Davide Masoero, Andrea Raimondo
We study the ODE/IM correspondence for all the states of the quantum Boussinesq model. We consider a particular class of third order linear ordinary differential operators and show that the generalised monodromy data of such operators provide solutions to the Bethe Ansatz equations of the Quantum Boussinesq model.
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In this paper we construct a map between a solution of supersymmetric Chern-Simons higher spin gravity based on the superalgebra $sl(3|2)$ with Lifshitz scaling and the $N=2$ super Boussinesq hierarchy. We show that under this map the time evolution equations of both theories coincide. In addition, we identify the Poisson structure of the Chern-Simons theory induced by gauge transformation with the second Hamiltonian structure of the super Boussinesq hierarchy.
We consider a non-self-adjoint third order operator on the interval $[0,2]$ with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We determine eigenvalue asymptotics at high energy and the trace formula for the operator.
Inverse problem for the L-operator in the Lax Pair of the Boussinesq equation on the circle
本文研究圆周上 Boussinesq 方程 Lax 对中三阶非自伴 L-算子的逆谱问题，构造系数到谱数据的映射，并证明其在零邻域内解析且一一对应。
Asymptotics of the divisor for the good Boussinesq equation
本文研究三点Dirichlet条件下三阶算子的谱，即好Boussinesq方程的辅助谱，用算子系数估计谱与归一化常数，为求解逆问题的系列研究之首篇。
Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations
本文基于高阶纤维流形上的 Lagrange 形式化与 Krupka–Betounes 型 Lepage 等价物，双场推导并发现含 $x,y$ 混合项的新 $2D$ 四阶、六阶修正 Boussinesq 型方程。
Inverse problem for the divisor of the good Boussinesq equation
本文研究了好 Boussinesq 方程的三阶算子在已知谱和规范常数下的逆问题，通过构造映射证明其解析双射性。
We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney-Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in <http://dx.doi.org/doi:10.1088/0305-4470/27/1/013>. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each othe…
We construct new integrable coupled systems of N=1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian structures. A fermionic extension of the Burgers equation is related with the Burgers flows on associative algebras. A Gardner's deformation is found for the bosonic super-field dispersionless Boussinesq equatio…
Resonant Geometry and Well-Posedness for the 2D Boussinesq Wave Kinetic Equation
针对源自Boussinesq动力学的各向异性向量值波动力学方程，引入适配共振流形角简并的角截断，证明其局部适定性，构建了首个严格解析框架。
In this paper we study the Yang-Baxter integrable structure of Conformal Field Theories with extended conformal symmetry generated by the W_3 algebra. We explicitly construct various T- and Q-operators which act in the irreducible highest weight modules of the W_3 algebra. These operators can be viewed as continuous field theory analogues of the commuting transfer matrices and Q-matrices of the integrable lattice systems associated with the quantum algebra U_q(\hat{sl}(3)). We formulate several conjectures detailing certain analytic characteristics of the Q-operators and propose exact asymptot…
In this work we present a solution of the Boussinesq equation. The derived formulas include solitons, Schwartz class solutions and solutions, possessing singularities on a closed set Z of the (x,t) domain, obtained from the zeros of the tau function. The idea for solving the Boussinesq equation is identical to the (unified) idea of solving the KdV and the evolutionary NLS equations: we use a theory of evolutionary vessels. But a more powerful theory of non-symmetric evolutionary vessels is presented, inserting flexibility into the construction and allowing to deal with complex-valued solutions…
$\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators
本文通过线性谱问题及其因子分解，构建了连续和离散的PGL(3)-不变可积系统，推导了Boussinesq系统的PGL(3)-不变形式，并发展了几何提升-解耦机制。
On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy
本文研究三线性 Boussinesq 方程及其从双线性晶格 AKP 方程的推导、对偶方程、守恒律与周期约化，并推广到高阶 Gel'fand-Dikii 晶格系统，构造了类 Somos 整数序列。
We consider Schrödinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are considered. These results are used to analyze the good Boussinesq equation.
We consider the third order operator with periodic coefficients on the real line. This operator is used in the integration of the non-linear evolution Boussinesq equation. For the minimal smoothness of the coefficients we prove that: 1) the operator is self-adjoint and it is decomposable into the direct integral, 2) the spectrum is absolutely continuous, fills the whole real axis, and has multiplicity one or three, 3) the Lyapunov function, analytic on a three-sheeted Riemann surface, is constructed and researched, 4) the spectrum of multiplicity three is bounded and it is described in terms o…
Mc'Kean's transformation for 3-rd order operators
本文研究 McKean 于 1981 年提出的一种变换，该变换将圆周上 good Boussinesq 方程 Lax 对中周期非自伴三阶算子的谱问题，归结为势解析依赖能量的 Hill 算子谱问题。
Soliton solutions associated with a class of third-order ordinary linear differential operators
本文通过求解三阶微分方程在无反射情况下的逆散射问题，利用透射系数束缚态极点及随时间演化的依赖常数，构造了相关可积非线性演化方程的显式解，并从物理上解释了 Hirota 双线性方法构造的 $\mathbf N$-孤子解中常数的起源。
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