arXiv subjects

Bao-Feng Feng

Publications and source records attributed to Bao-Feng Feng.

At least 19 recordsLinked to original sources

arXiv2026

Integrable full discretization of the multi-component short pulse equation

We propose a new formulation of the multi-component short pulse (MCSP) equation that includes the coupled complex short pulse (CCSP) equation as a reduction. Using Hirota's bilinear method, we construct its $N$-soliton solutions in Pfaffian form. We then derive integrable semi-discrete and fully discrete analogues of the MCSP equation admitting Pfaffian $N$-soliton solutions. The resulting fully discrete system provides a practical self-adaptive moving mesh scheme for numerical simulations. For the parameter sets considered, numerical simulations demonstrate excellent agreement between the numerical and exact solutions, confirming the robustness and high accuracy of the proposed scheme.

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arXiv2026

Soliton solutions to the coupled Sasa-Satsuma-mKdV equation

We consider the soliton solutions of a recently proposed coupled Sasa-Satsuma-mKdV equation using the Kadomtsev-Petviashvili reduction method. The system consists of a complex-valued component coupled with a real-valued one. Under zero or nonzero boundary conditions, we derive four distinct classes of soliton solutions: bright-bright, dark-dark, bright-dark, and dark-bright. These solutions are derived from the vector Hirota equation, for which the bright, dark, and bright-dark soliton solutions are provided in the Appendix. We perform asymptotic analysis of soliton collisions for each class of solutions, in which inelastic collisions are observed between bright-bright solitons. In the dark-dark case, we identify soliton profiles similar to the Sasa-Satsuma equation, including double-hole, Mexican hat, and anti-Mexican hat solutions; this study further explores the collisions between these structures and hyperbolic tangent shaped kink solitons. Regarding the bright-dark case, beyond the expected soliton-kink interactions, we report and analyze a notable collision occurring between kink solitons.

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arXiv2026

Soliton solutions to the coupled Sasa-Satsuma equation under mixed boundary conditions

In this paper, we derive general bright-dark soliton solutions to the coupled Sasa-Satsuma (CSS) equation using the Kadomtsev-Petviashvili (KP) reduction method. Since the CSS equation is a special case of the four-component Hirota equation, our approach begins with the construction of two-bright-two-dark soliton solutions for the four-component Hirota equation. By imposing specific parameter constraints, these solutions are subsequently reduced to the bright-dark soliton solutions of the CSS equation. Finally, the dynamical behaviors of the one- and two-bright-dark soliton solutions are thoroughly analyzed and illustrated.

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arXiv2025

Integrable discrete massive Thirring model

In this paper, we are concerned with integrable semi- and fully discrete analogues of the massive Thirring model in light core coordinates. By using the Hirota's bilinear approach and the KP reduction method, we propose both the semi- and fully discrete massive Thirring models and construct their multi-bright soliton solutions.

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arXiv2025

Integrable self-adaptive moving mesh schemes for multi-component short pulse type equations with nonzero boundary values

In this paper, we construct integrable self-adaptive moving mesh schemes for multi-component modified short pulse and short pulse equations with nonzero boundary values by using the consistency condition with the hodograph transformation. The essential point is that the edge point $x_{0}$ of the hodograph transformation cannot be kept fixed when the boundary flux is nonzero. We derive the evolution equation for $x_{0}$ and incorporate it into the semi-discrete moving mesh scheme. This supplies a moving-edge mechanism that extends the previously fixed-edge schemes and, in particular, allows periodic computations with nonzero boundary values. These schemes automatically adjust the mesh intervals according to the solution profile. We also derive multi-soliton solutions in Pfaffian form for the proposed schemes, which preserve the integrable structure in the discrete scheme. Numerical experiments for one- and two-soliton solutions demonstrate that the proposed schemes achieve high accuracy even in regions with rapid variation, while maintaining stability over long-time simulations, with small relative errors near peak amplitudes.

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arXiv2025

Pfaffian solution for dark-dark soliton to the coupled complex modified Korteweg-de Vries equation

In this paper, we study coupled complex modified Korteweg-de Vries (ccmKdV) equation by combining the Hirota's method and the Kadomtsev-Petviashvili (KP) reduction method. First, we show that the bilinear form of the ccmKdV equation under nonzero boundary condition is linked to the discrete BKP hierarchy through Miwa transformation. Based on this finding, we construct the dark-dark soliton solution in the pfaffian form. The dynamical behaviors for one- and two-soliton are analyzed and illustrated.

