arXiv subjects

Shou-Fu Tian

Publications and source records attributed to Shou-Fu Tian.

At least 19 recordsLinked to original sources

arXiv2026

Soliton resolution conjecture and soliton gases for the Benjamin-Ono equation with infinite-order multisolitons in the sharp well-posedness space

We prove the soliton resolution conjecture for infinite-order multisoliton solutions in the sharp well-posedness space $H^s(\mathbb R)$, $s>-\frac12$, and construct soliton gases in $H^{s}(\mathbb R)$ with arbitrary $s\in\mathbb R$, for the Benjamin--Ono equation. In particular, our work addresses two important open problems about inverse spectral for Lax operator, including the problem of spectral characterization of infinite-order multisolitons and that of determining the range of their asymptotic parameters. A central ingredient of our work is a spectral decomposition adapted to rough initial data. This extends the global soliton--radiation description to weakly weighted $H^s$ data while allowing infinite negative eigenvalues. Another important ingredient is a continuous norming action that extends finite inverse spectral shifts to gas limits while retaining a full continuous norming weight. Our framework provides a basis for studying infinite-order soliton resolution and the construction and asymptotics of soliton gases for nonlocal equations.

math.AP↗

arXiv2026

Sharp well-posedness and soliton resolution for the half-wave maps equation in $H^{1/2}_\bullet$

We prove sharp well-posedness and soliton resolution for the half-wave maps equation on the real line $\R$ in $H^{1/2}_\bullet$. This regularity threshold for global well-posedness is optimal, as we show by constructing examples of ill-posedness in $H^s_\bullet$ for any $s<1/2$. Moreover, with the spectral assumption satisfied by a dense set of initial data in the energy space, we prove that solutions in $H^{1/2}_\bullet$ asymptotically decouple as an infinite superposition of solitons. Our results extend the known global well-posedness and finite soliton resolution for rational data in $H^\infty_\bullet$ to sharp global well-posedness and soliton resolution for general data in $H^{1/2}_\bullet$. Different from rational solutions, general data can also involve an infinite discrete spectrum, which makes the reduction analysis framework on the finite rank no longer applicable. For sharp well-posedness, we present a new isometry criterion and stability results for resolvent transform in the explicit formula. For soliton resolution, we develop a spectral method for this equation based on direct resolvent analysis and a novel trace norm estimate that upgrades energy convergence to uniform convergence. Our argument also yields sharp well-posedness and soliton resolution for the generalized half-wave maps equation with targets in the complex Grassmannians, extending the classical sphere-valued case.

math.AP↗

arXiv2026

On the Cauchy problem for the Tzitzéica equation: Soliton resolution conjecture and asymptotic analysis

We prove the soliton resolution conjecture for the Tzitzéica equation, which is an important integrable model arising in affine differential geometry and characterizing proper affine spheres. Recently, Huang, Wang and Zhu (Math. Ann. \textbf{395}, 18 (2026)) reported the long-time asymptotic result for the Tzitzéica equation with purely continuous spectrum. Inspired by their work, we carry out a full asymptotic analysis of the Tzitzéica equation. Compared with the previous work, by developing the $\bar{\partial}$-nonlinear steepest descent method, we reveal that the solutions of the Tzitzéica equation exhibits three different asymptotic regions depending on $ξ:=x/t$. For the region $ξ\in(-\infty,-1]\cup[1,\infty)$, where there are no stationary phase points, we rigorously prove that the solution satisfies $u(x,t)=O_ξ(t^{-1})$ for $ξ\ge1$ and $u(x,t)=2πi+O_ξ(t^{-1})$ for $ξ\le-1$. In the region $ξ\in(-1,1)$, the phase function $θ(z)$ has two stationary phase points. The corresponding asymptotic approximations can be characterized with an $N$-soliton solution as well as an interaction term between soliton solutions and the dispersion term with a residual error of order $O(t^{-3/4+b_K})$ with $0\le b_K<1/4$. In the transition regions $\{(x,t):1-\varepsilon<|ξ|<1\}$, the corresponding asymptotic approximation can be characterized by the dispersion term with a residual error of order $O(t^{-3/4+b_\varepsilon})$ where $0\le b_\varepsilon<1/4$. Our results provide a rigorous analysis of the long-time asymptotic behavior of solutions of the Tzitzéica equation in different regions and confirm the soliton resolution conjecture for the Tzitzéica equation.

