserved quantities, Ito [37] found, in scaling weights 7 and 9 symmetries of the KdV, mKdV and SK equations, and new equations, the integrability of which was not ascertained.

An approach that is different from the ones reviewed above has been adopted in [38]. Instead of considering hierarchies of soliton equations stemming from a lower order integrable equations, the authors pose the question of finding all equations of a given scaling weight, which admit (at least single-) soliton solutions of the same structure as those of the original, lower scaling-weight equation. In [38], this problem is considered for the scaling-weight-7 equation, Eq. (3), with a restriction to KdV-like solitons of Eq. (2). Use of the methods of [38] in order to extend the results to other scaling weights and/or other types of solitary wave solutions, represents independent problems.

In the present study, we propose a simple, algorithmic approach for solving the problems of the kind considered in [38]. In our approach, the most general family of equations in a given scaling weight admitting the same single-soliton solution as a given, lower scaling weight equation, is constructed with the minimal number of necessary parameters. The universality and conceptual transparency of the approach allow us to solve the problem in a unified manner for different equations and for different types of soliton solutions. In particular, applying the approach to the most general scaling-weight-7 equation provides new insight for the results obtained in [38]. Applying the approach to other equations yields new results concerning the most general families of evolution equations in a given scaling weight, which possess solitary wave solutions. Finally, this approach may serve as a starting point for the search of integrable nonlinear evolution equations in higher scaling weights.

According to our approach, the most general family of evolution equations of higher scaling weights, which have the same single-soliton solution (modulo adaptation of soliton velocity to the scaling weight) as a given, low scaling-weight equation, may be represented (up to trivial rescaling) as a sum of a symmetry of the original evolution equation and a linear combination, with free coefficients, of all independent “local special polynomials” in the scaling weight considered. (Special polynomials are differential polynomials in u, the solution of an evolution equation, which vanish when u is a single-soliton solution [39]. They are local, if they only contain powers of u and of its spatial derivatives.) The number of independent local special polynomials determines the number of free parameters in the family.

The construction of special polynomials in a given scaling weight is an algorithmic process. One first writes the most general local differential polynomial as a combination of all possible monomials in powers of u and of its spatial derivatives in the scaling weight considered, with undetermined coefficients. Next, one requires that the polynomial vanish when u(t, x) is the single-soliton solution of the lower-scaling weight equation. Special polynomials exist if this requirement is satisfied for some values of the undetermined coefficients, or under some relations amongst these coefficients. If there is only one set of values of the coefficients satisfying that requirement, then only one local polynomial of the considered scaling weight exists. If the requirement that the polynomial vanishes when u(t, x) is the single-soliton solution cannot determine all the coefficients in the polynomial, then the number of undetermined coefficients is equal to the number of linearly independent local special polynomials (see Appendix A for details).

Before proceeding, the following remark is appropriate here. Single-soliton solutions depend on only one, traveling wave, argument. As special polynomials are defined through one-soliton solutions, one might think that it may be useful to convert higher-scaling weight PDE’s obtained through the use of special polynomials (as described above) into ODE’s, which are solved by the traveling wave solutions. However, special polynomials defined in terms of the travelling wave argument are useless; adding such special polynomials to an ODE produces only the ODE consisting of two (or more) parts, which vanish separately. On the other hand, a PDE obtained from the original PDE by adding special polynomials dependent on u(t, x) is useful both in the context of classification of soliton equations and, what is more important, in the search for equations possessing multi-soliton solutions for which reductions to ODEs are impossible (see discussion in Section 6).

We first study equations that generate the unidirectional solitary wave solutions of the KdV, KK and mKdV equations, and construct sequences of equations of higher scaling weights, which emanate from these equations. In the case of the sequence, which emanates from the KdV equation (scaling weight 5), in scaling-weight 7, there is only one local special polynomial [39] (see also the Appendix A). This naturally explains the results of [38] that the scaling-weight 7 equations, which have the same single-soliton solution as the KdV equation, form a single-parameter family. (Only solutions with a continuous spectrum of wave velocities, which are genuine counterparts of the soliton solutions of the KdV equation, are considered.) It also explains why, within a family, parameters of the solution (e.g., wave numbers and soliton velocities) do not depend on the equation parameters. In scaling weight 9, up to trivial rescaling, the most general family of scaling-weight 9 equations, which have the same single-KdV type soliton solution is a four-parameter family, because there are four linearly independent local special polynomials in this scaling weight [39]. In the case of the sequence, which emanates from the KK equation, Eq. (12), (scaling weight 7), there are two independent scaling-weight 9 local special polynomials, which vanish when computed for a single-KK soliton. Hence, the most general family of scaling-weight 9 equations that have the same KK single-soliton solution is a two-parameter family. In the case of the mKdV equation (scaling weight 4), the most general families of scaling-weight 6 and 8 equations that have the same mKdV single-soliton solution are, respectively, 5 and 13-parameters families.

Next, we show that the same methodology can be applied to evolution equations of conceptually different type, bidirectional equations, which include a second derivative with respect to time (instead of a first derivative). A hierarchy of independent bidirectional evolution equations, which emanates from the Boussinesq-like bidirectional KdV (bKdV) equation [40], [41]

$$u_{\mathit{tt}}-u_{\mathit{xx}}-{\partial }_x^2{A}^{(n)}=0\text{.}$$

(A⁽ⁿ⁾ are differential polynomials in u) was identified in [26]. The first two members in the hierarchy of Eq. (15) are the bKdV equation,

$$u_{\mathit{tt}}-u_{\mathit{xx}}-{\partial }_x{S}_2^{\text{KdV}}[u]=0\left(S_2^{\text{KdV}}[u]=6{\mathit{uu}}_1+u_3\right)\text{,}$$

and the bidirectional Sawada–Kotera (bSK) equation,

$$u_{\mathit{tt}}-u_{\mathit{xx}}-{\partial }_x{S}_2^{\text{SK}}[u]=0\text{.}$$

where $S_2^{\text{SK}}[u]$ is given in Eq. (9). In Eqs. (16), (17), the differential polynomials added to the wave equation have, respectively, scaling weights 6 and 8.

In scaling weights 10 and 12, new nonequivalent equations were obtained [26]. In this hierarchy, only the bKdV equation, Eq. (16) (also called the Boussinesq equation), is integrable, and has N-soliton solutions for any N. The higher members are only known to have one-and two-solitons solutions. In this paper, we study the families of higher order soliton equations, which emanate from the bKdV equation (scaling weight 6), Eq. (16). In scaling weights 8 and 10, the families that have the same single-soliton solutions as the bKdV equation are, respectively, 3-and 7-parameters families.

The construction of special polynomials is described in Appendix A; the case of the KdV equation is used as an example. In particular, we show that it is not possible to construct all the linearly independent special polynomials in a given scaling weight in terms of lower-scaling weight special polynomials.