Citation: Quting Chen, Yadong Shang. Direct similarity reductions and new exact solutions of the short pulse equation[J]. AIMS Mathematics, 2019, 4(2): 231-241. doi: 10.3934/math.2019.2.231
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The general short pulse equation (SPE) takes the form
uxt=αu+13β(u3)xx, | (1) |
where, u(x,t) represents the magnitude of electric field, while the subscripts t and x denote partial differentiations. The short pulse equation was derived by Schäfer and Wayne[1] as a model equation describing the propagation of ultr-short light pulses in silica optical fibres[1,2]. The short pulse equation represents an alternative approach in contrast with the slowly varying envelope approximation which leads to the nonlinear Schrödinger equation. The SPE is also known as the cubic Robelo equation that has been derived as an integrable differential equation associated with pseudo-spherical surfaces [3,4]. In last two decades, many researches have been carry out for the short pulse equation. The Lax pair of the SPE of the Wadati-Konno-Ichikawa type was rediscovered in [5]. The integrability, soliton solutions, bi-Hamiltonian structure, Lie symmetry analysis and conservation laws, and other features of the SPE have been investigated by many authors[6,7,8,9,10,11,12,13,14]. In recent years, the short pulse equation has been generalized in many different aspects [15,16,17,18,19,20,21,22,23,24,25,26,27,28]. For more new developments, please refer to [29,30,31] and the references therein.
It is well-known that finding the exact solutions of nonlinear partial differential equations plays an important role in the study of nonlinear physical systems. A wealth of methods have been developed to find these exact solutions of a partial differential equation. In the last decade, active research efforts have been made on the derivation of exact solutions for the SPE. In [9], the Lie symmetry analysis and the generalized method are performed for a short pulse equation. In [10], the invariance properties and conservation laws of the nonlinear short pulse equation are studied through Lie symmetry analysis. In [19], the complete group classification is performed on the extended short pulse equation. In 1989, Clarkson and Kruskal [32,33] presented a simple ansatz based method for seeking similarity reductions of nonlinear partial differential equations, which is referred to as the direct method. The direct method has been employed to investigate a number of nonlinear partial differential equations [34,35,36]. The great advantage of the direct method is that it does not invoke group theory. Several researchers have showed a close relationship between the symmetry group methods and the direct method [37,38].
The goal of this paper is find some new similarity reductions of the short pulse equation
uxt+u+2uu2x+u2uxx=0, | (2) |
by using the direct method due to Clarkson and Kruskal. Since this method does not involve group theory and the constant coefficients in equation can be easily handled, a large number of sophisticate calculations can be avoided. The results of the present paper show that the direct method is more than the Lie group similarity reduction method for symmetry similarity. One new traveling wave solutions are obtained by the method of undetermined coefficients.
The rest of this paper is organized as follows. Section 2 gives reductions of (2) using the direct method as well as the reduced ordinary differential equations. These reductions obtained in the present paper have a more general form than those by using the Lie group method. In addition, we also find a new similarity reduction. In section 3, one new traveling wave solutions will be given by the method of undetermined coefficients.
According to the direct method, we seek the similarity reduction of (2) in the form
u(x,t)=α(x,t)+β(x,t)w(z(x,t)), | (3) |
where z=z(x,t) is the similarity variable.
Substituting (3) into (2) and collecting coefficients of monomials of w and its derivatives yields
(βzxzt+α2βz2x)w″+(βzxt+βxzt+βtzx+α2βzxx+2α2βxzx+4αβαxzx)w′+(β+βxt+2βα2x+α2βxx+2αβαxx+4ααxβx)w+(2αβ2x+β2αxx+4βαxβx+2αββx)w2+(2ββ2x+β2βxx)w3+(β3zxx+6β2βxzx)w′w2+2β3z2xw′2w+β3z2xw″w2+2αβ2z2xw2+(2αβ2zxx+4β2αxzx+8αββxzx)w′w+2αβ2z2xw″w+(α+αxt+2αα2x+α2αxx)=0. | (4) |
In order that this equation (4) becomes an ordinary differential equation of for w(z) then the ratios of the coefficients of different derivatives and powers of w(z) have to be functions of z only. This gives an over-determined system of equations for α(x,t), β(x,t) and z(x,t), whose solutions yield the desired similarity solutions. Before doing this we will make some remarks about this direct method of seeking similarity reductions.
