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

^{1}, Jia-Liang Ji

^{2}, Guo-Fu Yu

^{3}

^{, *}

^{1}Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, P. R. China

^{2}School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, 333 Longteng Road, Shanghai, 201620, P. R. China

^{3}School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

^{*}Corresponding author. Email: gfyu@sjtu.edu.cn

- DOI
- https://doi.org/10.2991/jnmp.k.200922.002How to use a DOI?
- Keywords
- Short pulse equation, coupled integrable dispersionless equations, sine-Gordon equation, AKNS
- Abstract
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.

- Copyright
- © 2020 The Authors. Published by Atlantis Press B.V.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. INTRODUCTION

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

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

*n*-component system

Here, *c _{jk}* are arbitrary constants with the symmetric relation

*c*=

_{jk}*c*. It is easy to check that when

_{kj}*n*= 2 and

*u*

_{1}=

*u*,

*u*

_{2}=

*v*with

*c*

_{11}=

*c*

_{22}= 0,

*c*

_{12}= 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

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]

*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

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],

*F*is given by (1.4). For the special case

*n*= 2 with

*u*

_{1}=

*u*,

*u*

_{2}=

*v*and

*c*

_{12}= 1,

*c*

_{11}=

*c*

_{22}= 0, the

*n*-component mSP (1.8) reduces to

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

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]

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

One can check that the compatibility condition *U _{s}* −

*V*+ [

_{y}*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),

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

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

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.

## 2. FROM THE GCD TO THE COUPLED SP EQUATIONS BY FENG

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

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

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

Similarly, from (1.21), we can derive

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

## 3. FROM THE MODIFIED CD EQUATIONS TO THE MODIFIED SP EQUATION

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

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

It is obvious that

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

## 4. THE LINK OF THE MOTIONS OF SPACE CURVES TO THE mCD AND mSP

Let ** γ** (

*y*,

*s*) : [0,

*l*] × [0,

*S*] →

*R*

^{3}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

*y*. The equation for the orthogonal triad

**n**,

**t**,

**b**along the curve takes the form

While the temporal evolution can be expressed as

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

Specially, if we choose

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

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 [*L _{i}*,

*L*] ↔ [

_{j}*e*,

_{i}*e*], where

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

One can check that the compatibility condition *P _{t}* −

*Q*+ [

_{x}*P*,

*Q*] = 0 gives the mSP equation (1.7).

## 5. THE RELATION BETWEEN THE MODIFIED CD AND sG EQUATION

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

*ϕ*(

*y*,

*s*) satisfies sG equation

By making the use of the trigonometric identity, we have

We can verify the mCD equation by direct calculation,

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

*ϕ*= sin

_{ys}*ϕ*with

Transformations (5.1) and (5.6) hint the substitution

With the substitution *v*^{2} = *ρ*, we find that above equations are reduction of AKNS(−1) equations [25].

## 6. K–K GENERAL CD EQUATIONS AND TWO-COMPONENT MODIFIED SP EQUATION

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

The K–K general CD equations read

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

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

*ρ*→ 1,

*u*→ 0,

*v*→ 0 as

*y*→ ∞, the above conserved quantity is identically zero. The conversion relation (6.6) gives

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

Eliminating *ρ* by use of (6.8) follows

Similarly, from Eq. (6.3) we can derive

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

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

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

Similar as the focusing case, we define the hodograph transformation

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

## 7. FROM THE MULTI-COMPONENT REAL AND COMPLEX CD TO THE MULTI-COMPONENT REAL AND COMPLEX SP/mSP

Matsuno proposed the multi-component SP equation [11]

*c*are arbitrary constants with the symmetry

_{jk}*c*=

_{jk}*c*. When the matrix

_{kj}Feng proposed the multi-component complex SP equation [4]

*σ*is a constant and

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

*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

Thus we have

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

By use of Eq. (7.8), we get

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

When potentials *u _{j}* 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.

## 8. DETERMINANT EXPRESSION FOR MULTI-SOLITON SOLUTIONS OF THE mSP AND THE TWO-COMPONENT mSP SYSTEM

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

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

### Proposition 8.1.

*By means of the dependent variable transformations*

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

*where D is the Hirota D-operator defined by*

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

### Proposition 8.2.

*By means of the dependent variable transformation*

*and the hodograph transformation*

*the bilinear equations (8.4) and (8.5) derive the mSP equation (1.7)*.

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

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:*

*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*

*with*ξ i = p i − 1 y + p i s + ξ i 0
, p,

_{i}*α*

_{i}and*ξ*

_{i}_{0}

*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*

*the K–K general CD equations (6.1)–(6.3) are transformed into the following bilinear equations [21]*

### Proposition 8.4.

*With the dependent variable transformation*

*and the hodograph (reciprocal) transformation*

*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*

*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*

*where*

*and*Φ, Ψ, Φ,

*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).

## 9. CONCLUSION AND DISCUSSION

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

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

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

*c*= −

_{jk}*c*By the variable transformation

_{kj}*u*=

_{i}*g*/

_{i}*f*, the coupled PDEs (9.6) are transformed to the bilinear equations

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.

## CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

## ACKNOWLEDGMENTS

We are thankful to the reviewers for the careful reading and suggestions that improved the manuscript. The work is supported by

## REFERENCES

### Cite this article

TY - JOUR 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 - https://doi.org/10.2991/jnmp.k.200922.002 DO - https://doi.org/10.2991/jnmp.k.200922.002 ID - Hu2020 ER -