Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {}$$-symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.     

Finite plasticity in $$\varvec{P}^\top \! \varvec{P}$$. Part I: constitutive model

We address a finite-plasticity model based on the symmetric tensor \(\varvec{P}^\top \! \varvec{P}\) instead of the classical plastic strain \(\varvec{P}\). Such a structure arises by assuming that the material behavior is invariant with respect to frame transformations of the intermediate configuration. The resulting variational model is lower dimensional, symmetric and based solely on the reference configuration. We discuss the existence of energetic solutions at the material-point level as well as the convergence of time discretizations. The linearization of the model for small deformations is ascertained via a rigorous evolution-\(\Gamma \)-convergence argument. The constitutive model is combined with the equilibrium system in Part II where we prove the existence of quasistatic evolutions and ascertain the linearization limit (Grandi and Stefanelli in 2016).     

Linearly Stable Quasi-Periodic Breathers in a Class of Random Hamiltonian Systems

In this paper, we construct linearly stable quasi-periodic breathers for the Hamiltonian systems in the form \({{\rm i} \dot{q}_n+v_n q_n+\delta|q_n|^2q_n+\varepsilon_n \left(q_{n+1}+q_{n-1} \right)=0,\quad n \in \mathbb{Z}}\) where \({\{v_n\}_{n \in \mathbb{Z}}}\) is a family of time independent identically distributed (i.i.d) random variables with common distribution \({g = dv_n, v_n \in [0,1]}\) and \({|\varepsilon_n| \leq \varepsilon e^{-\varrho |n|}}\) with \({\varepsilon,\varrho > 0}\) . We prove that for \({\varepsilon, \delta}\) sufficiently small, the equation admits a family of small-amplitude and linearly stable, time quasi-periodic solutions for most of the parameters \({\{v_n\}_{n \in \mathbb{Z}}}\) .     

Oriented Shadowing Property and $$\Omega $$-Stability for Vector Fields

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.     

Stability of Fixed Points and Associated Relative Equilibria of the 3-Body Problem on $${\mathbb {S}}^1$$ and $${\mathbb {S}}^2$$S2

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to \({\mathbb {S}}^1\), but unstable if the bodies are considered in \({\mathbb {S}}^2\).     

Lumps and rogue waves of generalized Nizhnik–Novikov–Veselov equation

In this paper, via generalized bilinear forms, we consider the (\(2+1\))-dimensional bilinear p-Sawada–Kotera (SK) equation. We derive analytical rational solutions in terms of positive quadratic functions. Through applying the dependent transformation, we present a class of lump solutions of the (\(2+1\))-dimensional SK equation. Those rationally decaying solutions in all space directions exhibit two kinds of characters, i.e., bright lump wave (one peak and two valleys) and bright–dark lump wave (one peak and one valley). In addition, we also obtain three families of bright–dark lump wave solutions to the nonlinear p-SK equation for \(p=3\).     

A Note on the Relative Equilibria Bifurcations in the $$$$-Body Problem

Consider the planar Newtonian \((2N+1)\)-body problem, \(N\ge 1,\) with \(2N\) bodies of unit mass and one body of mass \(m\). Using the discrete symmetry due to the equal masses and reducing by the rotational symmetry, we show that solutions with the \(2N\) unit mass points at the vertices of two concentric regular \(N\)-gons and \(m\) at the centre at all times form invariant manifold. We study the regular \(2N\)-gon with central mass \(m\) relative equilibria within the dynamics on the invariant manifold described above. As \(m\) varies, we identify the bifurcations, relate our results to previous work and provide the spectral picture of the linearization at the relative equilibria.     

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.     

Rossby solitary waves excited by the unstable topography in weak shear flow

A new forced KdV equation including topography is derived and the numerical solutions are given. The topographic variable should be related with the temporal and spatial function, which is called unstable topography. The physical features of the solitary waves about the mass and energy are discussed by theoretical analysis. In further studies, the pseudo-spectral numerical methods are used to discuss the evolution of solitary wave generated by the topography when meridional wave number \(m=1\); in a similar way, we analyze the solitary wave when meridional wave number \(m=2\). At last, we make the comparison for the characteristics of waves between \(m=1\) and \(m=2\), the wave of meridional number \(m=1\) plays a leading role.     

