The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlevé equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.
The following coupled Schrödinger system with a small perturbation $\begin{array}{*{20}c} {u_{xx} + u - u^3 + \beta uv^2 + \varepsilon f(\varepsilon ,u,u_x ,v,v_x ) = 0 in \mathbb{R},} \\ {v_{xx} - v + v^3 + \beta u^2 v + \varepsilon g(\varepsilon ,u,u_x ,v,v_x ) = 0 in \mathbb{R}} \\ \end{array}$ is considered, where β and ∈ are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution (called the generalized heteroclinic solution thereafter).
In this paper, the authors study the homologically trivial symplectic group actions on homotopy elliptic surfaces E(n) and get some rigidity results.
Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ H seG, where H seG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.
For the Monge-Ampère equation detD 2 u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.
The authors study the Rayleigh-Taylor instability for two incompressible immiscible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eulerian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in (unstable) equilibrium is analyzed. By a general method of studying a family of modes, the smooth (when restricted to each fluid domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.
The Brio system is a 2 × 2 fully nonlinear system of conservation laws which arises as a simplified model in the study of plasmas. The present paper offers explicit solutions to this system subjected to initial conditions containing Dirac masses. The concept of a solution emerges within the framework of a distributional product and represents a consistent extension of the concept of a classical solution. Among other features, the result shows that the space of measures is not sufficient to contain all solutions of this problem.
The author considers the hyperbolic geometric flow $\tfrac{{\partial ^2 }}{{\partial t^2 }}g(t) = - 2Ric_{g(t)}$ introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to the hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.
For x = (x 1, x 2, ⋯, x n) ∈ ℝ+ n ∪ ℝ− n, the symmetric functions F n(x, r) and G n(x, r) are defined by $F_n (x,r) = F_n (x_1 ,x_2 , \cdots ,x_n ;r) = \sum\limits_{1 \leqslant i_1 < i_2 < \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }}{{x_{i_j } }}} }$ and $G_n (x,r) = G_n (x_1 ,x_2 , \cdots ,x_n ;r) = \sum\limits_{1 \leqslant i_1 < i_2 < \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - x_{i_j } }}{{x_{i_j } }}} } ,$ respectively, where r = 1, 2, ⋯, n, and i 1, i 2, ⋯, i n are positive integers. In this paper, the Schur convexity of F n(x, r) and G n(x, r) are discussed. As applications, by a bijective transformation of independent variable for a Schur convex function, the authors obtain Schur convexity for some other symmetric functions, which subsumes the main results in recent literature; and by use of the theory of majorization establish some inequalities. In particular, the authors derive from the results of this paper the Weierstrass inequalities and the Ky Fan’s inequality, and give a generalization of Safta’s conjecture in the n-dimensional space and others.
The weak finite determinacy of relative map-germs is studied. The authors first give the concept of weak finite determination, and then give several sufficient conditions for a relative map-germ to be weak finitely determined, which is an important complement to Mather’s work. Moreover, as an application, it is proven that the relative stable map-germs are weak finitely determined.