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arXiv2025

General soliton solutions to the coupled Hirota equation via the Kadomtsev-Petviashvili reduction

In this paper, we are concerned with various soliton solutions to the coupled Hirota equation, as well as to the Sasa-Satsuma equation which can be viewed as one reduction case of the coupled Hirota equation. First, we derive bright-bright, dark-dark, and bright-dark soliton solutions of the coupled Hirota equation by using the Kadomtsev-Petviashvili reduction method. Then, we present the bright and dark soliton solutions to the Sasa-Satsuma equation which are expressed by determinants of $N \times N$ instead of $2N \times 2N$ in the literature. The dynamics of first-, second-order solutions are investigated in detail. It is intriguing that, for the SS equation, the bright soliton for $N=1$ is also the soliton to the complex mKdV equation while the amplitude and velocity of dark soliton for $N=1$ are determined by the background plane wave. For $N=2$, the bright soliton can be classified into three types: oscillating, single-hump, and two-hump ones while the dark soliton can be classified into five types: dark (single-hole), anti-dark, Mexican hat, anti-Mexican hat and double-hole. Moreover, the types of bright solitons for the Sasa-Satsuma equation can be changed due to collision.

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arXiv2024

Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

In the present paper, an integrable semi-discretization of the modified Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev-Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by introducing dependent variables and discrete reciprocal transformation. It is also shown that the semi-discrete mCH equation converges to the continuous one in the continuum limit.

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arXiv2023

Integrable discretizations for a generalized sine-Gordon equation and the reductions to the sine-Gordon equation and the short pulse equation

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation $u_{t x}=\left(1+ν\partial_x^2\right) \sin u$. The bilinear equations of the discrete KP hierarchy and the proper definition of discrete hodograph transformations are the keys to the construction. Then we derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter $b\rightarrow0$. Especially, one full-discrete gsG equation is reduced to a semi-discrete gsG equation in the case of $ν=-1$ (Feng {\it et al. Numer. Algorithms} 2023). Furthermore, $N$-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are constructed. Dynamics of one- and two-soliton solutions for the discrete gsG equations are discussed with plots. We also investigate the reductions to the sine-Gordon (sG) equation and the short pulse (SP) equation. By introducing an important parameter $c$, we demonstrate that the gsG equation reduces to the sG equation and the SP equation, and the discrete gsG equation reduces to the discrete sG equation and the discrete SP equation, respectively, in the appropriate scaling limit. The limiting forms of the $N$-soliton solutions to the gsG equation also correspond to those of the sG equation and the SP equation.

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arXiv2022

Rogue waves and their patterns in the vector nonlinear Schrödinger equation

In this paper, we study the general rogue wave solutions and their patterns in the vector (or $M$-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev-Petviashvili hierarchy reduction method, we derived an explicit solution for the rogue wave expressed by $τ$ functions that are determinants of $K\times K$ block matrices ($K=1,2,\cdots, M$) with an index jump of $M+1$. Patterns of the rogue waves for $M=3,4$ and $K=1$ are thoroughly investigated. We find that when a specific internal parameter is large enough, the wave patterns are linked to the root structures of generalized Wronskian-Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system and many others. Moreover, the generalized Wronskian-Hermite polynomial hierarchy includes the Yablonskii-Vorob'ev polynomial hierarchy and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang {\it et al.} for the scalar NLS equation and the Manakov system. It is noted that the case $M=3$ displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of $M=3,4$. An excellent agreement is achieved.

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arXiv2022

Rogue waves in the massive Thirring model

In this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev-Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index- and two dimension-ones are proved to be consistent by only one constraint relation on parameters of tau-functions of the KP-Toda hierarchy.It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled integrable systems, the MT model only admits the rogue waves of bright-type, and the higher-order rogue waves represent the superposition of fundamental ones in which the non-reducible parameters determine the arrangement patterns of fundamental rogue waves. Particularly, the super rogue wave at each order can be achieved simply by setting all internal parameters to be zero, resulting in the amplitude of the sole huge peak of order $N$ being $2N+1$ times the background.Finally, rogue wave patterns are discussed when one of the internal parameters is large. Similar to other integrable equations, the patterns are shown to be associated with the root structures of the Yablonskii-Vorob'ev polynomial hierarchy through a linear transformation.

