Journal of Nonlinear Mathematical Physics

Volume 28, Issue 1, March 2021, Pages 14 - 26

On the Coupled Dispersionless-type Equations and the Short Pulse-type Equations

Juan Hu1, Jia-Liang Ji2, Guo-Fu Yu3, *
1Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, P. R. China
2School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, 333 Longteng Road, Shanghai, 201620, P. R. China
3School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
*Corresponding author. Email:
Corresponding Author
Guo-Fu Yu
Received 25 July 2019, Accepted 28 February 2020, Available Online 10 December 2020.
DOI to use a DOI?
Short pulse equation, coupled integrable dispersionless equations, sine-Gordon equation, AKNS

In this paper, we study the correspondence between the Coupled Dispersionless (CD)-type equations and the Short Pulse (SP)-type equations. From the real and complex modified CD equations, we construct the real and complex Modified Short Pulse (mSP) equations geometrically and algebraically. From the geometric point of view, we establish the link of the motions of space curves to the real and complex modified CD equations, then to the real and complex mSP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. By using hodograph transformation, we construct the two-component SP equation from the CD-type equations, the multi-component real and complex SP and mSP equations from the multi-component CD equations. The multi-soliton solutions in the determinant form for the mSP and two-component SP equations are provided.

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The Short Pulse (SP) equation was proposed by Schäfer and Wayne [16]

uxt=u+16(u3)xx, (1.1)
to describe the propagation of ultra-short optical pulses in nonlinear media [2]. Both the SP equation and the Nonlinear Schrödinger (NLS) equation are important in studying the dynamics of optical solitons in nonlinear optics. The efficiency and shortcoming of the SP compared with the NLS equation are explained in Feng [4]. The SP equation has received considerable attention in studies of the soliton theory and its integrable properties were investigated from various mathematical points of view, such as geometric meaning, soliton solutions and dynamics. The SP equation is integrable with Wadati-Konno-Ichikawa spectral problem [22], and it is also related to the Ablowitz–Kaup–Newell–Suger (AKNS) spectral problem under suitable hodograph transformation. It was shown that the SP equation can be transformed into the sine-Gordon (sG) equation through the appropriate hodograph transformation [15]. Recently, the links among the SP, AB system [5,13,19] and the first negative order AKNS(−1) system were clarified in Chen et al. [1] and nonlocal SP equation was also proposed therein.

Multi-component integrable SP equation was given when considering the effects of polarization or anisotropy. Matsuno proposed the two-component SP system [11]

uxt=u+12(uvux)x,vxt=v+12(uvvx)x, (1.2)
and more general n-component system
ui,xt=ui+12(Fui,x)x,i=1,2,,n, (1.3)
F=121j,kncjkujuk. (1.4)

Here, cjk are arbitrary constants with the symmetric relation cjk = ckj. It is easy to check that when n = 2 and u1 = u, u2 = v with c11 = c22 = 0, c12 = 1, the multi-component system (1.3) yields the two-component SP system (1.2).

Starting from the close relation between the SP and sG equation and using the Bäcklund transformation, Feng proposed an alternative integrable coupled SP system [3], which reads

uxt=u+16(u3)xx+12v2uxx,vxt=v+16(v3)xx+12u2vxx. (1.5)

When u = 0 or v = 0, the above coupled system is reduced to the SP equation (1.1). Note that the Matsuno’s two-component system (1.2) degenerates to the SP equation (1.1) when we identify v with u.

Recently, Sakovich considered the integrability of the following nonlinear PDE [14]

uxt=u+au2uxx+buux2, (1.6)
where a and b are arbitrary constants. When a/b = 1/2, Eq. (1.6) corresponds to the SP equation (1.1), whereas the case a/b = 1 yields, after rescaling the variable u, a so-called modified SP (mSP) equation
uxt=u+uux2+u2uxx. (1.7)

The mSP equation (1.7) can be derived directly from Feng’s coupled SP equation (1.5) by putting v = u and hence its integrability is assured.