math.AP↗

arXiv2026

On the Cauchy Problem for the Sawada-Kotera Equation: Soliton resolution conjecture, Painlevé transcendents and asymptotic stability

We derive long-time asymptotics in different space-time regions for the Sawada-Kotera (SK) equation with admissible data involving continuous spectrum and finite discrete spectrum. Our result proves the soliton resolution conjecture, yields the asymptotics governed by the F-XVIII Painlevé transcendent, and establishes the asymptotic stability of $N$-soliton solutions. More importantly, this extends the long-time asymptotic theory to the coexistence of solitons and radiation, whereas previous results were restricted to the purely continuous spectrum. One key ingredient in our argument is a new meromorphic Riemann-Hilbert framework that incorporates the discrete spectrum through complete six-point pole orbits while preserving cofactor analyticity and the singular structure at the spectral origin. A second ingredient is the combination of an orbitwise, region-dependent pole reduction mechanism and the $\bar{\partial}$-nonlinear steepest descent method, which allows us to select the solitons contributing in each region and separate them from the continuous radiation. Our framework also resolves the coalescing stationary points at the singular spectral origin through the regular modified SK problem and the Miura map.

math-ph↗

arXiv2026

Soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation in the scaling-critical space

We prove the soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation in the scaling-critical space $L^2_+(\mathbb R)$. For a natural class of initial data including a broad spectral Dini---moment class, global flow--Lax admissible solutions decompose into finitely many explicit modulated solitons plus a dispersive radiation, while finite-time blow-up solutions resolve into quantized zero-carrier $R$-bubbles and a strongly convergent endpoint remainder. This work extends soliton resolution to the optimal critical regularity. Previous results had required weighted $H^{1,1}$ initial data, and in the global case the solution was additionally required to remain in $H^{1,1}$ for all time, which amounts to an extra spatial decay assumption. One key ingredient in our argument is a new modular operator-theoretic framework that separates the discrete and continuous spectral channels and identifies the radiation profile through the distorted Fourier transform. A second ingredient is the development of several independent rigidity criteria for the discrete soliton profiles, which avoids inverse scattering and handles embedded eigenvalues. Our framework also yields a unified description of both global and finite-time asymptotics, with exact mass partition and mass-defect quantization.

math.AP↗

arXiv2026

Soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space

We prove the soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space $H^s\left(\mathbb{R}\right)$ for $s>-\frac12$. This extends soliton resolution to the sharp regularity, whereas previous results required weighted $H^{1,1}\left(\mathbb{R}\right)$ initial data. We first prove the soliton resolution conjecture in the sharp well-posedness space for the integrable equation. In particular, our work solves three important open problems, including the problem of embedded singular continuous spectrum, the extension of Gérard's explicit formula, and the inverse spectral problem for the Lax operator. A central ingredient of our work is a new modular operator-theoretic framework combining the closed quadratic-form theory of rough Lax operators, weighted limiting absorption, and distorted Fourier analysis, which yields a unified treatment of distributional spectral theory, continuous-spectrum scattering, and radiative asymptotics at the sharp well-posedness threshold. Another important ingredient is a Fourier-side regularity and stability result for the eigenfunctions of the rough Lax operator, established through a zero-frequency identity, resolvent commutator estimates, and an isolating-contour argument. This yields well-defined soliton centers that are stable under rough approximation. Our framework provides a broadly applicable framework for proving soliton resolution conjecture for integrable dispersive equations in low-regularity spaces.

