Further Dense Properties of the Space of Circle Diffeomorphisms with a Liouville Rotation Number

In continuation of Matsumoto’s paper (Nonlinearity 25:1495–1511, 2012) we show that various subspaces are \(C^{\infty }\)-dense in the space of orientation-preserving \(C^{\infty }\)-diffeomorphisms of the circle with rotation number \(\alpha \), where \(\alpha \in {\mathbb {S}}^1\) is any prescribed Liouville number. In particular, for every odometer \({\mathcal {O}}\) of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to \({\mathcal {O}}\).     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727–790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit \({\mathcal {O}}_{p}\) with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \). We prove that \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) is a three-dimensional \(C^{1}\)-submanifold of the phase space and admits a smooth global graph representation. Within \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \), there exist heteroclinic connections from \({\mathcal {O}}_{p}\) to three different periodic orbits. These connecting sets are two-dimensional \(C^{1}\)-submanifolds of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) and homeomorphic to the two-dimensional open annulus. They form \(C^{1}\)-smooth separatrices in the sense that they divide the points of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) into three subsets according to their \(\omega \)-limit sets.     

Delta Shock Waves in the Shallow Water System

We consider a Riemann problem for the shallow water system \(u_{t} +\big (v+\textstyle \frac{1}{2}u^{2}\big )_{x}=0\), \(v_{t}+\big (u+uv\big )_{x}=0\) and evaluate all singular solutions of the form \(u(x,t)=l(t)+b(t)H\big (x-\gamma (t)\big )+a(t)\delta \big (x-\gamma (t)\big )\), \(v(x,t)=k(t)+c(t)H\big (x-\gamma (t)\big )\), where \(l,b,a,k,c,\gamma :\mathbb {R}\rightarrow \mathbb {R}\) are \(C^{1}\)-functions of time t, H is the Heaviside function, and \(\delta \) stands for the Dirac measure with support at the origin. A product of distributions, not constructed by approximation processes, is used to define a solution concept, that is a consistent extension of the classical solution concept. Results showing the advantage of this framework are briefly presented in the introduction.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Initial and Boundary Blow-U Problem for $$$$-Lalacian Parabolic Equation with General Absortion

In this article, we investigate the initial and boundary blow-up problem for the \(p\)-Laplacian parabolic equation \(u_t-\Delta _p u=-b(x,t)f(u)\) over a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(N\ge 2\), where \(\Delta _pu=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) with \(p>1\), and \(f(u)\) is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.     

One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity

We consider positive classical solutions of

$$\begin{aligned} v_t=(v^{m-1}v_x)_x, \qquad x\in {\mathbb {R}}, \ t>0, \qquad (\star ) \end{aligned}$$

in the super-fast diffusion range \(m<-1\). Our main interest is in smooth positive initial data \(v_0=v(\cdot ,0)\) which decay as \(x\rightarrow +\infty \), but which are possibly unbounded as \(x\rightarrow -\infty \), having in mind monotonically decreasing data as prototypes. It is firstly proved that if \(v_0\) decays sufficiently fast only in one direction by satisfying

$$\begin{aligned} v_0(x) \le cx^{-\beta } \qquad \text{ for } \text{ all } ~x>0 \quad \hbox { with some }\quad \beta >\frac{2}{1-m} \end{aligned}$$

and some \(c>0\), then the so-called proper solution of (\(\star \)) vanishes identically in \({\mathbb {R}}\times (0,\infty )\), and accordingly no positive classical solution exists in any time interval in this case. Complemented by some sufficient criteria for solutions to remain positive either locally or globally in time, this condition for instantaneous extinction is shown to be optimal at least with respect to algebraic decay of the initial data. This partially extends some known nonexistence results for (\(\star \)) (Daskalopoulos and Del Pino in Arch Rat Mech Anal 137(4):363–380, 1997) in that it does not require any knowledge on the behavior of \(v_0(x)\) for \(x<0\). Next focusing on the phenomenon of extinction in finite time, we show that in this respect a mass influx from \(x=-\infty \) can interact with mass loss at \(x=+\infty \) in a nontrivial manner. Namely, we shall detect examples of monotone initial data, with critical decay as \(x\rightarrow +\infty \) and exponential growth as \(x\rightarrow -\infty \), that lead to solutions of (\(\star \)) which become extinct at a finite positive time, but which have empty extinction sets. This is in sharp contrast to known extinction mechanisms which are such that the corresponding extinction sets coincide with all of \({\mathbb {R}}\).

