# Generalizations of the short pulse equation

## Abstract

We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.

## Keywords

Short-wave limit Symmetries Recursion operator Lax pair## Mathematics Subject Classification

37K10 35G20## 1 Introduction

Short pulses and their properties are a subject of current interest in nonlinear optics and electrodynamics, both theoretically and experimentally. For instance, a rigorous justification of the short pulse equation, starting from a quasilinear Klein-Gordon equation (a toy model for Maxwell’s equations) was given in [25]. Moreover, for electrons accelerated in short laser pulses, it was shown recently that, due to quantum effects, the radiation reaction can be quenched by suitably tuning the pulse length, although the lengths required are currently out of experimental reach [15].

In this paper we are concerned with generalized short pulse equations of the form (1) from the viewpoint of integrability. The main result of the paper is the following.

### Theorem

### Remark

*a*,

*b*,

*c*, the derivative of (8), that is

*u*, and if it is rewritten as one, solving for \(u_t\), then it becomes nonlocal, involving the integration operator \(D_x^{-1}\). Physically, such equations appear in the description of the short-wave behavior of nonlinear systems. For example, the

*b*-family of equations

*u*, (10) is seen to be the

*x*derivative of an equation of the form (1). It is known from [22] that (for \(b\ne 0\)) the Eq. (9) is integrable, in the sense that it admits an infinite hierarchy of commuting symmetries, if and only if \(b=2\) (Camassa–Holm [3]) or \(b=3\) (Degasperis–Procesi [7]). Surprisingly, comparison with (2), (4) and (6) in the above theorem shows that in the short-wave limit, there are three integrable cases of Eq. (10): not only \(b=2\) (Hunter–Saxton [17]) and \(b=3\) (Vakhnenko [30]), but also the case \(b=3/2\), which appears to be new.

*u*and its derivatives only). This part of the proof requires the use of computer algebra, and further details are omitted. Once a finite list of equations has been obtained as above (by scaling \(c_j,d_j\) suitably), the remainder of the proof consists of explicitly constructing a recursion operator and associated infinite hierarchy of symmetries for each equation found. Thus, in the rest of the paper, we consider each equation on the list in turn, and for each one present the first higher symmetry, with flow variable \(\tau \), and a recursion operator \(\mathcal R\). The recursion operator is factored as \(\mathcal{R}=\mathcal{H}\mathcal{J}\), in terms of a compatible implectic–symplectic pair, with \(\mathcal{H}\) being a Hamiltonian operator such that the flow can be written as

Throughout the paper, subscripts with numbers are used to denote higher derivatives, so that \( u_{nx}=\frac{\partial ^n u}{\partial x^n}\) for \(n\ge 2\), but we also write, e.g., \(u_{xx}=u_{2x}\).

## 2 Properties of the generalized short pulse equations

### 2.1 Vakhnenko’s equation

*x*derivative of (2) arises as a short-wave, high-frequency limit of the Degasperis–Procesi equation. Sometimes (2) is also referred to as the reduced Ostrovsky equation [12, 13], since (up to rescaling dependent and independent variables) it is the special case \({\beta }= 0\) of the Ostrovksy equation

*Higher symmetry*The first higher symmetry of the Eq. (2) is

*Hamiltonian structure and recursion operator*In terms of the quantity

*Reciprocal transformation*Viewed as a short-wave limit of the Degasperis–Procesi equation, the

*x*derivative of (2) can be written in the form

*X*,

*T*by means of the reciprocal transformation

*W*to denote \(u_x\), then we have

*p*satisfies the Tzitzeica equation in the form

*Lax pair*In [32], a scalar Lax pair was presented for a reciprocally transformed version of (2), and in [18] this was used to obtain a \(3\times 3\) matrix Lax pair for the original equation, which is equivalent to the following Lax representation with spectral parameter \({\lambda }\):

### 2.2 The short pulse equation

The short pulse equation was first derived as an equation for pseudospherical surfaces with an associated inverse scattering problem [1, 26]. Its physical derivation in nonlinear optics came later [29] and led to the construction of alternative forms of the Lax pair, recursion operator and bi-Hamiltonian structure [2, 27].

*Higher symmetry*The first higher symmetry of the Eq. (3) is

*Hamiltonian structure and recursion operator*The above symmetry takes the Hamiltonian form

*Reciprocal transformation*The Eq. (3) has the conservation law

*X*,

*T*according to

*Lax pair*Equation (3) admits the Lax representation

### 2.3 Equation (4)

The Eq. (4) does not appear to have been considered before in the literature.

*Higher symmetry*The first higher symmetry of the Eq. (4) is

*Hamiltonian structure and recursion operator*Let \(w=(6 u_{2x}+1)^{-\frac{2}{3}}\). Then the symmetry (21) becomes

*w*, its symplectic operator \(\mathcal{J}\) has the same form as that for (2), being given by (13). Thus, the recursion operator \(\mathcal{R}=\mathcal{H}\mathcal{J}\) generates the symmetries for (4) and

*Reciprocal transformation*After rescaling

*u*and taking \(t\rightarrow -t\), the

*x*derivative of Eq. (4) can be rewritten in the form

*b*-family of peakon equations (9), with \(b=3/2\). The quantity \(m^{2/3}\) is a conserved density, and the conservation law

*Lax pair*Starting from a \(3\times 3\) Lax representation for the Tzitzeica equation, it is straightforward to obtain the following Lax representation for (23):

### 2.4 Equation (5)

To the best of our knowledge, the Eq. (5) has not been studied before.

