arXiv subjects

Yuqin Yao

Publications and source records attributed to Yuqin Yao.

14 recordsLinked to original sources

arXiv2025

Cauchy matrix structures and solutions to the nonisospectral three-component mKdV equations

Nonisospectral integrable systems can describe solitary waves in nonuniform media. In this paper, we apply the Cauchy matrix approach to construct three types of nonisospectral matrix modified Korteweg-de Vries (mKdV) eqautions and present their Cauchy matrix structures and solutions. Further, through complex reduction, we further obtain three nonisospectral three-component mKdV (NTCmKdV) equations, which can be regarded as novel members of the nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. In particular, the explicit solutions are given for the soliton solutions, and the double-pole solutions, respectively. The dynamical behaviors of these solutions are analyzed to reveal the influence of nonisospectral terms on the solution structure.

nlin.SI↗

arXiv2025

(3+1)-dimensional modified Kadomtsev-Petviashvili equation and its \bar{\partial}-formalism

Constructing integrable evolution nonlinear PDEs in three spatial dimensions is one of the most important open problems in the area of integrability. Fokas achieved progress in 2006 by constructing integrable nonlinear equations in 4+2 dimensions, but the reduction to 3+1 dimensions remained open until 2022 when he introduced a suitable nonlinear Fourier transform to achieve this reduction for the Kadomtsev-Petiashvili (KP) equation. Here, the integrable generalization of the modified KP (mKP) equation has been presented which has the novelty that it involves complex time and preserves Laplace's equation. The (3+1)-dimensional mKP equation is obtained by imposing the requirement of real time. Then, the spectral analysis of the eigenvalue equation is given and the solution of the $\bar{\partial}$-problem is demonstrated based on Cauchy-Green formula. Finally, a novel $\bar{\partial}$-formalism for the initial value problem of the (3+1)-dimensional mKP equation is worked out.

nlin.SI↗

arXiv2024

The symmetric (2+1)-dimensional Lotka-Volterra equation with self-consistent sources

The symmetric (2+1)-dimensional Lotka-Volterra equation with self-consistent sources is constructed and solved by employing the source generation procedure, whose solutions are expressed in terms of pfaffians. As special cases of the pfaffian solutions, different types of explicit solutions are obtained, including dromions, soliton solutions and breather solutions.

nlin.SI↗

arXiv2017

Completion of the integrable coupling systems

In this paper, we proposed an procedure to construct the completion of the integrable system by adding a perturbation to the generalized matrix problem, which can be used to continuous integrable couplings, discrete integrable couplings and super integrable couplings. As example, we construct the completion of the Kaup-Newell (KN) integrable coupling, the Wadati-Konno-Ichikawa (WKI) integrable couplingsis, vector Ablowitz-Kaup-Newell-Segur (vAKNS) integrable couplings, the Volterra integrable couplings, Dirac type integrable couplings and NLS-mKdV type integrable couplings.

nlin.SI↗

arXiv2013

The generalized Kupershmidt deformation for constructing new discrete integrable systems

KdV6 equation can be described as the Kupershmidt deformation of the KdV equation (see 2008, Phys. Lett. A 372: 263). In this paper, starting from the bi-Hamiltonian structure of the discrete integrable system, we propose a generalized Kupershmidt deformation to construct new discrete integrable systems. Toda hierarchy, Kac-van Moerbeke hierarchy and Ablowitz-Ladik hierarchy are considered. The Lax representations for these new deformed systems are presented. The generalized Kupershmidt deformation for the discrete integrable systems provides a new way to construct new discrete integrable systems.

nlin.SI↗

arXiv2012

A new $(γ_n,σ_k)-$ KP hierarchy and generalized dressing method

A new (γ_n,σ_k)-KP hierarchy with two new time series γ_n and σ_k, which consists of γ_n-flow, σ_k-flow and mixed γ_n and σ_k evolution equations of eigenfunctions, is proposed. Two reductions and constrained flows of (γ_n,σ_k)-KP hierarchy are studied. The dressing method is generalized to the (γ_n,σ_k)-KP hierarchy and some solutions are presented.

