On the basis of the spectral analysis of the
4
×
4
Lax pair associated with the spin-1 Gross–Pitaevskii equation and the scattering matrix, the solution to the Cauchy problem of the spin-1 ...Gross–Pitaevskii equation is transformed into the solution to the corresponding Riemann–Hilbert problem. The Deift–Zhou nonlinear steepest descent method is extended to the Riemann–Hilbert problem, from which a model Riemann–Hilbert problem is established with the help of distinct factorizations of the jump matrix for the Riemann–Hilbert problem and a decomposition of the matrix-valued spectral function. Finally, the long-time asymptotics of the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is obtained.
A vector general nonlinear Schrödinger equation with
(
m
+
n
)
components is proposed, which is a new integrable generalization of the vector nonlinear Schrödinger equation and the vector derivative ...nonlinear Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger equation such that the amplitude of the rogue wave is
M
times higher than its background wave.
•We propose a new matrix YO equation associated with a (2m+n)×(2m+n)matrix spectral problem and derive its Lax pair.•Multi-fold generalized Darboux transformations for the matrix YO equation are ...constructed by using the gauge transformation between Lax pairs and their Riccati equations.•Based on the reduced technique, the Lax pair for the (n+1)-component YO equation and its multi-fold generalized Darboux transformations are obtained.•We obtain explicit solutions for the lower-order matrix YO equation, including soliton solutions, rogue-wave solutions, wave solutions that are constant along the x-direction, and wave solutions traveling at varying speeds.
Based on an introduced (2m+n)×(2m+n) matrix spectral problem, a matrix Yajima–Oikawa long-wave-short-wave resonance equation is proposed, which can be reduced to an (n+1)-component Yajima–Oikawa long-wave-short-wave resonance equation. Multi-fold generalized Darboux transformations for these two equations are constructed by using the gauge transformation between Lax pairs and their Riccati equations. Every solution to the Riccati equations can be transformed into a new solution of the matrix Yajima–Oikawa long-wave-short-wave resonance equation through the Darboux transformations. As an application of the obtained Darboux transformation, first, we derive high-order rogue wave solutions of the Yajima–Oikawa long-wave-short-wave resonance equation. Second, we obtain explicit solutions for the lower-order matrix Yajima–Oikawa long-wave-short-wave resonance equation, including soliton solutions, rogue wave solutions, wave solutions that are constant along the x-direction, and wave solutions traveling at varying speeds.
Based on the Baker-Akhiezer functions on Riemann surfaces, a general method is proposed to construct rogue-wave solutions and breather solutions of integrable nonlinear evolution equations associated ...with higher-order matrix spectral problems on theta-function periodic backgrounds. The derivative Yajima-Oikawa long wave-short wave equation associated with a 3×3 matrix Lax pair is chosen as an illustrative example of our method. First, a theta-function seed solution for the derivative Yajima-Oikawa long wave-short wave equation is derived by using the Weierstrass functions. Then, the general solution of the 3×3 matrix spectral problem and its derivatives with respect to the spectral parameter are obtained with the help of the Baker-Akhiezer functions. Finally, by means of Darboux transformations, explicit exact solutions such as rogue waves and breathers on theta-function backgrounds for the derivative Yajima-Oikawa long wave-short wave equation are constructed. Taking the advantage of the Jacobi θ-functions, these new solutions are numerically calculated and illustrated. In addition, by observing the explicit solutions, it is found that two rogue waves of different shapes may appear in one solution, which is a new phenomenon different from the solutions on constant backgrounds.
The initial value problem for the Sasa–Satsuma equation is transformed to a 3×3 matrix Riemann–Hilbert problem with the help of the corresponding Lax pair. Two distinct factorizations of the jump ...matrix and a decomposition of the vector-valued function ρ(k) are given, from which the long-time asymptotics for the Sasa–Satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method.
A Darboux transformation for an integrable generalization of the coupled nonlinear Schrödinger equation is derived with the help of the gauge transformation between the Lax pair. As a reduction, a ...Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation is obtained, from which some new solutions for the integrable generalization of the nonlinear Schrödinger equation are given.
On the basis of a
3
×
3
matrix spectral problem, a second-order three-wave interaction system is proposed. A one-fold Darboux transformation is derived by using the gauge transformation between the
3
...×
3
matrix spectral problems. The compact determinant form of the
N
-fold Darboux transformation is obtained by iterating the onefold Darboux transformation and solving a complex linear algebraic system. Furthermore, the
N
-fold Darboux transformation and Taylor expansion are used to construct multifold generalized Darboux transforms of the second-order three-wave interaction system. As applications, the solutions of the interaction between a dark-bright soliton and a rogue wave (RW), the solutions of the interaction between two dark-bright solitons and a second-order rogue wave, the solutions of the eye-shaped rogue wave and triangle rogue wave are obtained.
•The general Nth-order periodic solutions, Nth-order degenerate periodic and Nth-order rational solutions of the reduced Maxwell–Bloch equations are derived from the Darboux transformation.•Explicit ...expressions of these solutions from first to second order are presented.•Typical nonlinear wave patterns such as single-peak, double-peak-double-dip, double-peak and single-dip structures in the second-order rational solutions are shown.
We investigate the reduced Maxwell–Bloch (RMB) equations which describe the propagation of short optical pulses in dielectric materials with resonant non-degenerate transitions. The general Nth-order periodic solutions are provided by means of the Darboux transformation. The Nth-order degenerate periodic and Nth-order rational solutions containing several free parameters with compact determinant representations are derived from two different limiting cases of the obtained general periodic solutions, respectively. Explicit expressions of these solutions from first to second order are presented. Typical nonlinear wave patterns for the four components of the RMB equations such as single-peak, double-peak-double-dip, double-peak and single-dip structures in the second-order rational solutions are shown. This kind of the rational solutions correspond to rogue waves in the reduced Maxwell–Bloch equations.
In this study, multiple higher-order pole solutions of spinor Bose–Einstein condensates are explored by means of the inverse scattering transform, which are associated with different higher-order ...pole pairs of the transmission coefficient and give solutions to the spin-1 Gross–Pitaevskii equation. First, a direct scattering problem is introduced to map the initial data to the scattering data, which includes discrete spectrums, reflection coefficients, and a polynomial that replaces the normalized constants. In order to analyze symmetries and discrete spectra in the direct scattering problem, a generalized cross product is defined in four-dimensional vector Space. The inverse scattering problem is then characterized in terms of the
4
×
4
matrix Riemann–Hilbert problem that is subject to the residual conditions of these higher-order poles. In the reflectionless case, the Riemann–Hilbert problem can be converted into a linear algebraic system, which has a unique solution and allows us to explicitly obtain multiple higher-order pole solutions to the spin-1 Gross–Pitaevskii equation.
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