Existence of traveling waves in the fractional bistable equation

We construct traveling waves of the fractional bistable equation by approximating the fractional Laplacian ${(D^{2})^{\alpha}, \alpha \in (0, 1)}$ , with operators ${J \ast u - (\int_{R} J)u}$ , where J is nonsingular. Since the resulting approximating equations are known to have traveling waves, the solutions are obtained by passing to the limit. This provides an answer to the statement (about existence and properties) “This construction will be achieved in a future work” before Assumption 2 in Imbert and Souganidis [6]. With a modification of a part of the argument, we also get the existence of traveling waves for the ignition nonlinearity in the case ${\alpha \in (1/2, 1)}$ .     

On the exact solitary wave solutions of a special class of Benjamin-Bona-Mahony equation

The general form of Benjamin-Bona-Mahony equation (BBM) is $u_t + au_x + bu_{xxt} + (g(u))_x = 0,a,b \in \mathbb{R},$ where ab ≠ 0 and g(u) is a C ₂-smooth nonlinear function, has been proposed by Benjamin et al. in [1] and describes approximately the unidirectional propagation of long wave in certain nonlinear dispersive systems. In this payer, we consider a new class of Benjamin-Bona-Mahony equation (BBM) $u_t + au_x + bu_{xxt} + (pe^u + qe^{ - u} )_x = 0,a,b,p,q \in \mathbb{R},$ where ab ≠ 0, and qp ≠ 0, and we obtain new exact solutions for it by using the well-known (G′/G)-expansion method. New periodic and solitary wave solutions for these nonlinear equation are formally derived.     

Study of the Equation of Partial Waves with Rapidly Oscillating Potential

In this paper, we consider the differential equation of partial waves $$\psi ''(x) + \left[ {k^2 - \frac{{\lambda ^2 - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}}}{{x^2 }} - V(x)} \right]\psi (x) = 0,$$ and the corresponding integral equations. We obtain estimates for the solutions of this differential equation with boundary conditions for x = 0 and x = ∞. The analitycity domains for the wave functions are established.     

Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in (0.1) $ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + . $ It is shown that two basic Riemann problems for Eq. (0.1) with the initial data $ S_ \mp (x) = \mp \operatorname{sgn} x $ exhibit a shock wave (u(x, t) ≡ S _?(x)) and a smooth rarefaction wave (for S ₊), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u _t + uu _x = 0, which was developed by O.A. Oleinik and S.N. Kruzhkov (for equations in x ? ? ^N ) in the 1950s–1960s.     

Structural Stability of Discontinuous Galerkin Schemes

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193?C2199, 2009; Chaos Solitons Fractals 41(2):655?C660, 2009).     

Abstract wave equations and associated Dirac-type operators

We discuss the unitary equivalence of generators G _A,R associated with abstract damped wave equations of the type ${\ddot{u} + R \dot{u} + A^*A u = 0}$ in some Hilbert space ${\mathcal{H}_1}$ and certain non-self-adjoint Dirac-type operators Q _A,R (away from the nullspace of the latter) in ${\mathcal{H}_1 \oplus \mathcal{H}_2}$ . The operator Q _A,R represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of A*A. In addition to the unitary equivalence results concerning G _A,R and Q _A,R , we provide a detailed study of the domain of the generator G _A,R , consider spectral properties of the underlying quadratic operator pencil ${M(z) = |A|^2 - iz R - z^2 I_{\mathcal{H}_1}, z\in\mathbb{C}}$ , derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The special example where R represents an appropriate function of |A| is treated in depth, and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator. The cases of undamped (R?=?0) and damped (R ≠ 0) abstract wave equations as well as the cases ${A^* A \geq \varepsilon I_{\mathcal{H}_1}}$ for some ${\varepsilon > 0}$ and ${0 \in \sigma (A^* A)}$ (but 0 not an eigenvalue of A*A) are separately studied in detail.     

