From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

Abstract

In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two-component extended Kadomtsew-Petviashvili (KP) hierarchy. As a by-product, the N -soliton solutions in the form of determinants for these equations are provided.

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

“…Such advantages can be observed in many analytical results related to the NLS equation, the complex short pulse equation and their coupled models. As is shown in [18,33], in contrast with the fact that one-soliton solution to the SP equation is always a loop soliton without physical meaning (1), the one-soliton solution to the CSP equation (2) is an envelope soliton with a few optical cycles. Compared to the SP equation, few results are known to the CSP equation (2).…”

Defocusing complex short-pulse equation and its multi-dark-soliton solution

“…Similar to the NLS equation, it is known that the complex-valued function has advantages in describing optical waves which have both the amplitude and phase information [1]. Following this spirit, one of the authors recently proposed a complex short pulse (CSP) equation [16,17] q xt + q + 1 2 |q| 2 q x x = 0.…”

Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation

“…Even though various solutions including bright soliton, dark soliton, breather, and rogue wave solutions to the focusing and defocusing CSP equation [1,5,6,16] were constructed, it is still intriguing to find all kinds of solutions to the coupled CSP equation, especially the one of mixed focusing and defocusing nonlinearity. In the last, the integrable discretizations for the CSP and the coupled CSP equations remain a topic to be explored in the future.…”

“…Equation (11) is also a reduction of a so-called vector sine-Gordon equation [36]. If Q = q s is a real-valued function, the complex sine-Gordon equation (11) It was pointed out in [5,7] that the focusing CCD system is linked to the focusing CSP equation by a hodograph (reciprocal) transformation, by which the Lax pair of the focusing CSP equation was established. Both the generalized coupled dispersionless (CD) equation and the focusing CSP equation were interpreted as the motion of space curve in Euclidean space [5].…”

“…Because the complex-valued function can contain the information of both amplitude and phase, it is more appropriate for the description of the optical waves [4]. It was shown that the CSP equation (1) is integrable in the sense that it admits a Lax pair and multisoliton solutions [1,[5][6][7].…”