Classical Darboux transformation and exact soliton solutions of a two-component complex short pulse equation

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, we show that the mCSP equation is linked to a complex coupled dispersionless equation of defocusing type via a hodograph transformation, thus, its Lax pair can be deduced. Then the bilinearization of the defocusing mCSP equation is formulated via dependent variable and hodograph transformations. One- and two-dark soliton solutions are found by Hirota’s bilinear method and their properties are analyzed. It is shown that, depending on the parameters, the dark soliton solution can be either smoothed, cusponed or looped one. More specifically, the dark soliton tends to be evolved into a singular (cusponed or looped) one due to the increase of the spatial wave number in background plane waves and the increase of the depth of the trough. In the last part of the paper, we derive the defocusing mCSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the N-dark soliton solution in the form of determinants for the defocusing mCSP is provided.

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

A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients

In this paper, we propose a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrödinger (NLS) equation in the ultrashort-pulse regime. Furthermore, we construct the multi-dark-soliton solution for the defocusing complex short-pulse equation through the Darboux transformation and reciprocal (hodograph) transformation. One- and two-dark-soliton solutions are given explicitly, whose properties and dynamics are analyzed and illustrated.

Darboux transformation of two novel two-component generalized complex short pulse equations

Darboux transformation and general soliton solutions for the reverse space–time nonlocal short pulse equation

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Linget al. 2016Physica D327, 13–29 (doi:10.1016/j.physd.2016.03.012); Fenget al. 2016Phys. Rev. E93, 052227 (doi:10.1103/PhysRevE.93.052227)).

A vector general nonlinear Schrodinger equation with $$(m+n)$$ components is proposed, which is a new integrable generalization of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrodinger equation. Resorting to the Riccati equations associated with the Lax pair and the gauge transformations between the Lax pairs, a general N-fold Darboux transformation of the vector general nonlinear Schrodinger equation with $$(m+n)$$ components is constructed, which can be reduced directly to the classical N-fold Darboux transformation and the generalized Darboux transformation without taking limits. As an illustrative example, some exact solutions of the two-component general nonlinear Schrodinger equation are obtained by using the general Darboux transformation, including a first-order rogue-wave solution, a fourth-order rogue-wave solution, a breather solution, a breather–rogue-wave interaction, two solitons and the fission of a breather into two solitons. It is a very interesting phenomenon that, for all $$M>0$$, there exists a rogue-wave solution for the two-component general nonlinear Schrodinger equation such that the amplitude of the rogue wave is M times higher than its background wave.

Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions

The phenomena of the trapped Bose–Einstein condensates related to matter waves and nonlinear atom optics can be governed by a variable-coefficient Korteweg–de Vries (vc-KdV) model with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condensate, and such a model can also be used to describe the water waves propagating in a channel with an uneven bottom and/or deformed walls. In this paper, with the help of symbolic computation, the bilinear form for the vc-KdV model is obtained and some exact solitonic solutions including the N-solitonic solution in explicit form are derived through the extended Hirota method. We also derive the auto-B¨ acklund transformation, nonlinear superposition formula, Lax pairs and conservation laws of this model. Finally, the integrability of the variable-coefficient model and the characteristic of the nonlinear superposition formula are discussed.

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.

We study Darboux transformation for the short pulse (SP) equation and obtain its soliton solutions. We express the multisoliton solutions in terms of Wronski determinants. We also obtain an explicit expression of single soliton solution of the SP equation using the Darboux transformation.

On the exact soliton solutions for a class of coupled field equations

In this paper, a compound equation which is a mix of the Wadati–Konno–Ichikawa (WKI) equation and the short‐pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compound WKI–SP equation into the mKdV–SG (modified Korteweg–de Vries and sine‐Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the ‐loop soliton solution may be decomposed exactly into separate soliton elements by using a Moloney–Hodnett‐type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of and are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI–SP‐type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time‐varying coefficients and the corresponding dispersion relation will have a time‐dependent velocity. The whole hierarchy of equations which include the WKI‐type equations, the SP‐type equations, and the compound generalized WKI–SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as equation so that its Lax pairs can be directly written out with the help of and . A unified hodograph transformation is established to relate the compound WKI–SP hierarchy with the mKdV–SG hierarchy.

Dynamic analysis on optical pulses via modified PINNs: Soliton solutions, rogue waves and parameter discovery of the CQ-NLSE