math.AP↗

arXiv2026

Stability of $N$-soliton solutions for the modified Camassa--Holm equation

In this work, we address the stability of $N$-soliton solutions to the completely integrable modified Camassa--Holm (mCH) equation. Recently, Li, Liu, and Zhu (Math. Ann. 392 (2025), 899--932) established the orbital stability of 2-soliton solutions in $H^4(\mathbb{R})$ with respect to the solution $u$ and highlighted the stability of mCH $N$-soliton solutions remains an urgent challenge. Motivated by their work, we systematically investigate the stability of mCH $N$-solitons. We first employ the bi-Hamiltonian structure of mCH to construct a novel hierarchy of explicit conservation laws with well-defined regularity domains. Then by formulating an appropriate Lyapunov functional, we apply the Inverse Scattering Transform to conduct a rigorous spectral analysis on the recursion operators. Finally, we demonstrate that the mCH $N$-solitons are non-isolated constrained minimizers of a variational problem. Our analysis proves that the $N$-soliton solutions of the mCH equation are both dynamically and orbitally stable in $H^{N+1}(\mathbb{R})$. Notably, when reduced to the 2-soliton case, our framework establishes stability in $H^3(\mathbb{R})$, which improves upon the existing regularity threshold.

math.AP↗

arXiv2026

Rogue wave statistics and integrable turbulence in the Gerdjikov-Ivanov equation

This paper numerically investigates the statistical properties of rogue waves and their generation mechanisms in integrable turbulence, taking the Gerdjikov-Ivanov (GI) equation as the research object. The eigenvalue spectra of the analytical solutions and the chaotic wave field are calculated using the Fourier collocation method. Subsequently, taking a plane wave with random noise as the initial condition, the evolution of chaotic wave fields is simulated using the split-step Fourier (SSF) method. Numerical results show that the larger the initial disturbance intensity, the faster the wave field converges to a chaotic state, and the higher the peak amplitude after convergence, the higher the tail of the probability density function, and the significantly higher probability of rogue wave occurrence. Moreover, as the initial disturbance intensity increases, the turbulence type transitions from breather turbulence to soliton turbulence. In addition, the evolution of the wave-action spectrum is studied. The research has found that the wave-action spectrum of the GI equation shows an asymmetric distribution during the time evolution process, and this asymmetry persists even after the system reaches a steady state.

nlin.PS↗

arXiv2025

Stability of $n$-soliton solutions for the Intermediate Long Wave equation

In this work, we focus on the stability of $n$-soliton solutions ($n\in \mathbb{N}, n\geq 1$) to the completely integrable intermediate long wave equation (ILW), which models long internal gravity waves in a stratified fluid of finite depth. We show that the $n$-soliton solutions of the ILW equation form non-isolated constrained minimizers of a variational problem associated with a non-local elliptic equation. To establish this result, we construct a suitable Lyapunov functional and utilize the inverse scattering transform to relate the infinite sequence of conservation laws to the scattering data. Furthermore, we employ the recursion operator derived from the bi-Hamiltonian structure to optimize our analysis. Our analysis demonstrates that the $n$-soliton solutions of the ILW equation are dynamically stable in the space $H^{\frac{n}{2}}(\mathbb{R})$ ($n\in \mathbb{N}, n\geq 1$). Additionally, we establish the orbital stability of double soliton solutions in $H^1(\mathbb{R})$.

math.AP↗

arXiv2025

On Cauchy problem to the modified Camassa-Holm equation: Painlevé asymptotics

We investigate the Painlevé asymptotics for the Cauchy problem of the modified Camassa-Holm (mCH) equation with decaying initial data \begin{align*}\nonumber &m_t+\left((u^2-u_x^2)m\right)_x+κu_{x}=0, \ (x,t)\in\mathbb{R}\times\mathbb{R}^+,\\ &u(x,0)=u_0(x), \end{align*} where $u_0(x)\in H^{4,2}(\mathbb{R})$ and $κ$ is a constant. Recently, Yang and Fan (Adv. Math. 402 (2022) 108340) reported the long-time asymptotic results for the mCH equation in the different solitonic regions. The main purpose of our work is to study the asymptotic behavior of the mCH equation in the transition regions, which are the critical regions between the different solitonic regions. The key is to establish a connection between the solution for the Cauchy problem of the mCH equation in the transition region and the Painlevé II equation. With the $\bar{\partial}$-generalization of the Deift-Zhou nonlinear steepest descent method and double scaling limit technique, in two transition regions defined by \begin{align}\nonumber \mathcal{P}_{I}:=\{(x,t):0\leqslant \left|\frac{x}{t}-2\right|t^{2/3}\leqslant C\},~~~~\mathcal{P}_{II}:=\{(x,t):0\leqslant \left|\frac{x}{t}+1/4\right|t^{2/3}\leqslant C\}, \end{align} where $C>0$ is a constant, we obtain that the leading order approximation to the solution of the mCH equation can be expressed in terms of the Painlevé II equation.