Remark 1. We will choose the coefficient of w″w2 as the normalizing coefficient and introduce a function Γ(z), where Γ is a function of z to be determined.
Remark 2. Whenever we use Γ(z) to denote a function to be determined, then it is a function, upon which we can perform any mathematical operation (e.g., differential, integration, logarithm, exponentiation, taking powers, rescaling, etc.) and then also call the resulting function Γ(z), without loss of generality.
In the determination of α, β, z, and Γ(z), there are following three remarks:
Remark Ⅰ: If α(x,t) is determined by α(x,t)=ˆα(x,t)+β(x,t)Γ(z), one can take Γ(z)≡0.
Remark Ⅱ: If β(x,t) has the form β(x,t)=ˆβ(x,t)Γ(z), one can choose Γ(z)≡1 by rescaling w(z)→w(z)Γ(z).
Remark Ⅲ: If z(x,t) is determined by an implicit form Γ(z)=ˆz(x,t), one can assume z=ˆz(x,t) by the transformation z→Γ−1(z).
The coefficients of w′2w and w″w yield the common constraint
β3z2xΓ(z)=αβ2z2x. | (5) |
By using Remark Ⅰ, we obtain Γ(z)≡0, and therefore
α≡0. | (6) |
Substituting (6) in equation (4), we have:
βzxztw″+(βzxt+βxzt+βtzx)w′+(β+βxt)w+(2ββ2x+β2βxx)w3+(β3zxx+6β2βxzx)w′w2+2β3z2xw′2w+β3z2xw″w2=0. | (7) |
To make an ordinary differential equation for w(z), we introduce functions Γn(z), n=1,2,…,5 in the following manner:
β3z2xΓ1(z)=βzxzt,β3z2xΓ2(z)=βzxt+βxzt+βtzx,β3z2xΓ3(z)=β+βxt,β3z2xΓ4(z)=2ββ2x+β2βxx,β3z2xΓ5(z)=β3zxx+6β2βxzx. |
In order to seek out much more similarity reductions of SPE, we consider the following two cases, respectively.
Case 1. βx=0:
Taking βx=0 in the above formulas yields
β3z2xΓ1(z)=βzxzt, | (8) |
β3z2xΓ2(z)=βzxt+βtzx, | (9) |
β3z2xΓ3(z)=β, | (10) |
Γ4(z)=0, | (11) |
β3z2xΓ5(z)=β3zxx. | (12) |
Taking into account Remark Ⅱ, from (10), we get
β=z−1x. | (13) |
Rewriting (12) and take notice of βx=0, we get
Γ5(z)=0. | (14) |
Due to (14) and (12), we have
z=θ(t)x+σ(t). | (15) |
Substituting (13) and (15) in (8), (9), we get
Γ1(z)=θ(t)[θ′(t)x+σ′(t)], | (16) |
Γ2(z)=0. | (17) |
(16) could be rewritten as
Γ1(z)=θ′(t)z+θ(t)σ′(t)−σ(t)θ′(t). | (18) |
In this case, (7) becomes
[θ′(t)z+θ(t)σ′(t)−σ(t)θ′(t)]w″(z)+w(z)+2w′2(z)w(z)+w″(z)w2(z)=0. | (19) |
In order to the equation (19) is an ordinary differential equation of w(z), θ(t) and σ(t) need to satisfy the system
θ′(t)=a1, | (20) |
θσ′(t)−σθ′(t)=c1, | (21) |
where a1, c1 are arbitrary constants.
Solving the one order ordinary differential equations (20) and (21) yields
θ=a1t+a2, | (22) |
σ=b1t+b2, | (23) |
where a1, a2, b1 and b2 are all arbitrary constants.
In summary, in this case we obtain the similarity transformation and similarity variables of the SPE (2) given by
u(x,t)=1a1t+a2w(z), | (24) |
z(x,t)=a1xt+a2x+b1t+b2, | (25) |
where w(z) satisfies
(a1z+a2b1−a1b2)w″+w+2w′2w+w″w2=0. | (26) |
Especially, setting a1=b2=0, a2=1 in (24) and (25), we obtain the traveling wave reduction
u=w(z),z=x+b1t, | (27) |
and the SPE can be reduced to the following ODE:
b1w″+w+2w′2w+w″w2=0. | (28) |
This similarity reduction is the same as the case (ⅰ) of Liu and Li [9] obtained by Lie symmetry analysis.