Displacement and Dissolution Characteristics of CO$$_{}$$/Brine System in Unconsolidated Porous Media

This study investigated the dynamic displacement and dissolution of \(\hbox {CO}_{2}\) in porous media at 313 K and 6/8 MPa. Gaseous (\(\hbox {gCO}_{2}\)) at 6 MPa and supercritical \(\hbox {CO}_{2 }(\hbox {scCO}_{2}) \) at 8 MPa were injected downward into a glass bead pack at different flow rates, following upwards brine injection. The processes occurring during \(\hbox {CO}_{2}\) drainage and brine imbibition were visualized using magnetic resonance imaging. The drainage flow fronts were strongly influenced by the flow rates, resulting in different gas distributions. However, brine imbibition proceeded as a vertical compacted front due to the strong effect of gravity. Additionally, the effects of flow rate on distribution and saturation were analyzed. Then, the front movement of \(\hbox {CO}_{2}\) dissolution was visualized along different paths after imbibition. The determined \(\hbox {CO}_{2}\) concentrations implied that little \(\hbox {scCO}_{2}\) dissolved in brine after imbibition. The dissolution rate was from \(10^{-8}\) to \(10^{-9}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) and from \(10^{-6}\) to \(10^{-8}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) for \(\hbox {gCO}_{2}\) at 6 MPa and \(\hbox {scCO}_{2 }\) at 8 MPa, respectively. The total time for the \(\hbox {scCO}_{2}\) dissolution was short, indicating fast mass transfer between the \(\hbox {CO}_{2}\) and brine. Injection of \(\hbox {CO}_{2}\) under supercritical conditions resulted in a quick establishment of a steady state with high storage safety.     

Existence of Traveling Waves of General Gray-Scott Models

This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m (\(m > 1\)) between two chemical species, a reactant A and an auto-catalyst B, \( A + m B \rightarrow (m+1) B\), and a super-linear decay of order \( n > 1\), \( B \rightarrow C\), where \( 1< n < m\). Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value \(u_0\) at \( - \infty \). The explicit bound is also derived in terms of \(u_0\) and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases.     

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.     

Application of linear and nonlinear vibration absorbers in micro-milling process in order to suppress regenerative chatter

In this paper, a generalized higher-order variable-coefficient nonlinear Schrödinger equation is studied, which describes the propagation of subpicosecond or femtosecond pulses in an inhomogeneous optical fiber. We derive a set of the integrable constraints on the variable coefficients. Under those constraints, via the symbolic computation and modified Hirota method, bilinear equations, one-, two-,three-soliton solutions and dromion-like structures are obtained. Properties and interactions for the solitons are studied: (a) effects on the solitons resulting from the wave number k, third-order dispersion \(\delta _1(z)\), group velocity dispersion \(\alpha (z)\), gain/loss \(\varGamma _2(z)\) and group-velocity-related \(\gamma (z)\) are discussed analytically and graphically where z is the normalized propagation distance along the fiber; (b) bound state with different values of \(\alpha (z)\), \(\delta _1(z)\), \(\gamma (z)\) and \(\varGamma _2(z)\) are presented where some periodic or quasiperiodic formulae are derived. Interactions between the two solitons and between the bound states and a single soliton are, respectively, discussed; and (c) single, double and triple dromion-like structures with different values of \(\alpha (z)\), \(\delta _1(z)\), \(\gamma (z)\) are also presented, distortions of which are found to be determined by those variable coefficients.     

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 \(f\).     