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arXiv2022

General rogue wave solutions to the Sasa-Satsuma equation

General rogue wave solutions to the Sasa-Satsuma equation are constructed by the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented in three different forms. The first form is expressed in terms of recursively defined differential operators while the second form shares a similar solution structure except that the differential operators are no longer recursively defined. Instead of using differential operators, the third form is expressed by Schur polynomials.

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arXiv2022

General rogue wave solutions to the discrete nonlinear Schrödinger equation

In the present paper, we attempt to construct both the general rogue wave solutions to the fully discrete nonlinear Schrödinger (fd-NLS) equation via the KP-Toda reduction method. First, we deduce the general breather solution of the fd-NLS equation starting from a pair of bilinear equations. We then derive the general rogue wave solution by taking a limit to the breather solution.

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arXiv2021

General bright and dark soliton solutions to the massive Thirring model via KP hierarchy reductions

In the present paper, we are concerned with the tau function and its connection with the Kadomtsev-Petviashvili (KP) theory for the massive Thirring (MT) model. First, we bilinearize the massive Thirring model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP-Toda hierarchy, we derive the multi-bright solution to the MT model by the KP hierarchy reductions. Then, we show that the discrete KP equation can generate a set of bilinear equations of a deformed KP-Toda hierarchy through Miwa transformation. By imposing constraints on the parameters of the tau function, the general dark soliton solution to the MT model is constructed from the tau function of the discrete KP equation. Finally, the dynamics and properties of one- and two-soliton for both the bright and dark cases are analyzed in details.

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arXiv2021

Multi-breather solutions to the Sasa-Satsuma equation

General breather solution to the Sasa-Satsuma (SS) equation is systematically investigated in this paper. We firstly transform the SS equation into a set of three Hirota bilinear equations under proper plane wave background. Starting from a specially arranged tau-function of the Kadomtsev-Petviashvili hierarchy and a set of eleven bilinear equations satisfied, we implement a series steps of reduction procedure, i.e., C-type reduction, dimension reduction and complex conjugate reduction, and reduce these eleven equations to three bilinear equations for the SS equation. Meanwhile, general breather solution to the SS equation is found in determinant of even order. The one- and two-breather solutions are calculated and analyzed in details.

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arXiv2021

Darboux transformation and solitonic solution to the coupled complex short pulse equation

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis to understand the soliton solution better. Breather and rogue wave solutions are constructed in case of non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution.

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arXiv2021

An integrable semi-discretization of the modified Camassa-Holm equation with linear dispersion term

In the present paper, we are concerned with integrable discretization of a modified Camassa-Holm equation with linear dispersion term. The key of the construction is the semi-discrete analogue for a set of bilinear equations of the modified Camassa-Holm equation. Firstly, we show that these bilinear equations and their determinant solutions either in Gram-type or Casorati-type can be reduced from the discrete KP equation through Miwa transformation. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solution in Gram-type or Casorati-type determinant form. Finally, by defining dependent variables and discrete hodograph transformations, we are able to derive an integrable semi-discrete analogue of the modified Camassa-Holm equation. It is also shown that the semi-discrete modified Camassa-Holm equation converges to the continuous one in the continuum limit.

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arXiv2019

A focusing and defocusing semi-discrete complex short pulse equation and its varioius soliton solutions

In this paper, we are concerned with a a semi-discrete complex short pulse (CSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz-Ladik (AL) lattice in the ultra-short pulse regime. By using a generalized Darboux transformation method, various solutions to this newly integrable semi-discrete equation are studied with both zero and nonzero boundary conditions. To be specific, for the focusing CSP equation, the multi-bright solution (zero boundary condition), multi-breather and high-order rogue wave solutions (nonzero boudanry conditions) are derived, while for the defocusing CSP equation with nonzero boundary condition, 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 (see Physica D, 327 13-29 and Phys. Rev. E 93 052227)

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