Taking a1=1, a2=b1=b2=0 in (24) and (25), we have the following similarity variables
u=1tw(z),z=xt, | (29) |
and the SPE is reduced to the following ODE:
zw″+w+2w′2w+w″w2=0. | (30) |
This similarity reduction is the same as the case (ⅱ) of Liu and Li [9] obtained by Lie symmetry analysis.
Compared with the literature [9], we can see that the similarity reduction obtained by the direct method here have more general form. It is shown that the similarity reduction of the SPE obtained by the Lie group method can be regarded as a special case of those obtained by using the direct method developed by Clarkson and Kruskal. Our result indicates that the direct method is more general than the Lie method for symmetry reductions.
Case 2. βx≠0:
In this case, we consider the original constraints:
β3z2xΓ1(z)=βzxzt, | (31) |
β3z2xΓ2(z)=βzxt+βxzt+βtzx, | (32) |
β3z2xΓ3(z)=β+βxt, | (33) |
β3z2xΓ4(z)=2ββ2x+β2βxx, | (34) |
β3z2xΓ5(z)=β3zxx+6β2βxzx. | (35) |
Applying Remark Ⅱ to (31), we obtain
β=(ztzx)12, | (36) |
and
Γ1(z)=1. | (37) |
Substituting (36) in (35), we arrive
zxΓ5(z)=3zxtzt−2zxxzx. | (38) |
Because Γ5(z) is a function of z, we can divide it into the sum of two functions of z as
Γ5(z)=Γ6(z)+Γ7(z), | (39) |
where Γ6(z) and Γ7(z) are functions to be determined. In this way, we can assume that
zxΓ6(z)=3zxtzt, | (40) |
zxΓ7(z)=−2zxxzx, | (41) |
which still satisfies (38).
Integrating (40) with respect to x once, then integrating with respect to t once and using the Remark 2 gives rise to
Γ6(z)=σ(x)+γ1(t), | (42) |
where σ(x) and γ1(t) are functions of integration.
Integrating (41) with respect to x twice and using the Remark 2 gives rise to
Γ7(z)=θ(t)x+γ2(t), | (43) |
where σ(x) and γ2(t) are functions of integration.
From (39), we have
Γ5(z)=θ(t)x+σ(x)+γ(t), | (44) |
where γ(t)=γ1(t)+γ2(t) is a function of integration.
By Remark Ⅲ, we have
z=θ(t)x+σ(x)+γ(t). | (45) |
Substituting (45) in (31)-(35), we obtain
Γ1(z)=1,Γ2(z)=θ′(θ′x+γ′)(θ+σ′)+θ″x+γ″2(θ′x+γ′)2−σ″2(θ+σ′)2,Γ3(z)=1(θ′x+γ′)(θ+σ′)−12(θ′x+γ′)2⋅[−θ″θ+σ′+θ′2+θ″σ″x+σ″γ″(θ+σ′)2−2θ′σ″(θ′x+γ′)(θ+σ′)3]−14(θ′x+γ′)3(θ+σ′)[θ′−σ″(θ′x+γ′)θ+σ′][θ″x+γ″θ+σ′−θ′(θ′x+γ′)(θ+σ′)2],Γ4(z)=5σ″24(θ+σ′)4+θ′24(θ′x+γ′)2(θ+σ′)2−12(θ′x+γ′)(θ+σ′)2(θ′σ‴x+3θ′σ″+σ‴γ′),Γ5(z)=3θ′(θ′x+γ′)(θ+σ′)−2σ″(θ+σ′)2. |
From Γn(z), n=1,2,…,5, we can see both zx=θ+σ′ and zt=θ′x+γ′ are indispensable in denominators, so we assume that
z=zxzt=θθ′x+θ′σ′x+σ′γ′+θγ′. | (46) |
Comparing (45) with (46), setting
θ′(t)[θ(t)+σ′(x)]=θ(t), | (47) |
σ′(x)γ′(t)=σ(x), | (48) |
θ(t)γ′(t)=γ(t). | (49) |
Taking notice of (49), we find that θ(t) cannot be equal to zero (otherwise, there will be γ(t)=0, so there will be z=σ(x) from (45), and therefore there will be β=0 from (36)). From (47) we find that
θ′(t)≠0, | (50) |
σ′(x)=c0, | (51) |
where c0 is a constant. Solving (51), yields
σ(x)=c0x+c3, | (52) |
where c3 is a constant. Substituting (52) into (48), yields
c0γ′(t)=c0x+c3. | (53) |
Due to (53), there will be c0=c3=0, so
σ(x)=0. | (54) |
Considering (47) and (50), then the one order ordinary differential equations (45) and (46) yields
θ(t)=t+c1, | (55) |
γ(t)=c2t+c1c2, | (56) |
where c1 and c2 are two arbitrary constants.