Matsuno proposed an integrable multi-component generalization of the mSP equation [12],

ui,xt=ui+(Fui,x)x12(1j,kncjkuj,xuk,x)ui=0,i=1,2,,n, (1.8)
where F is given by (1.4). For the special case n = 2 with u1 = u, u2 = v and c12 = 1, c11 = c22 = 0, the n-component mSP (1.8) reduces to
uxt=u+v(uux)x,vxt=v+u(vvx)x. (1.9)

The further degeneration u = v yields the mSP equation (1.7). Especially, by imposing complex conjugate condition v* = u(≡q), the two-component mSP system (1.9) reduces to the so-called modified complex short pulse equation

qxt=q+q*(qqx)x. (1.10)

Similar to the complex short pulse equation proposed by Feng [4], that admits both the focusing and defocusing type, the modified complex short pulse equation (1.10) has also the defocusing type [18]

qxt=qq*(qqx)x. (1.11)

It was shown in Shen et al. [17] that the SP equation is closely related to the Coupled Dispersionless (CD) equations,

ρs+2uuy=0, (1.12)
uys=2ρu, (1.13)
proposed by Konno and Oono [9]. The CD equations are integrable with Lax pair
Ψy=UΨ,Ψs=VΨ, (1.14)
U=iλ(ρuyuyρ),V=(i2λuui2λ). (1.15)

One can check that the compatibility condition UsVy + [U, V] = 0 gives the CD equations. Soon after the proposition of the CD equations, it was pointed out that the CD equations were closely related to the sG equation [6]. The complex version of the CD equations were presented and solved by the inverse scattering method [8].

Shortly, Kakuhata and Konno [7] proposed a more general CD equations (K–K general CD for short),

ρs+(uv)y=0, (1.16)
uys=(2ρ1)u, (1.17)
vys=(2ρ1)v, (1.18)
with Lax pair
Ψy=UΨ,Ψs=VΨ, (1.19)
U=iλ(ρ12uyvyρ+12),V=(i2λuvi2λ). (1.20)

By using the idea to derive Feng’s coupled SP system, an integrable Generalized CD (GCD) equations were proposed [10],

uys4ρuv+(ρv)1rryuy=0, (1.21)
rys4ρrv+(ρv)1uuyry=0, (1.22)
ρs+v1uuy=0, (1.23)
vs+ρ1rry=0. (1.24)

When v = 1/2, r = 0, the GCD equations degenerate into the CD equations (1.12) and (1.13).

Upon the reduction r = u and v = ρ, we obtain from the above GCD equations

ρρs+uuy=0, (1.25)
uys4uρ2+ρ2uuy2=0, (1.26)
that could be called the modified CD (mCD) equations.

In this paper, we prove that there exists general correspondence between the SP- and CD-type equations. The rest of the paper is organized as follows. In Section 2, we show that the coupled SP equations proposed by Feng can be derived from the GCD equations under suitable hodograph transformation. In Section 3, we study the connection between the mCD and mSP equations. The link of the motions of space curves to the mCD and mSP equations is derived in Section 4. The relation between the mSP and sG equation is given in Section 5. Section 6 is devoted to the connection between the K–K general CD and two-component SP. We construct the multi-component real and complex SP/mSP from the multi-component CD in Section 7. Determinant expressions for multi-soliton solutions of the mSP and two-component mSP are given in Section 8. Finally, the concluding remarks and discussion are presented.


From Eqs. (1.23) and (1.24), we have the conservation law

(ρv)s+(r2+u22)y=0, (2.1)
that hints the hodograph transformation
dx=2ρvdy(r2+u2)ds,dt=2ds, (2.2)
or equivalently,
s=(r2+u2)x+2t,y=2ρvx. (2.3)

By using the above transformation and Eqs. (1.23) and (1.24), we have

(r2+u2)(qv)x=2(ρv)t+ρv(r2+u2)x. (2.4)

With Eq. (2.4) and the hodograph transformation (2.3), Eq. (1.22) is transformed into

uxt=u+12(r2+u2)uxx+uux2. (2.5)

Similarly, from (1.21), we can derive

rxt=r+12(r2+u2)rxx+rrx2. (2.6)

Thus we get Feng’s coupled SP equation (1.5) from the GCD equation (1.21)(1.24).


