Blow-up solutions and peakons to a generalized μ\mu-Camassa-Holm integrable equation

Consideration here is a generalized $\mu$-type integrable equation, which can be regarded as a generalization to both the $\mu$-Camassa-Holm and modified $\mu$-Camassa-Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling-wave solutions and formation of singularities of solutions for the equation are investigated. It is found that the equation admits a single peaked soliton and multi-peakon solutions. The effects of varying $\mu$-Camassa-Holm and modified $\mu$-Camassa-Holm nonlocal nonlinearities on blow-up criteria and wave breaking are illustrated in detail. Our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates.Comment: 36 page

Bosonic Super Liouville System: Lax Pair and Solution

We study the bosonic super Liouville system which is a statistical transmutation of super Liouville system. Lax pair for the bosonic super Liouville system is constructed using prolongation method, ensuring the Lax integrability, and the solution to the equations of motion is also considered via Leznov-Saveliev analysis.Comment: LaTeX, no figures, 11 page

A new integrable two-component system with cubic nonlinearity

In this paper, a new integrable two-component system, mt=[m(uxvx−uv+uvx−uxv)]x,nt=[n(uxvx−uv+uvx−uxv)]x, where m=u−uxx and n=v−vxx, is proposed. Our system is a generalized version of the integrable system mt=[m(u2x−u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained

Massive Thirring Model: Inverse Scattering and Soliton Resolution

In this paper the long-time dynamics of the massive Thirring model is investigated. Firstly the nonlinear steepest descent method for Riemann-Hilbert problem is explored to obtain the soliton resolution of the solutions to the massive Thirring model whose initial data belong to some weighted-Sobolev spaces. Secondly, the asymptotic stability of multi-solitons follow as a corollary. The main difficulty in studying the massive Thirring model through inverse scattering is that the corresponding Lax pair has singularities at the origin and infinity. We overcome this difficulty by making use of two transforms that separate the singularities.Comment: arXiv admin note: text overlap with arXiv:2009.04260, arXiv:1907.0711

Solutions to the SU(N\mathcal{N}) self-dual Yang-Mills equation

In this paper we aim to derive solutions for the SU($\mathcal{N}$) self-dual Yang-Mills (SDYM) equation with arbitrary $\mathcal{N}$. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the (matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU$(\mathcal{N})$ SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.Comment: 26 page