In this case, we obtain the similarity reduction of the SPE given by
u(x,t)=√x+c2t+c1w(z), | (57) |
z(x,t)=xt+c1x+c2t+c1c2, | (58) |
where c1 and c2 are two arbitrary constants, and w(z) satisfies
w″+1zw′+(1z−14z2)w+14z2w3+3zw′w2+2w′2w+w″w2=0. | (59) |
As far as our known, this is a new similarity reduction that have not been reported in any other paper by Lie symmetry analysis. It is shown that there exists exact solutions of the short pulse equation which can be obtained using the direct method developed by Clarkson and Kruskal, but which are not obtained by using the Lie symmetry similarity method. Our results indicate again that the direct method is more general than the Lie method for symmetry reductions.
In this section, we look for the traveling wave solutions of the SPE equation.
In section 2, we have got the similarity reduction u(x,t)=φ(ξ),ξ(x,t)=x+ct, where c>0 is the propagating wave velocity. And the traveling solution satisfies the following ordinary differential equation (ODE)
cφ″+φ+2φ′2φ+φ″φ2=0, | (60) |
where φ′=dφdξ. Let X=φ,Y=φ′, Eq. (60) can be converted to an equivalent planar system
X′=Y,Y′=−(1+2Y2)Xc+X2. | (61) |
Making the transformation
dξ=(b1+X2)dζ, | (62) |
system (61) becomes
dXdζ=(c+X2)Y, dYdζ=−X(1+2Y2). | (63) |
In the literature [6], the author considered the solitary wave solutions of the SPE. In the literature [7,8], Parkes obtained loop-soliton solutions and some periodic traveling wave solutions of the SPE. Liu and Li [9] solved the traveling wave equation by the bifurcation theory method of dynamical systems and obtained some traveling wave solutions. Xie, Hong and Gao in [11] investigated the periodic solution of the SPE by using the dynamical system theory. Ma and Li [14] got some new by Jacobi elliptic function solutions by a direct symbolic computation method combined with variable transformations. Next, we will get the traveling wave solution of the SPE (2) by the method of undetermined coefficients.
Now let us search for the solution of the SPE (2) in form of
u(x,t)=Ax+Bt+C, | (64) |
where A, B and C are three constants to be determined.
Substituting (64) into (2) yields
Ax+Bt+C+2A3x+2A2Bt+2A2C=0. | (65) |
Due to x and t are independent variables, so A, B and C satisfy a set of nonlinear algebraic equations
A+2A3=0, | (66) |
B+2A2B=0, | (67) |
C+2A2C=0. | (68) |
Solving this system of nonlinear algebraic equations, one obtain A=±1√2i, while B and C are arbitrary constants.
Then there is a solution of (2)
u(x,t)=±1√2(xi+Bt)+C, | (69) |
where B and C are arbitrary constants. Although this solution is simple, it has not been reported in any other literature. It is expected that this special solution can be applied for the numerical simulations.
In the present paper, we establish some similarity reductions of the short pulse equation(SPE) based on the direct similarity reduction method proposed by Clarkson and Kruskal. Some of these similarity reductions have a more general form than those obtained by using the Lie group method. Especially, we obtain some new similarity reductions which can not be obtained by Lie group symmetry analysis method. Finally, we derive one new exact analytic solutions which may be studied for the numerical simulations.
This work is supported the Program for the Innovation Research Grant for the postgraduates of Guangzhou University (NO:2017GDJC-D08). The authors wish to express their gratitude to the anonymous referees for the careful reading, the helpful suggestions and comments of the paper.
The authors declare no conflict of interest.
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