The first equation of the modified CD system represents a conservation law

(ρ2)s+(u2)y=0. (3.1)

We introduce the hodograph transformation (y, s) → (x, t) by

dx=2ρ2dy2u2ds,dt=2ds. (3.2)

It is obvious that

xy=2ρ2,xs=2u2, (3.3)
or equivalently,
s=2u2x+2t,y=2ρ2x. (3.4)

Substitution of (3.4) into Eq. (1.26) yields

2ρ2x(2u2x+2t)u4uρ2+ρ2u(2ρ2xu)2=0, (3.5)
or in the simplified form
uxtuuux2u2uxx=0, (3.6)
which is exactly the modified SP equation (1.7).


Let γ (y, s) : [0, l] × [0, S] → R3 be a family of smooth space curve parameterized by the arc length y ∈[0, l] at each time s. The unit tangent vector t(y, s), principal normal vector n(y, s) and binormal vector b(y, s) are defined by

respectively. Here ′ denotes the differentiation with respect to y. The equation for the orthogonal triad n, t, b along the curve takes the form
[tnb]y=[0kgknkg0τknτ0][tnb]. (4.1)

While the temporal evolution can be expressed as

[tnb]s=[0αβα0γβγ0][tnb]. (4.2)

The compatibility condition of (4.1) and (4.2) implies a more general class of integrable system

kg,s=αyknγ+τβ, (4.3)
kn,s=βykgγ+τα, (4.4)
τs=γy+kgβknα. (4.5)

Specially, if we choose

α=4u,β=0,γ=λ1, (4.6)
kg=0,kn=4λuy,τ=2λ(2ρ212uy2ρ2), (4.7)

Equation (4.3) holds automatically and Eqs. (4.4) and (4.5) become

uys4uρ2+ρ2uuy2=0, (4.8)
ρρs+uuy=0. (4.9)

Thus the link between the motion of space curves and the modified CD system is established. It is well known that the real Lorentz Lie algebra so(2, 1) in the plane is isomorphic to sl(2, R) and the correspondence reflects as [Li, Lj] ↔ [ei, ej], where

L1=(001000100),L2=(010100000),L3=(000001010), (4.10)
is the basis of so(2, 1) and
e1=12(0110),e2=12(0110),e3=12(1001). (4.11)
is the basis of sl(2, R). Based on this fact, we can construct the Lax pair for the modified CD equations geometrically as follows
Ψy=UΨ,Ψs=VΨ, (4.12)
U=kne1kge2+τe3=λ(2ρ2uy22ρ22uy2uyuy22ρ22ρ2), (4.13)
V=βe1αe2+γe3=(12λ2u2u12λ). (4.14)

By using the hodograph transformation between mCD and mSP equation, we can construct the Lax pair for the mSP equation (1.7) as

Ψx=PΨ,Ψt=QΨ, (4.15)
P=12ρ2U=λ(1ux22ux2uxux21), (4.16)
Q=12(V+u2ρ2U)=(14λ+λu2(1ux2)2λu2uxuu+2λu2ux14λλu2(1ux2)). (4.17)

One can check that the compatibility condition PtQx + [P, Q] = 0 gives the mSP equation (1.7).