nlin.SI↗

arXiv2011

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in \cite{kd} offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax Pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.

nlin.SI↗

arXiv2010

The new integrable deformations of short pulse equation and sine-Gordon equation, and their solutions

We first derive an integrable deformed hierarchy of short pulse equation and their Lax representation. Then we concentrated on the solution of integrable deformed short pulse equation (IDSPE). By proposing a generalized reciprocal transformation, we find a new integrable deformed sine-Gordon equation (IDSGE) and its Lax representation. The multisoliton solutions, negaton solutions and positon solutions for the IDSGE and the N-loop soliton solutions, N-negaton and N-positon solutions for the IDSPE are presented. In the reduced case the new N-positon solutions and N-negaton solutions for short pulse equation are obtained.

nlin.SI↗

arXiv2010

A new extended matrix KP hierarchy and its solutions

With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $τ_B$ flow. The extended KP hierarchy contains two time series ${t_A}$ and ${τ_B}$ and eigenfunctions and adjoint eigenfunctions as components. The extended matrix KP hierarchy and its $t_A$-reduction and $τ_B$ reduction include two types of matrix KP hierarchy with self-consistent sources and two types of (1+1)-dimensional reduced matrix KP hierarchy with self-consistent sources. In particular, the first type and second type of the 2+1 AKNS equation and the Davey-Stewartson equation with self-consistent sources are deduced from the extended matrix KP hierarchy. The generalized dressing approach for solving the extended matrix KP hierarchy is proposed and some solutions are presented. The soliton solutions of two types of 2+1-dimensional AKNS equation with self-consistent sources and two types of Davey-Stewartson equation with self-consistent sources are studied.

nlin.SI↗

arXiv2010

The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems

Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a $t$-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.

nlin.SI↗

arXiv2009

A new extended discrete KP hierarchy and generalized dressing method

Inspired by the squared eigenfunction symmetry constraint, we introduce a new $\ta_k$-flow by ``extending'' a specific $t_n$-flow of discrete KP hierarchy (DKPH). We construct extended discrete KPH (exDKPH), which consists of $t_n$-flow, $\ta_k$-flow and $t_n$ evolution of eigenfunction and adjoin eigenfunctions, and its Lax representation. The exDKPH contains two types of discrete KP equation with self-consistent sources (DKPESCS). Two reductions of exDKPH are obtained. The generalized dressing approach for solving the exDKPH is proposed and the N-soliton solutions of two types of the DKPESCS are presented.

nlin.SI↗

arXiv2008

On Camassa-Holm equation with self-consistent sources and its solutions

Regarded as the integrable generalization of Camassa-Holm (CH) equation, the CH equation with self-consistent sources (CHESCS) is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.

nlin.SI↗

arXiv2008

The bi-Hamiltonian structure and new solutions of KdV6 equation

We show that the KdV6 equation recently studied in [1,2] is equivalent to the Rosochatius deformation of KdV equation with self-consistent sources (RD-KdVESCS) recently presented in [9]. The $t$-type bi-Hamiltonian formalism of KdV6 equation (RD-KdVESCS) is constructed by taking $x$ as evolution parameter. Some new solutions of KdV6 equation, such as soliton, positon and negaton solution, are presented.

nlin.SI↗

arXiv2008

Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources

We propose a systematic method to generalize the integrable Rosochatius deformations for finite dimensional integrable Hamiltonian systems to integrable Rosochatius deformations for infinite dimensional integrable equations. Infinite number of the integrable Rosochatius deformed higher-order constrained flows of some soliton hierarchies, which includes the generalized integrable H$\acute{e}$non-Heiles system, and the integrable Rosochatius deformations of the KdV hierarchy with self-consistent sources, of the AKNS hierarchy with self-consistent sources and of the mKdV hierarchy with self-consistent sources as well as their Lax representations are presented.

nlin.SI↗