Optik - International Journal for Light and Electron Optics xxx (2016) xxx-xxx

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Original research article

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Electron Optics

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Symmetry reductions and conservation laws of the short pulse equation
K. Fakhar ⁠a⁠, ⁠b, Gangwei Wang ⁠c⁠, ⁠⁎, A.H. Kara ⁠d
a

Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Department of Mathematics, Faculty of Science and Horticulture, Kwantlen Polytechnic University, Surrey, British Columbia V3W 2MB, Canada
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China
d
School of Mathematics, University of the Witwatersrand, P Bag 3 Wits, Johannesburg 2050 South Africa
b
c

ABSTRACT

Article history:
Received 7 June 2016
Accepted 8 August 2016
Available online xxx

In this letter, we study invariance properties of the nonlinear short pulse equation through Lie symmetry
analysis. We show that this leads to several reductions yielding solutions of the short pulse equation. Furthermore, we obtain two conservation laws of the equation through the direct method. We show that two
resulting nonlocally related systems yield no nonlocal symmetries of the short pulse equation. Some remarks
and appropriate conclusions are drawn at the end.
© 2016 Published by Elsevier Ltd.

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Keywords:
Short pulse equation
Lie symmetries
Conservation laws
Exact solutions

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ARTICLE INFO

1. Introduction

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Although the nonlinear Schrödinger equation (NLSE) [1] forms the basis for optimizing existing fiber links, it is inadequate
for describing ultra-short pulses. This leads to the derivation of the short pulse equation, as a model equation, to approximate the
evolution of very short optical pulses in nonlinear media [2,3]. The short pulse equation (SPE) serves as a basic governing equation for many physical/mathematical models describing different processes in many scientific areas [4–7]. Several authors have
analyzed the SPE in different contexts by utilizing various methods and obtained diverse classes of solutions [8–19]. References
[11,15] have obtained traveling wave solutions. In [15], authors have used Lie symmetries: i.e., translation symmetries and did
detailed analysis, however, they only listed series type solutions through scaling symmetries and no analysis regarding combination of different symmetries, etc. Although extensive literature is available on the SPE, the quest to explore the nonlinearity of
the equation is still far from being complete. In the last few decades, active research efforts have been made on the derivation
of conservation laws for PDEs. Consequently, several methods such as Noether's theorem [20] for variational problems, multiplier approach (direct method) [21–24], symmetry action on a known conservation law [25], and partial Noether approach [26],
etc., have been employed to construct these. To the best of our knowledge, so far there has been just one attempt [27], to obtain
the conservation laws of SPE, which however is incomplete. Therefore, the purpose of the letter is to complete and extend the
study of SPE through Lie symmetry analysis and conservation laws. Here, we study the SPE through the classical Lie symmetry
approach [21,28–30], finding its conservation laws through the direct method as well as seeking nonlocal symmetries from nonlocally related systems arising from conservation laws. We consider the SPE given by [2]

Corresponding author.
Email address: pukai1121@163.com, wanggangwei@bit.edu.cn (G. Wang)

http://dx.doi.org/10.1016/j.ijleo.2016.08.013
0030-4026/© 2016 Published by Elsevier Ltd.

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Optik - International Journal for Light and Electron Optics xxx (2016) xxx-xxx

(1)

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which represents a model equation describing the propagation of ultra-short light pulses in silica optical fibers with α and β as
parameters. For the case β = 0, (1) reduces to the Klein–Gordon equation that possesses an infinite-dimensional Lie symmetry
algebra.
2. Reduction and exact solutions of SPE via Lie point symmetries

The non-zero commutators for the infinitesimal generators of the SPE are

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where [X⁠i, X⁠j] = X⁠iX⁠j − X⁠jX⁠i.
Remark 1
Using the relation

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The Lie point symmetries generators of SPE, obtained in [15], are

(2)

where ϵ is a real number and [, ] is the Lie bracket, we obtain the adjoint transformations and the following adjoint table from the
infinitesimal generators of the Lie algebra of (1).