It is known that we can solve the SP equation by use of solutions of the sG equation. Here we show that it is also true for the mSP case. We first introduce the variable transformation

v=12cos(ϕ2),u=14ϕs, (5.1)
and suppose the function ϕ (y, s) satisfies sG equation
ϕys=sinϕ. (5.2)

By making the use of the trigonometric identity, we have


We can verify the mCD equation by direct calculation,

(v2)s+(u2)y=18ϕssinϕ+18ϕsϕys=0, (5.3)
uys4uv2+uuy2v2=14[ϕscosϕϕscos2ϕ2+14ϕssin2ϕcos2ϕ2]=0. (5.4)

Thus, solutions of the sG equation give solutions of the mCD equations through the transformation (5.1). It was shown in Hirota and Tsujimoto [6] that CD equations

qs+2rry=0,rys2qr=0, (5.5)
were solved by the sG equation ϕys = sin ϕ with
q=12cosϕ,r=12ϕs. (5.6)

Transformations (5.1) and (5.6) hint the substitution

r=2u,q=4v212, (5.7)
that leads to the system
(v2)s+(u2)y=0, (5.8)
uys8uv2+u=0. (5.9)

With the substitution v2 = ρ, we find that above equations are reduction of AKNS(−1) equations [25].


6.1. From K–K General CD to Two-component mSP

The K–K general CD equations read

ρs+(uv)y=0, (6.1)
uys=(2ρ1)u, (6.2)
vys=(2ρ1)v. (6.3)

Equation (6.1) represents a conservation law and can be used to define a reciprocal transformation (y, s) → (x, t) as

dx=ρdyuvds,dt=ds, (6.4)
or equivalently,
y=ρx,s=uvx+t. (6.5)

Upon using (6.2) and (6.3), we have

ρs=(uyv+uvy)=12ρ1(uyvys+vyuys), (6.6)
which implies a conserved quantity
ρρ2uyvy (6.7)
for the K–K general CD equations. Under the boundary conditions ρ → 1, u → 0, v → 0 as y → ∞, the above conserved quantity is identically zero. The conversion relation (6.6) gives
ρ2uxvx=uyvy=ρρ2. (6.8)

Note that from the conversion relation (6.5), Eq. (6.2) can be rewritten into

ρx(utuvux)=(2ρ1)u. (6.9)

Eliminating ρ by use of (6.8) follows

uxt=u+v(ux2+uuxx). (6.10)

Similarly, from Eq. (6.3) we can derive

vxt=v+u(vx2+vvxx). (6.11)

Thus, the two-component mSP equations are derived from the K–K general CD equations. In the degenerated case u = v, we obtain the mSP equation from (6.1)(6.3) by the hodograph transformation (6.5).

6.2. From Complex Generalized CD Equations to Complex mSP Equation in Focusing Type

By setting the relation v = u* in the K–K general CD equations (6.1)(6.3), we get the complex generalized CD equations in focusing type

ρs+(|u|2)y=0, (6.12)
uys=(2ρ1)u. (6.13)

From the first equation (6.12) of the complex generalized CD equations, which stands for a conservation law, we can define a hodograph transformation

dx=ρdy|u|2ds,dt=ds, (6.14)
and it follows
y=ρx,s=|u|2x+t. (6.15)

Through above hodograph transformation, the complex focusing mSP equation (1.10) can be derived from the complex generalized CD equations (6.12) and (6.13).

6.3. From the Complex Generalized CD Equations to Complex mSP Equation in Defocusing Type

By setting v = −u* in K–K general CD equations (6.1)(6.3), we have complex generalized CD equations in defocusing type

ρs(|u|2)y=0, (6.16)
uys=(2ρ1)u. (6.17)

Similar as the focusing case, we define the hodograph transformation

dx=ρdy+|u|2ds,dt=ds, (6.18)
and it follows
y=ρx,s=|u|2x+t. (6.19)

Through above hodograph transformation, the complex defocusing mSP equation (1.11) is obtained from the complex generalized CD equations (6.16) and (6.17).


