On the Coupled Dispersionless-type Equations and the Short Pulse-type Equations
- https://doi.org/10.2991/jnmp.k.200922.002How 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.
- © 2020 The Authors. Published by Atlantis Press B.V.
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The Short Pulse (SP) equation was proposed by Schäfer and Wayne 
Multi-component integrable SP equation was given when considering the effects of polarization or anisotropy. Matsuno proposed the two-component SP system 
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 , which reads
Recently, Sakovich considered the integrability of the following nonlinear PDE 
Matsuno proposed an integrable multi-component generalization of the mSP equation ,
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
It was shown in Shen et al.  that the SP equation is closely related to the Coupled Dispersionless (CD) equations,
One can check that the compatibility condition Us − Vy + [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 . The complex version of the CD equations were presented and solved by the inverse scattering method .
Shortly, Kakuhata and Konno  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 ,
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
Similarly, from (1.21), we can derive
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
4. THE LINK OF THE MOTIONS OF SPACE CURVES TO THE mCD AND mSP
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
While the temporal evolution can be expressed as
Specially, if we choose
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
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 Pt − Qx + [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
By making the use of the trigonometric identity, we have
We can verify the mCD equation by direct calculation,
With the substitution v2 = ρ, we find that above equations are reduction of AKNS(−1) equations .
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
Eliminating ρ by use of (6.8) follows
Similarly, from Eq. (6.3) we can derive
6.2. From Complex Generalized CD Equations to Complex mSP Equation 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
6.3. From the Complex Generalized CD Equations to Complex mSP Equation in Defocusing Type
Similar as the focusing case, we define the hodograph transformation
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 
Feng proposed the multi-component complex SP equation 
Under the hodograph transformation, Eq. (7.5) was transformed into
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
Remark: Matrix AKNS(−1) was given in Chen et al. , 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
The bilinearization of the mCD equations is established by the following proposition.
By means of the dependent variable transformations
The proof can be found in Zhang et al. .
By means of the dependent variable transformation
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.
The proof can be found in Shen et al. . So we can express solutions of the mSP equation in the determinant form.
By means of the dependent variable transformations
With the dependent variable transformation
The proof is similar as that of Proposition 2.
The proof can be completed by using the reduction technique from the extended KP hierarchy (see  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  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
It would be interesting to find the correspondence to the SP-type equations. When considering Sp(m) invariant systems, the coupled system
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 , GCD  and mCD equations . 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.
We are thankful to the reviewers for the careful reading and suggestions that improved the manuscript. The work is supported by
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 -