Visual Investigation of Improvement in Extra-Heavy Oil Recovery by Borate-Assisted $$\hbox {CO}_{2}$$-Foam Injection

The significant reduction in heavy oil viscosity when mixed with \(\hbox {CO}_{2}\) is well documented. However, for \(\hbox {CO}_{2}\) injection to be an efficient method for improving heavy oil recovery, other mechanisms are required to improve the mobility ratio between the \(\hbox {CO}_{2}\) front and the resident heavy oil. In situ generation of \(\hbox {CO}_{2}\)-foam can improve \(\hbox {CO}_{2}\) injection performance by (a) increasing the effective viscosity of \(\hbox {CO}_{2}\) in the reservoir and (b) increasing the contact area between the heavy oil and injected \(\hbox {CO}_{2}\) and hence improving \(\hbox {CO}_{2}\) dissolution rate. However, in situ generation of stable \(\hbox {CO}_{2}\)-foam capable of travelling from the injection well to the production well is hard to achieve. We have previously published the results of a series of foam stability experiments using alkali and in the presence of heavy crude oil (Farzaneh and Sohrabi 2015). The results showed that stability of \(\hbox {CO}_{2}\)-foam decreased by addition of NaOH, while it increased by addition of \(\hbox {Na}_{2}\hbox {CO}_{3}\). However, the highest increase in \(\hbox {CO}_{2}\)-foam stability was achieved by adding borate to the surfactant solution. Borate is a mild alkaline with an excellent pH buffering ability. The previous study was performed in a foam column in the absence of a porous medium. In this paper, we present the results of a new series of experiments carried out in a high-pressure glass micromodel to visually investigate the performance of borate–surfactant \(\hbox {CO}_{2}\)-foam injection in an extra-heavy crude oil in a transparent porous medium. In the first part of the paper, the pore-scale interactions of \(\hbox {CO}_{2}\)-foam and extra-heavy oil and the mechanisms of oil displacement and hence oil recovery are presented through image analysis of micromodel images. The results show that very high oil recovery was achieved by co-injection of the borate–surfactant solution with \(\hbox {CO}_{2}\), due to in-situ formation of stable foam. Dissolution of \(\hbox {CO}_{2}\) in heavy oil resulted in significant reduction in its viscosity. \(\hbox {CO}_{2}\)-foam significantly increased the contact area between the oil and \(\hbox {CO}_{2}\) significantly and thus the efficiency of the process. The synergy effect between the borate and surfactant resulted in (1) alteration of the wettability of the porous medium towards water wet and (2) significant reduction of the oil–water IFT. As a result, a bank of oil-in-water (O/W) emulsion was formed in the porous medium and moved ahead of the \(\hbox {CO}_{2}\)-foam front. The in-situ generated O/W emulsion has a much lower viscosity than the original oil and plays a major role in the observed additional oil recovery in the range of performed experiments. Borate also made \(\hbox {CO}_{2}\)-foam more stable by changing the system to non-spreading oil and reducing coalescence of the foam bubbles. The results of these visual experiments suggest that borate can be a useful additive for improving heavy oil recovery in the range of the performed tests, by increasing \(\hbox {CO}_{2}\)-foam stability and producing O/W emulsions.     

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 new type of functional chemical sensitizer $$\hbox {MgH}_{2}$$ for improving pressure desensitization resistance of emulsion explosives

In millisecond-delay blasting and deep water blasting projects, traditional emulsion explosives sensitized by the chemical sensitizer \(\hbox {NaNO}_{2}\) often encounter incomplete explosion or misfire problems because of the “pressure desensitization” phenomenon, which seriously affects blasting safety and construction progress. A \(\hbox {MgH}_{2}\)-sensitized emulsion explosive was invented to solve these problems. Experimental results show that \(\hbox {MgH}_{2}\) can effectively reduce the problem of pressure desensitization. In this paper, the factors which influence the pressure desensitization of two types of emulsion explosives are studied, and resistance to this phenomenon of \(\hbox {MgH}_{2}\)-sensitized emulsion explosives is discussed.     

Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions

We consider the elliptic equation \(-\Delta u +u =0\) with nonlinear boundary condition \(\frac{\partial u}{\partial n}= \lambda u + g(\lambda ,x,u), \) where \(\frac{g(\lambda ,x,s)}{s} \rightarrow 0, \hbox { as }|s|\rightarrow \infty \) and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.     

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.