Matsuno proposed the multi-component SP equation [11]

ui,xt+ui+12[(1jkncjkujuk)ui,x]x=0,i=1,2,,n, (7.1)
where cjk are arbitrary constants with the symmetry cjk = ckj. When the matrix (cjk)j,k=1n is positive definite, we can recast (7.1) into
ui,xt+ui+12[(k=1nuk2)ui,x]x=0,i=1,2,,n. (7.2)

Feng proposed the multi-component complex SP equation [4]

uj,xt+uj+σ2(|u|2uj,x)x=0,j=1,2,,n, (7.3)
where the parameter σ is a constant and |u|2=j=1n|uj|2. From the complex CD equations
ρy+σ2(|u|2)s=0, (7.4)
uj,ys=ρuj,j=1,2,,n, (7.5)
we define a hodograph transformation
or equivalently,

Under the hodograph transformation, Eq. (7.5) was transformed into

ρx(uj,tσ2|u|2uj,x)=ρuj, (7.6)
which gives the multi-component complex SP equation (7.3). When u is real, we can modify our deduction and derive the real multi-component SP equation (7.2).

Based on the second form of the multi-component complex CD equations

ρy=(|u|2)s, (7.7)
uj,ys=(2ρ1)uj,j=1,2,,n, (7.8)
we define the hodograph transformation

Thus we have

ρs=(k=1n|uk|2)y=12ρ1(k=1nuk,yuk,ys*+uk,y*uys), (7.9)
that is

Under the zero boundary condition and the hodograph transformation, we obtain

ρρ2=k=1nuk,yuk,y*=ρ2k=1nuk,xuk,x*. (7.10)

By use of Eq. (7.8), we get

ρx[(t|u|2x)uj]=(2ρ1)uj. (7.11)

Eliminating ρ from (7.10) and (7.11) results in the multi-component complex modified SP equation

uj,xt=uj+(|u|2uj,x)x(k=1nuk,xuk,x*)uj. (7.12)

When potentials uj are real, the system (7.12) becomes the multi-component mSP proposed by Matsuno [12].

Remark: Matrix AKNS(−1) was given in Chen et al. [1], and the links between the vector AKNS(−1) and SP equation were presented therein.


By setting u = v in the K–K general CD equations (6.1)(6.3), we obtain

ρs+(u2)y=0, (8.1)
uys=(2ρ1)u. (8.2)

The bilinearization of the mCD equations is established by the following proposition.

Proposition 8.1.

By means of the dependent variable transformations

u=gf,ρ=1(lnf)ys, (8.3)

Equations (8.1) and (8.2) are transformed into the bilinear equations

Ds2ff=2g2, (8.4)
DyDsgf=fg, (8.5)
where D is the Hirota D-operator defined by
DsmDynfg=mɛmnδnf(s+ɛ,y+δ)g(sɛ,yδ)|ɛ=0,δ=0. (8.6)

The proof can be found in Zhang et al. [25].

Proposition 8.2.

By means of the dependent variable transformation

u=gf, (8.7)
and the hodograph transformation
x=y(lnf)s,t=s, (8.8)
the bilinear equations (8.4) and (8.5) derive the mSP equation (1.7).

Proof. From the hodograph transformation and bilinear equations, we have

xy=1(lnf)ys=ρ,xs=(lnf)ss=u2, (8.9)
which implies
y=ρx,s=u2x+t. (8.10)

Thus, the mSP equation is derived from the K–K general CD system based on the discussion in Section 6.1.

Remark: We emphasize that the SP and mSP are connected with the same bilinear form but different hodograph links.

Based on the reduction of the two-component KP hierarchy [17], we have the Gram determinant solution for the bilinear equations (8.4) and (8.5).

Theorem 8.1.

The bilinear equations (8.4) and (8.5) admit the following determinant solution:

f=|AIIB|,g=|AIΦTIB0T0C0|, (8.11)
where I is an N × N identity matrix, 0 is an N-component zero row vector, A and B are N × N matrices, Φ, C are N-component row vectors whose elements are defined as
aij=1pi+pjeξi+ξj,bij=αiαjpi+pj, (8.12)
Φ=(eξ1,eξ2,,eξN),C=(α1,α2,,αN), (8.13)
with ξi=pi1y+pis+ξi0 , pi, αi and ξi0 are constants.

The proof can be found in Shen et al. [17]. So we can express solutions of the mSP equation in the determinant form.

Proposition 8.3.