math.AP↗

arXiv2025

On soliton resolution to Cauchy problem of the spin-1 Gross-Pitaevskii equation

We investigate the Cauchy problem for the spin-1 Gross-Pitaevskii(GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng $etal.$ (Commun. Math. Phys. 382, 585-611 (2021)) reported the long-time asymptotic result with error $\mathcal{O}(\frac{\log t}t)$ for the spin-1 GP equation that only exists in the continuous spectrum. The main purpose of our work is to further generalize and improve Geng's work. Compared with the previous work, our asymptotic error accuracy has been improved from $\mathcal{O}(\frac{\log t}t)$ to $\mathcal{O}(t^{-3/4})$. More importantly, by establishing two matrix valued functions, we obtained effective asymptotic errors and successfully constructed asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including two cases: (i)coexistence of discrete and continuous spectrum; (ii)only continuous spectrum which considered by Geng's work with error $\mathcal{O}(\frac{\log t}t)$. For the case (i), the corresponding asymptotic approximations can be characterized with an $N$-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$. For the case (ii), the corresponding asymptotic approximations can be characterized with the leading term on the continuous spectrum and the residual error order $\mathcal{O}(t^{-3/4})$. Finally, our results confirm the soliton resolution conjecture for the spin-1 GP equation.

math.AP↗

arXiv2025

The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: long-time asymptotics

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to0$ as $x\to-\infty$ and $q(x,0)\simeq Ae^{2\I Bx}$ as $x\to\infty$, where $A>0$ and $B\in\mathbb{R}$. The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that $q(x,0)$ is close, in a certain spectral sense, to the ``step-like'' function $q_{0,R}(x)= \begin{cases} 0, &x\leq R,\\ Ae^{2\I Bx}, &x>R, \end{cases}$ with $R>0$. A special attention is paid to how $B\ne0$ affects the asymptotics.

math.AP↗

arXiv2024

Exponential and algebraic double-soliton solutions of the massive Thirring model

The newly discovered exponential and algebraic double-soliton solutions of the massive Thirring model in laboratory coordinates are placed in the context of the inverse scattering transform. We show that the exponential double-solitons correspond to double isolated eigenvalues in the Lax spectrum, whereas the algebraic double-solitons correspond to double embedded eigenvalues on the imaginary axis, where the continuous spectrum resides. This resolves the long-standing conjecture that multiple embedded eigenvalues may exist in the spectral problem associated with the massive Thirring model. To obtain the exponential double-solitons, we solve the Riemann--Hilbert problem with the reflectionless potential in the case of a quadruplet of double poles in each quadrant of the complex plane. To obtain the algebraic double-solitons, we consider the singular limit where the quadruplet of double poles degenerates into a symmetric pair of double embedded poles on the imaginary axis.

nlin.SI↗

arXiv2023

Characteristics of rogue waves in the scalar and vector nonlocal nonlinear Schrödinger equations

In this paper, general higher-order rogue wave solutions of the parity-time ($\mathcal {P}\mathcal {T}$) symmetric scalar and coupled nonlocal nonlinear Schrödinger equations (NLSEs) are calculated theoretically via a Darboux transformation by a separation of variable technique. Furthermore, in order to understand these solutions better, the main characteristics of the obtained solutions are explored clearly and conveniently. Our results show that the dynamics of these solutions exhibits rich patterns, most of which have no counterparts in the corresponding local equations.

math-ph↗

arXiv2023

On long-time asymptotics to the nonlocal Lakshmanan -Porsezian-Daniel equation with step-like initial data