     

Synchronization transitions induced by partial time delay in a excitatory–inhibitory coupled neuronal network

Information transmission delays are an inherent factor of neuronal systems as a consequence of the finite propagation speeds and time lapses occurring by both dendritic and synaptic processes. In real neuronal systems, some delay between two neurons is too small and can be ignored, which results in partial time delay. In this paper, we focus on investigating influences of partial time delay on synchronization transitions in a excitatory–inhibitory (E–I) coupled neuronal networks. Here, we suppose time delay between two neurons equals to \(\tau \) with probability \(p_{\mathrm{delay}}\) and investigate effect of partial time delay on synchronization transitions of the neuronal networks by controlling \(\tau \) and \(p_{\mathrm{delay}}\) under three cases. In these three cases, excitatory synapses are always considered to delayed with probability \(p_{\mathrm{delay}}\), while inhibitory synapses are considered to be without delays (case I), delayed with probability \(p_{\mathrm{delay}}\) (case II), and always delayed (case III), respectively. It is revealed that, in the first two cases, partial time delay has little influences on synchronization of the neuronal network for small \(p_{\mathrm{delay}}\), while it could induce synchronization transitions at \(\tau \) around integer multiples of the period of individual neuron T when \(p_{\mathrm{delay}}\) is large enough, while in the case III, partial time delay could induce synchronization transitions at \(\tau \) being around odd integer multiples of T / 2 for small \(p_{\mathrm{delay}}\) and at \(\tau \) being around integer multiples of T for large \(p_{\mathrm{delay}}\). Most interesting observation is that partial time delay could induce frequent synchronization transitions at \(\tau \) being around integer multiples of T / 2 for intermediate \(p_{\mathrm{delay}}\). Moreover, effect of rewiring probability on synchronization transitions induced by partial time delay has been discussed. It is found that synchronization transitions induced by partial time delay are robust to rewiring probability for large \(p_{\mathrm{delay}}\) under the three cases.     

Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).     

On the Damped Harmonic Oscillator with Time Dependent Damping Coefficient

Conditions guaranteeing asymptotic stability for the differential equation

$$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$

are studied, where the damping coefficient \(h:[0,\infty )\rightarrow [0,\infty )\) is a locally integrable function, and the frequency \(\omega >0\) is constant. Our conditions need neither the requirement \(h(t)\le \overline{h}<\infty \) (\(t\in [0,\infty )\); \(\overline{h}\) is constant) (“small damping”), nor \(0< \underline{h}\le h(t)\) (\(t\in [0,\infty )\); \(\underline{h}\) is constant) (“large damping”); in other words, they can be applied to the general case \(0\le h(t)<\infty \) (\(t\in [0,\infty \))). We establish a condition which combines weak integral positivity with Smith’s growth condition

$$\begin{aligned} \int ^\infty _0 \exp [-H(t)]\int _0^t \exp [H(s)]\,\mathrm{{d}}s\,\mathrm{{d}}t=\infty \qquad \left( H(t):=\int _0^t h(\tau )\,\mathrm{{d}}\tau \right) , \end{aligned}$$

so it is able to control both the small and the large values of the damping coefficient simultaneously.

     

Uniaxial experimental study of the acoustic emission and deformation behavior of composite rock based on 3D digital image correlation (DIC)

In this paper, uniaxial compression tests were carried out on a series of composite rock specimens with different dip angles, which were made from two types of rock-like material with different strength. The acoustic emission technique was used to monitor the acoustic signal characteristics of composite rock specimens during the entire loading process. At the same time, an optical non-contact 3 D digital image correlation technique was used to study the evolution of axial strain field and the maximal strain field before and after the peak strength at different stress levels during the loading process. The effect of bedding plane inclination on the deformation and strength during uniaxial loading was analyzed. The methods of solving the elastic constants of hard and weak rock were described. The damage evolution process, deformation and failure mechanism, and failure mode during uniaxial loading were fully determined. The experimental results show that the θ = 0?–45?specimens had obvious plastic deformation during loading, and the brittleness of the θ = 60?–90?specimens gradually increased during the loading process. When the anisotropic angle θincreased from 0?to 90?, the peak strength, peak strain,and apparent elastic modulus all decreased initially and then increased. The failure mode of the composite rock specimen during uniaxial loading can be divided into three categories:tensile fracture across the discontinuities(θ = 0?–30?), slid-ing failure along the discontinuities(θ = 45?–75?), and tensile-split along the discontinuities(θ = 90?). The axial strain of the weak and hard rock layers in the composite rock specimen during the loading process was significantly different from that of the θ = 0?–45?specimens and was almost the same as that of the θ = 60?–90?specimens. As for the strain localization highlighted in the maximum principal strain field, the θ = 0?–30?specimens appeared in the rock matrix approximately parallel to the loading direction,while in the θ = 45?–90?specimens it appeared at the hard and weak rock layer interface.     

On the Convexity of Nonlinear Elastic Energies in the Right Cauchy-Green Tensor

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies \(W(F)=\widehat{W}(F^{T}F)=\widehat{W}(C)\) which are convex with respect to the right Cauchy-Green tensor \(C=F^{T}F\), where \(F\) denotes the gradient of deformation. Examples of such energies exhibiting a blow up for \(\det F\to0\) are given.     

