arXiv subjects

Bao-Feng Feng

Publications and source records attributed to Bao-Feng Feng.

At least 37 records · Page 2Linked to original sources

arXiv2018

Multi-breathers and high order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.

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arXiv2018

High-order rogue waves of a long wave-short wave model

The long wave-short wave model describes the interaction between the long wave and the short wave. Exact higher-order rational solution expressed by determinants is calculated via the Hirota's bilinear method and the KP hierarchy reduction. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate and dark ones, whereas the rogue wave for the long wave is always bright type. The higher-order rogue waves correspond to the superposition of fundamental rogue waves. The modulation instability analysis show that the condition of the baseband modulation instability where an unstable continuous-wave background corresponds to perturbations with infinitesimally small frequencies, coincides with the condition for the existence of rogue-wave solutions.

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arXiv2017

General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

General soliton solutions to a nonlocal nonlinear Schrödinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions {are considered} via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. First, general $N$-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.

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arXiv2017

Breather to the Yajima-Oikawa system

The Yajima-Oikawa (YO) system describes the resonant interaction between long and short waves under certain condition. In this paper, through the KP hierarchy reduction, we construct the breather solutions for the YO system in one- and two-dimensional cases. Similar to Akhmediev and Kuznetsov-Ma breather solutions (the wavenumber k_i->ik_i) for the nonlinear Schrodinger equation, is shown that the YO system have two kinds ofbreather solutions with the relations p_{2k-1}->ip_{2k-1}, p_{2k}->-ip_{2k}, q_{2k-1}->iq_{2k-1} and q_{2k}->-iq_{2k}, in which the homoclinic orbit and dark soliton solutions are two special cases respectively. Furthermore, taking the long wave limit, we derive the rational and rational and rational-exp solutions which contain lump, line rogue wave, soliton and their mixed cases. By considering the further reduction, such solutions can be reduced to one-dimensional YO system.

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arXiv2017

General high-order rogue waves of the (1+1)-dimensional Yajima-Oikawa system

General high-order rogue wave solutions for the (1+1)-dimensional Yajima-Oikawa (YO) system are derived by using Hirota's bilinear method and the KP-hierarchy reduction technique. These rogue wave solutions are presented in terms of determinants in which the elements are algebraic expressions. The dynamics of first and higher-order rogue wave are investigated in details for different values of the free parameters. It is shown that the fundamental (first-order) rogue waves can be classified into three different patterns: bright, intermediate and dark ones. The high-order rogue waves correspond to the superposition of fundamental rogue waves. Especially, compared with the nonlinear Schodinger equation, there exists an essential parameter αto control the pattern of rogue wave for both first- and high-order rogue waves since the YO system does not possess the Galilean invariance.

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arXiv2017

$N$-Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation

In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic behavior is analysed for two-soliton solutions.

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arXiv2017

Inverse scattering transform for the nonlocal reverse space-time Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations with nonzero boundary conditions

The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations were recently introduced and shown to be integrable infinite-dimensional dynamical systems. The inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, IST for these equations with nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complicated due to the associated branching structure of the associated linear eigenfunctions. With constant amplitude at infinity, four cases are analyzed; they correspond to two different signs of nonlinearity and two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1-soliton and 2-soliton solutions are found. In terms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations and the RST NLS equation is the time dependence of the scattering data. Spatially dependent boundary conditions are also briefly considered.

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arXiv2016

General soliton solutions to a coupled Fokas-Lenells equation

In this paper, we firstly establish the multi-Hamiltonian structure and infinite many conservation laws for the vector Kaup-Newell hierarchy of the positive and negative orders. The first nontrivial negative flow corresponds to a coupled Fokas-Lenells equation. By constructing a generalized Darboux transformation and using a limiting process, all kinds of one-soliton solutions are constructed including the bright-dark soliton, the dark-anti-dark soliton and the breather-like solutions. Furthermore, multi-bright and multi-dark soliton solutions are derived and their asymptotic behaviors are investigated.

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arXiv2016

Geometric formulation and multi-dark soliton solution to the defocusinig complex short pulse equation

In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space $\mathbf{R}^{2,1}$, then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the the defocusing CSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the $N$-dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.

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arXiv2016

A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

In the present paper, we propose a two-component generalization of the reduced Ostrovsky equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and $N$-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. Especially, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their $N$-soliton solutions in terms of pffafians are also provided.

nlin.SI↗

arXiv2016

A defocusing complex short pulse equation and its multi-dark soliton solution by Darboux transformation

In this paper, we propose a complex short pulse equation of both focusing and defocusing types, which governs the propagation of ultra-short pulses in nonlinear optical fibers. It can be viewed as an analogue of the nonlinear Schrödinger (NLS) equation in the ultra-short 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.