By means of the dependent variable transformations

u=gf,v=hf,ρ=1(lnf)ys, (8.14)
the K–K general CD equations (6.1)(6.3) are transformed into the following bilinear equations [21]
Ds2ff=2gh, (8.15)
DyDsfg=fg, (8.16)
DyDsfh=fh. (8.17)

Proposition 8.4.

With the dependent variable transformation

u=gf,v=hf, (8.18)
and the hodograph (reciprocal) transformation
x=y(lnf)s,t=s, (8.19)
the two-component mSP equation shares the same bilinear equations (8.15)(8.17).

The proof is similar as that of Proposition 2.

Theorem 8.2.

The bilinear equations (8.15)(8.17) admit the determinant solution

f=|AIIB|,g=|AIΦTIB0T0Ψ¯0|,h=|AI0TIBΨTΦ¯00|, (8.20)
where I is an N × N identity matrix, 0 is an N-component zero row vector, A and B are N × N matrices with entries
aij=1pi+p¯jeξi+ξ¯j,bij=αiα¯jp¯i+pj, (8.21)
and Φ, Ψ, Φ, Ψ¯ are N-component row vectors defined by

The proof can be completed by using the reduction technique from the extended KP hierarchy (see [17] for reference). So solutions of the two-component mSP system can be expressed in the compact determinant form. We note that Theorem 3.1 in Matsuno [11] gives different determinant form of the multi-soliton solutions to the bilinear equations (8.15)(8.17).


The correspondence between CD- and SP-type equations is established by using appropriate hodograph transformation. Especially, we present the link of the motions of space curves to the modified CD equations and mSP equation. Determinant solutions of the mSP and the two-component mSP equations are expressed by using Hirota method.

The so-called derivative CD equations

qt+1k<lMckl(rk,trlrkrl,t)=0, (9.1)
rk,xt2qxrk=0,k=1,2,,M (9.2)
were proposed [20]. Through the variable transformation
q=x(lnF)t,rk=Gk/F, (9.3)

Equations (9.1) and (9.2) are transformed into the system of bilinear equations

Dt2FF21k<lMcklDtGkGl=0, (9.4)
(DxDt2)FGi=0,i=1,2,,M. (9.5)

It would be interesting to find the correspondence to the SP-type equations. When considering Sp(m) invariant systems, the coupled system

ui,xt=ui1j<kncjk(uj,xukujuk,x)ui,i=1,2,,n, (9.6)
was proposed [24] with skew-symmetric coupling constants cjk = −ckj By the variable transformation ui = gi/f, the coupled PDEs (9.6) are transformed to the bilinear equations
DxDtfgi=fgi,(i=1,2,,n), (9.7)
DxDtff=1j<kncjkDxgjgk. (9.8)

Note that bilinear equations (9.7) and (9.8) are similar to (9.4) and (9.5). One question is whether there exists some link between the Sp(m) invariant system (9.6) and the derivative CD system (9.1) and (9.2). Besides, the integrable discretization procedure has been applied to CD [21], GCD [24] and mCD equations [26]. Since there exists the correspondence between CD-type equations and SP-type equations, it is possible to construct integrable discrete versions of the modified SP and multi-component mSP equations. We shall investigate these topics in future.


The authors declare they have no conflicts of interest.


We are thankful to the reviewers for the careful reading and suggestions that improved the manuscript. The work is supported by National Natural Science Foundation of China (Grant nos. 11871336, 11771395, 11901381).


[13]J Pedlosky, Finite-amplitude baroclinic wave packets, J. Atmos. Sci., Vol. 29, 1972, pp. 680-686.
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Cite this article

AU  - Juan Hu
AU  - Jia-Liang Ji
AU  - Guo-Fu Yu
PY  - 2020
DA  - 2020/12
TI  - On the Coupled Dispersionless-type Equations and the Short Pulse-type Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 14
EP  - 26
VL  - 28
IS  - 1
SN  - 1776-0852
UR  -
DO  -
ID  - Hu2020
ER  -