Bifurcations of a creeping air–water flow in a conical container

This numerical study describes the eddy emergence and transformations in a slow steady axisymmetric air–water flow, driven by a rotating top disk in a vertical conical container. As water height \(H_{\mathrm{w}}\) and cone half-angle \(\beta \) vary, numerous flow metamorphoses occur. They are investigated for \(\beta =30^{\circ }, 45^{\circ }\), and \(60^{\circ }\). For small \(H_{\mathrm{w}}\), the air flow is multi-cellular with clockwise meridional circulation near the disk. The air flow becomes one cellular as \(H_{\mathrm{w}}\) exceeds a threshold depending on \(\beta \). For all \(\beta \), the water flow has an unbounded number of eddies whose size and strength diminish as the cone apex is approached. As the water level becomes close to the disk, the outmost water eddy with clockwise meridional circulation expands, reaches the interface, and induces a thin layer with anticlockwise circulation in the air. Then this layer expands and occupies the entire air domain. The physical reasons for the flow transformations are provided. The results are of fundamental interest and can be relevant for aerial bioreactors.     

Models of Elastic Shells in Contact with a Rigid Foundation: An Asymptotic Approach

We consider a family of linearly elastic shells with thickness \(2\varepsilon\) (where \(\varepsilon\) is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface \(S\), and may enter in contact with a rigid foundation along the bottom face.We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements \(\boldsymbol{u}^{\varepsilon}\) of covariant components \(u_{i}^{\varepsilon}\)) when \(\varepsilon\) tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is \(O(\varepsilon^{2})\) and surface tractions density is \(O(\varepsilon^{3})\), the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found.     

A Shilnikov Phenomenon Due to State-Dependent Delay,by Means of the Fixed Point Index

The first part of this paper is a general approach towards chaotic dynamics for a continuous map \(f:X\supset M\rightarrow X\) which employs the fixed point index and continuation. The second part deals with the differential equation

$$\begin{aligned} x'(t)=-\alpha \,x(t-d_{{\varDelta }}(x_t)). \end{aligned}$$

with state-dependent delay. For a suitable parameter \(\alpha \) close to \(5\pi /2\) we construct a delay functional \(d_{{\varDelta }}\), constant near the origin, so that the previous equation has a homoclinic solution, \(h(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \), with certain regularity properties of the linearization of the semiflow along the flowline \(t\mapsto h_t\). The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion.

     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\):

$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$

where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.

     

Stability of Gaussian-type light bullets in the cubic-quintic-septimal nonlinear media with different diffractions under $${\mathcal {}}$$-symmetric potentials

From the governing equation \(-(3+1)\)-dimensional nonlinear Schrödinger equation with cubic-quintic-septimal nonlinearities, different diffractions and \({\mathcal {PT}}\)-symmetric potentials, we obtain two kinds of analytical Gaussian-type light bullet solutions. The septimal nonlinear term has a strong impact on the formation of light bullets. The eigenvalue method and direct numerical simulation to analytical solutions imply that stable and unstable evolution of light bullets against white noise attributes to the coaction of cubic-quintic-septimal nonlinearities, dispersion, different diffractions and \({\mathcal {PT}}\)-symmetric potential.     

The stability of two-dimensional spatial solitons in cubic–quintic–septimal nonlinear media with different diffractions and $${\varvec{\mathcal {}}}$$-symmetric potentials

A (2+1)-dimensional nonlinear Schrödinger equation in cubic–quintic–septimal nonlinear media with different diffractions and \({\mathcal {PT}}\)-symmetric potentials is studied, and (2+1)-dimensional spatial solitons are derived. The stable region of analytical spatial solitons is discussed by means of the eigenvalue method. The direct numerical simulation indicates that analytical spatial soliton solutions stably evolve within stable region in the media of focusing septimal and focusing or defocusing cubic nonlinearities with disappearing quintic nonlinearity under the 2D extended Scarf II potential. However, under the extended \({\mathcal {PT}}\)-symmetric potential with \(p=2\) and \(p=3\), analytical spatial soliton solutions stably evolve within stable region in the media of focusing quintic and septimal nonlinearities with defocusing cubic nonlinearity. In other cases, analytical spatial soliton solutions cannot sustain their original shapes, and they are distorted and broken up and finally decay into noise.     

Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

We study the Neumann boundary value problem for the second order ODE

$$\begin{aligned} u^{\prime \prime } + (a^+(t)-\mu a^-(t))g(u) = 0, \qquad t \in [0,T], \end{aligned}$$

(1)

where \(g \in {\mathcal {C}}^1({\mathbb {R}})\) is a bounded function of constant sign, \(a^+,a^-: [0,T] \rightarrow {\mathbb {R}}^+\) are the positive/negative part of a sign-changing weight \(a(t)\) and \(\mu > 0\) is a real parameter. Depending on the sign of \(g^{\prime }(u)\) at infinity, we find existence/multiplicity of solutions for \(\mu \) in a “small” interval near the value

$$\begin{aligned} \mu _c = \frac{\int _0^T a^+(t) \, dt}{\int _0^T a^-(t) \, dt}\,. \end{aligned}$$

The proof exploits a change of variables, transforming the sign-indefinite Eq. (1) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for \(\mu \rightarrow 0^+\) and \(\mu \rightarrow +\infty \) are given, as well.