*Higher symmetry*The first higher symmetry of the Eq. (5) is

*Hamiltonian structure and recursion operator*The above symmetry takes the form

*p*is given by

*Reciprocal transformation*The quantity

*p*in (26) is a conserved density for (5), with the conservation law

*X*derivative of the latter, using \(u_X=W/p\), and taking the difference of the two equations in (31), an equation for \(\psi \) alone results, namely

*Lax pair*The Eq. (5) has the \(3\times 3\) Lax representation

*F*is an arbitrary function. However, upon making a point transformation in

*X*, so that

*F*can be removed by choosing \(G(X) =\int (1+F(X))^{1/3}\,\mathrm {d}X\).

*V*is not specified a priori, then the compatibility conditions for the scalar linear system are

*V*in terms of \(\psi \) in the first equation of (37) and integrating produces

*T*, this becomes the Tzitzeica equation in the form (32).

### 2.5 The Hunter–Saxton equation

*x*derivative of the Hunter–Saxton equation corresponds to geodesic flow on an infinite-dimensional homogeneous space with constant positive curvature (see [21] and references).

*Higher symmetry*The first higher symmetry of the Eq. (6) is

*Hamiltonian structure and recursion operator*Notice that

*Reciprocal transformation*Considered as a short-wave limit of the Camassa–Holm equation, the

*x*derivative of (6) can be written in the form

*X*,

*T*according to

*p*in terms of

*m*. Now from the second equation in (41) and the definition of

*W*it follows that \((\log p)_T = 2pu_X=2W\), so that upon differentiating the latter with respect to

*X*and using the third equation to eliminate \(W_X\), an equation for

*p*alone results, namely

*Lax pair*A Lax pair for the Hunter–Saxton equation in the form (38) was found in [20]. For Eq. (6), with the inclusion of linear dispersion, a Lax representation is

### 2.6 The single-cycle pulse equation

The Eq. (7) was obtained recently by Sakovich [28] as a reduction of a coupled integrable short pulse system due to Feng [11]. Sakovich showed that the envelope soliton solution of (7) can only be as short as one cycle of its carrier frequency, and hence called it the single-cycle pulse equation.

*Higher symmetry*The first higher symmetry of the Eq. (7) is

*Hamiltonian structure and recursion operator*Notice that

*Reciprocal transformation*From the conservation law

*Lax pair*Using the inverse of the reciprocal transformation (43) to rewrite the Lax pair (45) in terms of the original independent variables

*x*,

*t*gives a Lax representation for Eq. (7), namely

### 2.7 Equation (8)

As noted in the remark above, the Eq. (8) combines the nonlinear terms from (6) and (7), but cannot be directly reduced to either equation.

*Higher symmetry*The first higher symmetry of the Eq. (8) is

*Hamiltonian structure and recursion operator*Similarly, to the previous case, we have

*Reciprocal transformation* In this subsection, we will make use of the higher symmetry above to show that the Eq. (8) has a reciprocal link to an equation of third order, given by (56) below, which is a symmetry of the Calogero–Degasperis–Fokas equation. For our purposes, it will be necessary to consider a solution \(u=u(x,t,\tau )\), which depends on the time \(\tau \) of the higher flow, in addition to *x*, *t*. Our ultimate goal will be to show that, by means of a further change of dependent and independent variables, the third-order equation we obtain can itself be reduced to the sine-Gordon equation.

*u*and

*p*are transformed to

*X*derivative of both sides, this yields

*X*and using \({P}_X=2{\beta }WW_X\) together with (54) and (55) leads to the third-order equation

*T*flow takes the form

*y*is given by (52) and

*T*so that \(\partial _T \rightarrow \sqrt{G(T)}\, \partial _T\), we see that \(\varphi =\mathrm {i}\log \eta \) satisfies the sine-Gordon equation in the form \(\varphi _{XT}+4\sin \varphi = 0\).

*Lax pair*In order to obtain a Lax pair for the Eq. (8), it is sufficient to rewrite (57) in terms of the original independent variables

*x*,

*t*. However, due to the dependence on

*p*, this does not directly produce matrices which are rational functions of the original field

*u*and its derivatives. To obtain a rational Lax pair, it is convenient to put (57) into scalar form and carry out a gauge transformation, which leads to the scalar linear system

*m*is given in (48),

*v*is as in (47), and

## 3 Conclusions

The list of integrable generalized short pulse equations appears to contain three new equations, namely (4), (5), and also (8), which combines the nonlinear terms of the Hunter–Saxton equation and the single-cycle pulse equation. All of the equations considered here are related by a reciprocal transformation to either the sine-Gordon equation or the Tzitzeica equation. However, although Eqs. (2), (4) and (5) are all related to the Tzitzeica equation, by comparing the expression for the differential \(\mathrm {d}X=p\, \mathrm {d}x+\cdots \) in each case, it is apparent that there are no direct links of Bäcklund type between these three equations, without changing the independent variable *x* via a hodograph-type transformation; the same remark applies to the Eqs. (3), (6) and (7). In the case of Eq. (8), the link to the sine-Gordon equation is rather indirect, and the equation that arises directly is the symmetry (56) of the Calogero–Degasperis–Fokas equation, which does not seem to have been considered before. These reciprocal links should be examined further, in order to derive explicit solutions of the new equations in parametric form. Since the reciprocal transformation is only defined for sufficiently smooth solutions, it is worth investigating situations where it breaks down: these equations may admit interesting weak solutions, e.g., distributions with non-empty singular support, as is the case for the *b*-family (9) mentioned above.

## Notes

### Acknowledgements

ANWH is supported by Fellowship EP/M004333/1 and JPW is partially supported by EP/P012698/1 from the Engineering and Physical Sciences Research Council (EPSRC). We are grateful to the referees for their comments on the original version of the article.

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