Oscillations of difference equations with several deviated arguments

We find the solutions ${f,g,h \colon G \to X,\,\varphi \colon G \to \mathbb{K}}$ of each of the functional equation $$\sum\limits_{\lambda \in K} f(x + \lambda y) = |K| \varphi (y) g(x) + |K|h(y), \quad x, y \in G$$ , where (G, + ) is an abelian group, K is a finite, abelian subgroup of the automorphism group of G, X is a linear space over the field ${\mathbb{K} \in \{ \mathbb{R},\mathbb{C}\}}$ .     

Boundedness of solutions of second order nonlinear dynamic equations

This paper deals with the boundedness of the solutions of the following dynamic equations(r(t)x△(t))△+a(t)f(xσ(t))+b(t)g(xσ(t))=0and(r(t)x△(t))△+a(t)xσ(t)+b(t)f(x(t-τ(t)))=e(t)on a time scale T.By using the Bellman integral inequality,we establish some suffcient conditions for boundedness of solutions of the above equations.Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.     

Traveling wave front and stability as planar wave of reaction diffusion equations with nonlocal delays

This paper is concerned with traveling wave front and the stability as planar wave of reaction diffusion system on ${\mathbb{R}^{n}}$ , where n ≥ 2. Existence and asymptotic behavior of traveling wave front are discussed firstly. The stability as planar wave is established secondly by using super-sub solution method. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar wave as ${t \rightarrow {\infty}}$ and the convergence is uniform in ${\mathbb{R}^{n}}$ .     

Notes on Homoclinic Solutions of the Steady Swift-Hohenberg Equation

This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a small positive constant. This slightly complements Santra and Wei＇s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C （0,β0） where β0 = 0.9342 ....     

Lower bounds on the number of maximal subgroups in a finite group

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, ${|m(G)| \geq |\pi(G)|+p}$ | m ( G ) | ≥ | π ( G ) | + p , where ${p \in \pi(G)}$ p ∈ π ( G ) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and ${q \in \pi(G)}$ q ∈ π ( G ) is the smallest prime such that ${Q \in {\rm syl}_q(G)}$ Q ∈ syl q ( G ) is noncyclic, then ${|m(G)| \geq |\pi(G)|+q}$ | m ( G ) | ≥ | π ( G ) | + q . Both lower bounds are best possible.     

Nordhaus-Gaddum results for the sum of the induced path number of a graph and its complement

The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path.Broere et al.proved that if G is a graph of order n,then n~(1/2) ≤ρ(G) + ρ(■) ≤ [3n/2].In this paper,we characterize the graphs G for which ρ(G) + ρ(■) = [3n/2],improve the lower bound on ρ(G) + ρ(■) by one when n is the square of an odd integer,and determine a best possible upper bound for ρ(G) + ρ(■) when neither G nor ■ has isolated vertices.     

Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in ${\mathbb{R}}^{n}$ according to a constraint vector field D(·) specified on the boundary $\partial G$ of the domain. Specifically, we consider equations of the form $\phi = \theta + \eta + u , \quad \dot{\theta}(t) \in F(\phi(t)), \quad \mbox{a.e. } t$ for u in an appropriate class of functions, where η is the “constraining term” in the Skorokhod problem specified by (G, D) and F is the set-valued upper semicontinuous envelope of f. The case $G ={\mathbb{R}}^{n}$ (when there is no constraining mechanism) and u is absolutely continuous corresponds to the well known setting of differential inclusions (DI). We provide a general sufficient condition for uniqueness of solutions and Lipschitz continuity of the solution map, in the form of existence of a Lyapunov set. Here we assume (i) G is convex and admits the representation $G=\cup_i\overline{C_i}$ , where $\{C_i,i\in {\mathbb{I}}\}$ is a finite collection of disjoint, open, convex, polyhedral cones in ${\mathbb{R}}^{n}$ , each having its vertex at the origin; (ii) f =  b +  f ^c is a vector field defined on G such that b assumes a constant value on each of the given cones and f ^c is Lipschitz continuous on G; and (iii) D is an upper semicontinuous, cone-valued vector field that is constant on each face of ?G. We also provide existence results under much weaker conditions (where no Lyapunov set condition is imposed). For stochastic differential equations (SDE) (possibly, reflected) that have drift coefficient f and a Lipschitz continuous (possibly degenerate) diffusion coefficient, we establish strong existence and pathwise uniqueness under appropriate conditions. Our approach yields new existence and uniqueness results for both DI and SDE even in the case $G = {\mathbb{R}}^{n}.$ The work has applications in the study of scaling limits of stochastic networks.     

Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.     

A Motzkin–Straus Type Result for 3-Uniform Hypergraphs

Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in [8]. They showed that if G is a 2-graph in which a largest clique has order l then ${\lambda(G)=\lambda(K^{(2)}_l),}$ where λ(G) denotes the Lagrangian of G. It is interesting to study a generalization of the Motzkin–Straus Theorem to hypergraphs. In this note, we give a Motzkin–Straus type result. We show that if m and l are positive integers satisfying ${{l-1 \choose 3} \le m \le {l-1 \choose 3} + {l-2 \choose 2}}$ and G is a 3-uniform graph with m edges and G contains a ${K_{l-1}^{(3)}}$ , a clique of order l?1, then ${\lambda(G) = \lambda(K_{l-1}^{(3)})}$ . Furthermore, the upper bound ${{l-1 \choose 3} + {l-2 \choose 2}}$ is the best possible.     

Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

A two-parameter family of Harnack type inequalities for non-negative solutions of a class of singular, quasilinear, homogeneous parabolic equations is established, and it is shown that such estimates imply the Hölder continuity of solutions. These classes of singular equations include p-Laplacean type equations in the sub-critical range ${1 < p \le\frac{2N}{N+1}}$ and equations of the porous medium type in the sub-critical range ${0 < m \le\frac{(N-2)_+}{N}}$ .     

On the SIG-Dimension of Trees Under the L ∞-Metric

Let ${{\mathcal P},}$ where ${|{\mathcal P}| \geq 2,}$ be a set of points in d-dimensional space with a given metric ρ. For a point ${p \in {\mathcal P},}$ let r _p be the distance of p with respect to ρ from its nearest neighbor in ${{\mathcal P}.}$ Let B(p,r _p ) be the open ball with respect to ρ centered at p and having the radius r _p . We define the sphere-of-influence graph (SIG) of ${{\mathcal P}}$ as the intersection graph of the family of sets ${\{B(p,r_p)\ | \ p\in {\mathcal P}\}.}$ Given a graph G, a set of points ${{\mathcal P}_G}$ in d-dimensional space with the metric ρ is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of ${{\mathcal P}_G.}$ It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L _∞-metric in some space of finite dimension. The SIG-dimension under the L _∞-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L _∞-metric. It is denoted by SIG _∞(G). We study the SIG-dimension of trees under the L _∞-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447–460, 2003). Let T be a tree with at least two vertices. For each ${v\in V(T),}$ let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leaf-degree as ${\alpha(T) = \max_{x \in V(T)}}$ leaf-degree(x). Let ${ S = \{v\in V(T)\|\}}$ leaf-degree{(v) = α}. If |S| = 1, we define β(T) = α(T) ? 1. Otherwise define β(T) = α(T). We show that for a tree ${T, SIG_\infty(T) = \lceil \log_2(\beta + 2)\rceil}$ where β = β (T), provided β is not of the form 2 ^k ? 1, for some positive integer k ≥ 1. If β = 2 ^k ? 1, then ${SIG_\infty (T) \in \{k, k+1\}.}$ We show that both values are possible.     

Classification of hypersurfaces with constant M?bius curvature in {\mathbb{S}^{m+1}}

Let ${x: M^{m} \rightarrow \mathbb{S}^{m+1}}$ be an m-dimensional umbilic-free hypersurface in an (m?+?1)-dimensional unit sphere ${\mathbb{S}^{m+1}}$ , with standard metric I?= dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function ${\rho=\sqrt{\frac{m}{m-1}}\|II-\frac{1}{m}tr(II)I\|}$ . Then positive definite (0,2) tensor ${\mathbf{g}=\rho^{2}I}$ is invariant under conformal transformations of ${\mathbb{S}^{m+1}}$ and is called M?bius metric. The curvature induced by the metric g is called M?bius curvature. The purpose of this paper is to classify the hypersurfaces with constant M?bius curvature.     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.