We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the Willmore conjecture.
由于它与低维共形几何的相关性，被称为广义Weierstrass诱导，修正的Veselov-Novikov（mVN）层次最近引起了新的兴趣。在文献中已经明确表明，一个外在的弦作用量à la Polyakov（Willmore泛函）在与mVN层次的第一个成员相关的变形下是不变的。在本说明中，我们更进一步，并显示显式不变性的功能下的变形与所有更高的成员的层次。
We investigate the generalized (2 + 1) Nizhnik-Novikov-Veselov equation and construct its linear eigenvalue problem in the coordinate space from the results of singularity structure analysis thereby dispelling the notion of weak Lax pair. We then exploit the Lax-pair employing Darboux transformation and generate lumps and rogue waves. The dynamics of lumps and rogue waves is then investigated.
Using the reality condition of the solutions, one constructs the real Pfaffian N-solitons solutions of the Novikov-Veselov (NV) equation using the $\tan$ function and the Schur identity. By the minor-summation formula of the Pfaffian, we can study the interactions of solitons in the Novikov-Veselov equation from the Kadomtsev-Petviashvili (KP) equation's point of view, that is, the totally non-negative Grassmannian. Especially, the Y-shape resonance, O-type and the P-type interactions of X-shape are investigated. Also, the maximum amplitude of the intersection of the line solitons and the crit…
The (2+1)-dimensional generalized Nizhnik-Novikov-Veselov equations (GNNVEs) are investigated in order to search the influence of initial solution to exact solutions. The GNNVEs are converted into the combined equations of differently two bilinear forms by means of the homogeneous balance of undetermined coefficients method. Accordingly, the two class of exact N-soliton solutions and three wave solutions are obtained respectively by using the Hirota direct method combined with the simplified version of Hereman and the three wave method. The proposed method is also a standard and computable met…
We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the "energy" parameter $ E $. We show that as $ |E| \to \infty $, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when $ | E | $ is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.
We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two-dimensional $O(N)$ sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions a…
We construct a hierarchy of pairwise commuting flows $d/dt_{i,n}$ indexed by $i \in \{1,2 \}$ and $n \in \mathbb{Z}_{\geq 0}$ on triples $(\mathcal{L}_1, \mathcal{L}_2, \mathcal{H})$ where $\partial_1$ and $\partial_2$ are two commuting derivations, $\partial_i \mathcal{L}_i$ is a self-adjoint pseudodifferential operator in $\partial_i$ and $\mathcal{H}$ is the formal Schrödinger operator $\mathcal{H}=\partial_1 \partial_2 +u$. $\mathcal{L}_1, \mathcal{L}_2$ and $\mathcal{H}$ are coupled by the relations $\mathcal{H} \mathcal{L}_i+\mathcal{L}_i^* \mathcal{H}=0$. We show that the flows $d/dt_{1…
Localized stem structures in soliton reconnection of the asymmetric Nizhnik-Novikov-Veselov system
College of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院）；Institute for Advanced Study, Shenzhen University（深圳大学高等研究院）；School of Mathematical Sciences, USTC（中国科学技术大学数学科学学院）
We develop the inverse scattering transform method for the Novikov equation $u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx}$ considered on the line $x\in(-\infty,\infty)$ in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a $3\times 3$ matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus…
This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation ((alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equ…
Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）
In the field of mathematical physics, there exist many physically interesting nonlinear dispersive equations with peakon solutions, which are solitary waves with discontinuous first-order derivative at the wave peak. In this paper, we apply the multi-layer physics-informed neural networks (PINNs) deep learning to successfully study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations with initial-boundary value conditions such as the Camassa-Holm (CH) equation, Degasperis-Procesi equation, modified CH equation with cubic nonlinearity, Novikov e…
We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa-Holm (mCH) equation and Novikov's equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov's equation to a negative mKdV flow and a negative Sawada-Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second ord…
A general family of peakon equations is introduced, involving two arbitrary functions of the wave amplitude and the wave gradient. This family contains all of the known breaking wave equations, including the integrable ones: Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation, and FORQ/modified Camassa-Holm equation. One main result is to show that all of the equations in the general family possess weak solutions given by multi-peakons which are a linear superposition of peakons with time-dependent amplitudes and positions. In particular, neither an integrability structure nor…
