arXiv subjects

Guangxiong Zhang

Publications and source records attributed to Guangxiong Zhang.

5 recordsLinked to original sources

arXiv2026

Vector rogue wave patterns associated with generalized Hermite and Okamoto polynomials

We establish rogue wave patterns associated with the fourth Painlevé equation $(\mathrm{P}_{\mathrm{IV}})$ in the multi-component nonlinear Schrödinger and Hirota equations. The generalized Hermite and generalized Okamoto polynomials arise in representations of rational solutions of $\mathrm{P}_{\mathrm{IV}}$, and we show that their roots determine two classes of rogue wave patterns when one of the internal parameters of rogue wave solutions is large. Specifically, the generalized Hermite polynomials arise from rogue wave solutions represented by Schur-polynomial determinants with consecutive indices, whereas the generalized Okamoto polynomials arise from analogous determinants with index jumps of three. Numerical examples for both equations agree with the predictions.

nlin.SI↗

arXiv2026

Zeros of the generalized Wronskian-Hermite polynomials

In this paper, we study generalized Wronskian-Hermite (WH) polynomials associated with arithmetic-progression index sets. We verify the conjecture that all nonzero roots are simple for three subclasses of these polynomials, which, in a natural sense, cover more than half of the relevant parameter range. We also provide an interpretation of the root multiplicity at $z=0$ in terms of Young diagrams. In addition, we show that certain members of generalized WH polynomials provide representations of rational solutions of the Noumi-Yamada systems, a family of higher-order Painlevé equations. Finally, we apply our results to the large-parameter asymptotic analysis of rogue wave patterns for the multi-component Hirota equation.

math.CV↗

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.

nlin.SI↗

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.

nlin.SI↗

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.

nlin.SI↗