In this work, the nonlinear steepest descent method is employed to study the long-time asymptotics of the integrable nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation with a step-like initial data: $q_{0}(x)\rightarrow0$ as $x\rightarrow-\infty$ and $q_{0}(x)\rightarrow A$ as $x\rightarrow+\infty$, where $A$ is an arbitrary positive constant. Firstly, we develop a matrix Riemann-Hilbert (RH) problem to represent the Cauchy problem of LPD equation. To remove the influence of singularities in this RH problem, we introduce the Blaschke-Potapov (BP) factor, then the original RH problem can be transformed into a regular RH problem which can be solved by the parabolic cylinder functions. Besides, under the nonlocal condition with symmetries $x\rightarrow-x$ and $t\rightarrow t$, we give the asymptotic analyses at $x>0$ and $x<0$, respectively. Finally, we derive the long-time asymptotics of the solution $q(x,t)$ corresponding to the complex case of three stationary phase points generated by phase function.

math.AP↗

arXiv2023

Inverse scattering transform of the general coupled Hirota system with nonzero boundary conditions

The initial value problem for the general coupled Hirota system with nonzero boundary conditions at infinity is solved by reporting a rigorous theory of the inverse scattering transform. With the help of a suitable uniformization variable, both the inverse and the direct problems are analyzed which allows us to develop the inverse scattering transform on the complex $z$-plane. Firstly, analyticity of the scattering eigenfunctions and scattering data, properties of the discrete spectrum, symmetries, and asymptotics are discussed in detail. Moreover, the inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is presented. Finally, the main characteristics of these obtained soliton solutions are graphically discussed in the $2\times2$ self-focusing case. This family of solutions contains novel Akhmediev breather and Kuznetsov-Ma soliton. These results would be of much importance in understanding and enriching breather wave phenomena arising in nonlinear and complex systems, especially in Bose-Einstein condensates.

math-ph↗

arXiv2022

On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions

In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions \begin{align*} m_t+2κq_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+κ,\\ q(x,0)=q_0(x)\in H^{4,2}(\mathbb R),~~x\in\mathbb R, ~~t>0, \end{align*} where $κ$ is a positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the CH equation with the initial boundary value condition. Furthermore, by developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, different long-time asymptotic expansions of the solution $q(x,t)$ are derived. Four asymptotic regions are divided in this work: For $ξ\in\left(-\infty,-\frac{1}{4}\right)\cup(2,\infty)$, the phase function $θ(z)$ has no stationary point on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete spectrum with residual error up to $O(t^{-1+2τ})$; For $ξ\in\left(-\frac{1}{4},0\right)$ and $ξ\in\left(0,2\right)$, the phase function $θ(z)$ has four and two stationary points on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$. Our results also confirm the soliton resolution conjecture for the CH equation with weighted Sobolev initial data in space-time solitonic regions.

math.AP↗

arXiv2022

On the long-time asymptotic of the modified Camassa-Holm equation with nonzero boundary conditions in space-time solitonic regions

We investigate the long-time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with nonzero boundary conditions in different regions \begin{align*} &m_{t}+\left((u^2-u_x^2)m\right)_{x}=0,~~ m=u-u_{xx}, ~~ (x,t)\in\mathbb{R}\times\mathbb{R}^{+},\\ &u(x,0)=u_{0}(x),~~\lim_{x\to\pm\infty} u_{0}(x)=1,~~u_{0}(x)-1\in H^{4,1}(\mathbb{R}), \end{align*} where $m(x,t=0):=m_{0}(x)$ and $m_{0}(x)-1\in H^{2,1}(\mathbb{R})$. Through spectral analysis, the initial value problem of the mCH equation is transformed into a matrix RH problem on a new plane $(y,t)$, and then using the $\overline{\partial}$-nonlinear steepest descent method, we analyze the different asymptotic behaviors of the four regions divided by the interval of $ξ=y/t$ on plane $\{(y,t)|y\in(-\infty,+\infty), t>0\}$. There is no steady-state phase point corresponding to the regions $ξ\in(-\infty,-1/4)\cup(2,\infty)$. We prove that the solution of mCH equation is characterized by $N$-soliton solution and error on these two regions. In $ξ\in(-1/4,0)$ and $ξ\in(0,2)$, the phase function $θ(z)$ has eight and four steady-state phase points, respectively. We prove that the soliton resolution conjecture holds, that is, the solution of the mCH equation can be expressed as the soliton solution on the discrete spectrum, the leading term on the continuous spectrum, and the residual error. Our results also show that soliton solutions of the mCH equation with nonzero condition boundary are asymptotically stable.

math.AP↗