X⁠1
X⁠1
X⁠2
X⁠3

X⁠2
X⁠1
X⁠1
e⁠ϵX⁠1

X⁠3

X⁠2
X⁠2
e⁠−ϵX⁠2

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Ad

−ϵX⁠1 + X⁠3
ϵX⁠2 + X⁠3
X⁠3

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Furthermore, from the commutators of the Lie point symmetries of (1) given through (2) and the adjoint representations of the
symmetry group on its Lie algebra given in the above table, we can obtain an optimal system of one-dimensional subalgebras [28]
given by {X⁠1, X⁠2 + δX⁠1, X⁠3}, where δ = 0, ± 1.
Now, we use the Lie symmetries of the SPE (1) as well as their linear combinations to obtain some reductions leading to invariant solutions of (1). We first use the combination of generators X⁠1 and X⁠2 that is, the obvious invariance of the SPE (1) under
translations in x and t. The change of variables
, where M is a constant, reduces (1) to the ODE
(3)

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where a prime (′) denotes d/dy. Now, (3) may be reduced further from its invariance under translation in y represented by the
symmetry generator Y⁠1 = (∂/∂y). Consequently, ODE (3) is reduced to the first order ODE

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which after some simplifications reduces to

which can be put in terms of the original variable as

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(4)

which after multiplication by t and then rearrangement can be written as

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where c⁠1, c⁠2 are constant of integrations and parameters α, β are the same as given in (1). The solution of (4) yields u as a function
of x and t. The traveling wave solutions obtained in [14] can be recovered from solution (4) by scaling both u and t, appropriately. Also, the arbitrariness of α and β in solution (4), offers the freedom of choosing any appropriate values of these parameters.
This fact will enable us to explicitly observe the effect of the nonlinear term given through β in (1). Fig. 1 presents a numerical
simulation for (4) of SPE (1), using Mathematica, where the parameter values are chosen as M = 1, α = 1, β = (1/2) with initial
conditions
.
Now, we consider the reduction of the SPE (1) through its invariance under scalings corresponding to the infinitesimal generator X⁠3. In particular (1), in terms of new variables y = xt, u = vx, reduces to the ODE

corresponding to an extreme nonlinear

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where D = (d/dy). Now we consider α = 0, for which we get the following first ODE for
situation:

(5)

(6)

where c⁠3 is a constant of integration. By using the special transformation

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(6) reduces to

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which further can be re-written as

and has the following solution

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where c⁠4 is a constant of integration. From (7) and (8), one gets the solution
SPE (1) for α = 0, which holds for any non-zero β.

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(7)

Fig. 1. Plot of the solution (3).

(8)
for (6) which ultimately leads to the solution of the

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has the solution

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which after simplification in terms of

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Moreover, in the special case when c⁠3 = 0 in (8), one obtains

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In term of the original variables of the SPE (1), Eq. (9) yields

(9)

(10)

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The bounded solution (10) of SPE (1) holds for any non-zero β. Furthermore, it has an obvious singularity as t → 0. Note that if
necessary, one can generalize the solution (10) by translating the variables t, x, i.e., t → t + A, x → x + B, to avoid a singularity in
a region of interest. Hence, the most general boundary conditions for (10) in a generic case will be

Now, consider reduction of the SPE (1) through a combination of generators X⁠1, X⁠2, X⁠3, in particular, X⁠1 + X⁠2 + X⁠3, though this
algebra is not in the optimal list. Under this algebra (1), in terms of invariance variables
, reduces
to the ODE

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which after rearrangement can be written as

(11)

where D = (d/dy). Now, we consider α = 0, for which (11) reduces to

. Thus in terms of its variables, the corresponding solution of the SPE (1) turns out to

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which has the particular solution
be

(12)

(13)

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The time dependent solution (13) is independent of β. In order to find a β-dependent solution of the SPE (1), we first note its
invariance under scalings, corresponding to its infinitesimal generator

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Correspondingly, we obtain the invariant solution of ODE (12) given by

In terms of the original variables of SPE (1), Eq. (14) yields the solution of (1) given by

(14)

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(15)

