Generalized traveling-wave solutions of nonlinear reaction–diffusion equations with delay and variable coefficients

The paper presents a number of new exact solutions to nonlinear reaction–diffusion equations with delay of the form $c\left(x\right)u_t={\left[a\left(x\right)u_x\right]}_x+b\left(x\right)F(u,w),w=u(x,t-\tau ),$where $\tau >0$ is the delay time, and $F(u,w)$ is an arbitrary function of two arguments. Solutions are sought in the form of a generalized traveling-wave, $u=U\left(z\right)$ with $z=t+\theta \left(x\right)$. It is shown that one of the two functional coefficients $a\left(x\right)$ and $b\left(x\right)$ of the equation considered can be specified arbitrarily. Examples of delay reaction–diffusion equations and their solutions are given. New exact solutions of few other nonlinear delay PDEs are also obtained.     

Solitary waves of a generalized Ostrovsky equation

We consider the existence and stability of traveling waves of a generalized Ostrovsky equation ${(u_t-\beta u_{xxx}-f{\left(u\right)}_x)}_x=\gamma u$, where the nonlinearity $f\left(u\right)$ satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for computing ground states based on their variational properties. The class of nonlinearities considered includes sums and differences of distinct powers.     

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, $u_t+u_{xxxx}+{?}_{xx}{u}_x^2=0$, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier–Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition $u_x\in L^{q^{\prime }}{L}^q(Q)$ is satisfied, where $q,q^{\prime }\in [1,\infty ]$ are such that either $1/q+4/q^{\prime }<1$ or $1/q+4/q^{\prime }=1$, $q^{\prime }<\infty$.     

A note on uniqueness of the ground state for nonlinear Schrödinger equations

We are concerned with the following nonlinear Schrödinger equation $-{\epsilon }^2\Delta u+V\left(x\right)u={\left|u\right|}^{p-2}u,u\in H^1\left({\mathbb{R}}^N\right),$where $N\ge 3$, $2<p<\frac{2N}{N-2}$. For $\epsilon$ small enough and a class of $V\left(x\right)$, we show the uniqueness of the positive ground state under certain assumptions on asymptotic behavior of $V\left(x\right)$ and its first derivatives. Here our results are suitable for a kind of $V\left(x\right)$ which has different increasing rates at different directions.     

Equitable 2-partitions of the Hamming graphs with the second eigenvalue

The eigenvalues of the Hamming graph $H(n,q)$ are known to be ${\lambda }_i(n,q)=(q-1)n-qi$, $0\le i\le n$. The characterization of equitable 2-partitions of the Hamming graphs $H(n,q)$ with eigenvalue ${\lambda }_1(n,q)$ was obtained by Meyerowitz (2003). We study the equitable 2-partitions of $H(n,q)$ with eigenvalue ${\lambda }_2(n,q)$. We show that these partitions are reduced to equitable 2-partitions of $H(3,q)$ with eigenvalue ${\lambda }_2(3,q)$ with the exception of two constructions.     

On the critical points of the flight return time function of perturbed closed orbits

We deal here with planar analytic systems $\dot{x}=X(x,\epsilon )$ which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by $T^{\mathrm{\Sigma }}(q,\epsilon )$ the time needed by a perturbed orbit that starts from $q\in \mathrm{\Sigma }$ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of $\dot{x}=X(x,0)$ is a continuable critical orbit in a family of the form $\dot{x}=X(x,\epsilon )$ if, for any $q\in \mathrm{\Gamma }$ and any Σ that passes through q, there exists $q^{\epsilon }\in \mathrm{\Sigma }$ a critical point of $T^{\mathrm{\Sigma }}(?,\epsilon )$ such that $q^{\epsilon }\to q$ as $\epsilon \to 0$. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of $\dot{x}=X(x,0)$ is a continuable critical orbit in any family of the form $\dot{x}=X(x,\epsilon )$. We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center $\dot{x}=X(x,0)$.     

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

We consider the following Schrödinger equation $-{\hslash }^2\Delta u+V\left(x\right)u=\Gamma \left(x\right)f\left(u\right)\mathrm{in}{\mathbb{R}}^N,$where $u\in H^1\left({\mathbb{R}}^N\right)$, $u>0$, $\hslash >0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $\Gamma$ attains global maximum at the same point or $V$ attains global minimum and $\Gamma$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash >0$.     

Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term

This paper discusses the quasilinear Schrödinger equation $-\Delta u+V\left(x\right)u-\Delta \left[{(1+u^2)}^{\frac{1}{2}}\right]\frac{u}{2{(1+u^2)}^{\frac{1}{2}}}=K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N,$where $N⩾3$. Under appropriate assumptions on the potentials $V$ and $K$ and local sublinear growth assumptions on the nonlinear term $f$, we get the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution.     

Irreducible polynomials from a cubic transformation

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field ${\mathbb{F}}_q$. We count the irreducible polynomials in ${\mathbb{F}}_q[x]$, of a given degree, that have the form $h{(x)}^{\mathrm{deg}f}\cdot f(R(x))$ for some $f(x)\in {\mathbb{F}}_q[x]$. As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.     

Daisy cubes and distance cube polynomial

Let $X\subseteq {\{0,1\}}^n$. The daisy cube $Q_n\left(X\right)$ is introduced as the subgraph of $Q_n$ induced by the union of the intervals $I(x,0^n)$ over all $x\in X$. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If $u$ is a vertex of a graph $G$, then the distance cube polynomial $D_{G,u}(x,y)$ is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to $Q_k$ at a given distance from the vertex $u$. It is proved that if $G$ is a daisy cube, then $D_{G,0^n}(x,y)=C_G(x+y-1)$, where $C_G\left(x\right)$ is the previously investigated cube polynomial of $G$. It is also proved that if $G$ is a daisy cube, then $D_{G,u}(x,-x)=1$ holds for every vertex $u$ in $G$.