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arXiv2015

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

In the present paper, we are concerned with the general localized solutions for the complex short pulse equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the $N$-bright soliton solution in a compact determinant form, then the $N$-breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first- and second-order rogue wave solutions are given explicitly and illustrated by graphs. The asymptotic analysis is performed rigorously for both the $N$-soliton and the $N$-breather solutions. All three forms of the localized solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation is different from the one to the nonlinear Schrödinger (NLS) equation, which could be a cusponed- or a looped one.

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arXiv2015

An integrable semi-discrete Degasperis-Procesi equation

Based on our previous work to the Degasperis-Procesi equation (J. Phys. A 46 045205) and the integrable semi-discrete analogue of its short wave limit (J. Phys. A 48 135203), we derive an integrable semi-discrete Degasperis-Procesi equation by Hirota's bilinear method. Meanwhile, $N$-soliton solution to the semi-discrete Degasperis-Procesi equation is provided and proved. It is shown that the proposed semi-discrete Degasperis-Procesi equation, along with its $N$-soliton solution converge to ones of the original Degasperis-Procesi equation in the continuous limit.

nlin.SI↗

arXiv2015

An integrable semi-discretization of the coupled Yajima--Oikawa system

In the present paper, an integrable semi-discrete analogue of the one-dimensional coupled Yajima--Oikawa system, which is comprised of multicomponent short-wave and one component long-wave, is proposed by using Hirota's bilinear method. Based on the reductions of the Bäcklund transformations of the semi-discrete BKP hierarchy, both the bright and dark soliton (for the short-wave components) solutions in terms of pfaffians are constructed.

nlin.SI↗

arXiv2015

Multi-dark soliton solutions of the two-dimensional multi-component Yajima-Oikawa systems

We present a general form of multi-dark soliton solutions of two-dimensional multi-component soliton systems. Multi-dark soliton solutions of the two-dimensional (2D) and one-dimensional (1D) multi-component Yajima-Oikawa (YO) systems, which are often called the 2D and 1D multi-component long wave-short wave resonance interaction systems, are studied in detail. Taking the 2D coupled YO system with two short wave and one long wave components as an example, we derive the general $N$-dark-dark soliton solution in both the Gram type and Wronski type determinant forms for the 2D coupled YO system via the KP hierarchy reduction method. By imposing certain constraint conditions, the general $N$-dark-dark soliton solution of the 1D coupled YO system is further obtained. The dynamics of one dark-dark and two dark-dark solitons are analyzed in detail. In contrast with bright-bright soliton collisions, it is shown that dark-dark soliton collisions are elastic and there is no energy exchange among solitons in different components. Moreover, the dark-dark soliton bound states including the stationary and moving ones are discussed. For the stationary case, the bound states exist up to arbitrary order, whereas, for the moving case, only the two-soliton bound state is possible under the condition that the coefficients of nonlinear terms have opposite signs.

nlin.SI↗

arXiv2015

Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse equation

In the present paper, integrable semi-discrete and fully discrete analogues of a coupled short pulse (CSP) equation are constructed. The key of the construction is the bilinear forms and determinant structure of solutions of the CSP equation. We also construct Nsoliton solutions for the semi-discrete and fully discrete analogues of the CSP equations in the form of Casorati determinant. In the continuous limit, we show that the fully discrete CSP equation converges to the semi-discrete CSP equation, then further to the continuous CSP equation. Moreover, the integrable semi-discretization of the CSP equation is used as a selfadaptive moving mesh method for numerical simulations. The numerical results agree with the analytical results very well.

nlin.SI↗

arXiv2015

Multi-soliton solution to the two-component Hunter-Saxton equation

In this paper, we study the bilinear form and the general N-soliton solution for a two-component Hunter-Saxton (2-HS) equation, which is the short wave limit of a twocomponent Camassa-Holm equation. By defining a hodograph transformation based on a conservation law and appropriate dependent variable transformations, we propose a set of bilinear equations which yields the 2-HS equation. Furthermore, we construct the N-soliton solution to the 2-HS equation based on the tau functions of an extended two-dimensional Toda-lattice hierarchy through reductions. One- and two-soliton solutions are calculated and analyzed.

nlin.SI↗

arXiv2015

General Mixed Multi-Soliton Solutions to One-Dimensional Multicomponent Yajima-Oikawa System

In this paper, we derive a general mixed (bright-dark) multi-soliton solution to a one-dimensional multicomponent Yajima-Oikawa (YO) system, i.e., the (M+1)-component YO system comprised of M-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients including positive, negative and mixed types. With the help of the KP-hierarchy reduction method, we firstly construct two types of general mixed N-soliton solution (two-bright-one-dark soliton and one-bright-two-dark one for SW components) to the (3+1)-component YO system in detail. Then by extending the corresponding analysis to the (M+1)-component YO system, a general mixed N-soliton solution in Gram determinant form is obtained. The expression of the mixed soliton solution also contains the general all bright and all dark N-soliton solution as special cases. Besides, the dynamical analysis shows that the inelastic collision can only take place among SW components when at least two SW components have bright solitons in mixed type soliton solution. Whereas, the dark solitons in SW components and the bright soliton in LW component always undergo usual elastic collision.

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