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The bounded solutions (13) and (15) become singular when t → 1. Again, this singularity, if necessary can be shifted by translating the variable t.
Now consider the differential invariants of the scaling symmetry given by

(16)

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In terms of these variables, Eq. (12) transforms to the first ODE

Approaches such as homotopy analysis and numerical techniques can be used to find solutions of (16) and hence by the process
of back substitutions to corresponding solutions to SPE (1).
3. Conservation laws of SPE

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Since the SPE (1) does not have a variational principle, it follows that the classical Noether's theorem cannot be used to obtain
conservation laws. We seek conservation laws using the direct method [22–24]. The multipliers and the conserved fluxes were
obtained through use of the GeM package [31]. Omitting details of the calculation, we directly obtain two conservation laws with
the flows given by
(i)

arising from multipliers Q⁠1 = u⁠t − βu⁠2u⁠x and

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(ii)

, respectively. The first conservation law is equiva-

lent to the conservation law obtained in [27], where as the second conservation is not obtained previously.

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4. Lie symmetries of SPE via nonlocally related systems

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The potential symmetry approach [22,32,33] provides a mechanism for seeking nonlocal symmetries of a PDE system from
conservation laws of the system. Each conservation law allows the introduction of auxiliary potential variables which are nonlocally defined with respect to the original dependent variables. A Lie symmetry of the resulting potential system yields a nonlocal
symmetry of a given PDE system provided its infinitesimals for the variables of the given PDE system have an essential dependence on potential variables. We start with the first conservation law. The procedure outlined in [22,32,33] yields the nonlocally
related system

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which has Lie symmetries

The second conservation law yields the nonlocally related system

Optik - International Journal for Light and Electron Optics xxx (2016) xxx-xxx

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with Lie symmetries

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Clearly, these symmetries correspond to Lie point symmetries of the SPE (1).
Remark 2
One can also use the symmetry-based method [34] to obtain a nonlocally related PDE system, which unlike the conservation
law-based method (as above) does not require the existence of a nontrivial local conservation law of a given PDE system.
5. SPE via double reduction

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The recently developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately to exact/invariant solutions for higher-order nonlinear partial differential
equations. The procedure is straightforward, and therefore we omit the details and directly write the results. For the details of the
results and the procedure, readers are strongly refereed to [35,36].
The possible associations between Lie point symmetries and conservation laws are presented in the form of table given below.

X⁠1
T⁠1
T⁠2

X⁠2

0
0

0
0

X⁠3
≠0
≠0

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It is evident from the table, that we can reduce SPE by combination of X⁠1 and X⁠2 on either of the conservation laws. The reduction
of SPE by combination of X⁠1 and X⁠2 on the first conservation law generates the following solution [36]
(17)

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Acknowledgements

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where c⁠5, c⁠6, c⁠7 are constant of integrations and parameters α, β are the same as given in (1). However, for the case of second
conservation laws, the reductions yields the trivial solution [35].
In summary, from the invariance of the short pulse equation under translations and scalings, we have obtained several reductions leading to a class of exact solutions of the SPE. The solution (4) is a traveling wave solution, for arbitrary values of its
parameters α and β. Due to the structure of the solution (4), previously known solutions can be recovered, by a suitable scaling
of u and t. The solutions (8) and (10), obtained through scaling symmetries, have not been obtained previously. Through a combination of translating and scaling symmetries, we have obtained previously unknown solutions (13) and (15). The solution (13)
is independent of β, whereas solution (15) holds for β > 0. Moreover, the reduced ODE in (16) can be further pursued by suitable
numerical techniques. We have also calculated conservation laws of the SPE, which could be useful for appropriate boundary
conditions for the SPE. Furthermore, we used the conservation law based method to construct nonlocally related systems (potential systems) of the SPE but the Lie symmetries of the constructed systems yielded no nonlocal symmetries of the SPE. Also, we
analyzed the SPE through double reduction procedure and able to obtain a solution (17), which could not be obtainable through
Lie symmetry method.

Authors like to thank Professor George Bluman at UBC for his useful comments and suggestions.
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