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.     

One-dimensional optical solitons in cubic–quintic–septimal media with $$\varvec{\mathcal {}}$$-symmetric potentials

A (\(1+1\))-dimensional inhomogeneous cubic–quintic–septimal nonlinear Schrödinger equation with \(\mathcal {PT}\)-symmetric potentials is studied, and two families of soliton solutions are obtained. From soliton solutions, the amplitude of soliton is independent of the \(\mathcal {PT}\)-symmetric potential parameter k; however, the phase depends on the parameter k. The phase of soliton alters from negative to positive values at the location of center. Moreover, the evolutional behaviors of these solitons are discussed.     

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.     

Remarks on the Generalized Reflectionless Schrödinger Potentials

We discuss certain compact, translation-invariant subsets of the set \({\mathcal {R}}\) of the generalized reflectionless potentials for the one-dimensional Schrödinger operator. We determine a stationary ergodic subset of \({\mathcal {R}}\) whose Lyapunov exponent is discontinuous at a point. We also determine an almost automorphic, non-almost periodic minimal subset of \(\mathcal {R}\).     

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.     

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 \({\mathcal {O}}\).     

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.     

Representation of a Smooth Isometric Deformation of a Planar Material Region into a Curved Surface

We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset \({\mathcal{D}}\) of two-dimensional Euclidean point space \(\mathbb{E}^{2}\) into a surface \({\mathcal{S}}\) in three-dimensional Euclidean point space \(\mathbb{E}^{3}\). To be isometric, such a deformation must preserve the length of every possible arc of material points on \({\mathcal{D}}\). Characterizing the curves of zero principal curvature of \({\mathcal{S}}\) is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in \(\mathbb{E}^{2}\), based upon an à priori chosen pre-image form of the curves of zero principal curvature in \({\mathcal{D}}\), and use that coordinate system to construct the most general isometric deformation of \({\mathcal{D}}\) to a smooth surface \({\mathcal{S}}\). A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of \({\mathcal{S}}\) are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).     

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.     

Dynamics of light bullets in inhomogeneous cubic-quintic-septimal nonlinear media with $${\varvec{{\mathcal {}}}}$$-symmetric potentials

A (3+1)-dimensional nonlinear Schrödinger equation with variable-coefficient dispersion/diffraction and cubic-quintic-septimal nonlinearities is studied, two families of analytical light bullet solutions with two types of \({{\mathcal {PT}}}\)-symmetric potentials are obtained. The coefficient of the septimal nonlinear term strongly influences the form of light bullet. The direct numerical simulation indicates that light bullet solutions in different cubic-quintic-septimal nonlinear media exhibit different property of stability, and under different \({\mathcal {PT}}\)-symmetric potentials they also show different stability against white noise. These stabilities of evolution originate from subtle interplay among dispersion, diffraction, nonlinearity and \({\mathcal {PT}}\)-symmetric potential. Moreover, compression and expansion of light bullets in the hyperbolic dispersion/diffraction system and periodic modulation system are investigated numerically. The evolution of light bullet in periodic modulation system is more stable than that in the hyperbolic dispersion/diffraction system.     

A novel approach for generating multi-direction multi-double-scroll attractors

Attention is focused in this work on quasiperiodic motion of nonlinear systems whose spectrum contains uniformly spaced sideband frequencies with a distance \(\omega _{d}\) apart, around a frequency \(\omega \) with \(\omega \gg \omega _{d}\) and its integer multiples, which are referred to as carrier frequencies. The ratio of the two frequencies \(\omega \) and \(\omega _{d}\) is an irrational number. A new method based on the traditional incremental harmonic balance (IHB) method with multiple timescales, referred to as Lau method, where two timescales, \(\tau _{1}=\omega t\) (a fast timescale) and \(\tau _{2}=\omega _{d}t\) (a slow timescale), are introduced, is presented to analyze quasiperiodic motion of nonlinear systems. An amplitude increment algorithm is adapted to deal with cases where the two frequencies \(\omega \) and \(\omega _{d}\) are    unknown a priori, in order to automatically trace frequency response of quasiperiodic motion of nonlinear systems and accurately calculate all frequency components and their corresponding amplitudes. Results of application of the present IHB method to quasiperiodic free vibration of a hinged–clamped beam with internal resonance between two transverse modes are shown and compared with previously published results with Lau method and those from numerical integration. While differences are noted between results predicted by the present IHB method and Lau method, excellent agreement is achieved between results from the present IHB method and numerical integration even in cases of strongly nonlinear vibration. The present IHB method is also used to analyze quasiperiodic free vibration of high-dimensional models of the hinged–clamped beam.     

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.     

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.

     

Comparative Study of Five Outcrop Chalks Flooded at Reservoir Conditions: Chemo-mechanical Behaviour and Profiles of Compositional Alteration

This study presents experimental results from a flooding test series performed at reservoir conditions for five high-porosity Cretaceous onshore chalks from Denmark, Belgium and the USA, analogous to North Sea reservoir chalk. The chalks are studied in regard to their chemo-mechanical behaviour when performing tri-axial compaction tests while injecting brines (0.219 mol/L \(\hbox {MgCl}_{2}\) or 0.657 mol/L NaCl) at reservoir conditions for 2–3 months (T = 130 \(^\circ \hbox {C}\); 1 PV/d). Each chalk type was examined in terms of its mineralogical and chemical composition before and after the mechanical flooding tests, using an extensive set of analysis methods, to evaluate the chalk- and brine-dependent chemical alterations. All \(\hbox {MgCl}_{2}\)-flooded cores showed precipitation of Mg-bearing minerals (mainly magnesite). The distribution of newly formed Mg-bearing minerals appears to be chalk-dependent with varying peaks of enrichment. The chalk samples from Aalborg originally contained abundant opal-CT, which was dissolved with both NaCl and \(\hbox {MgCl}_{2}\) and partly re-precipitated as Si-Mg-bearing minerals. The Aalborg core injected with \(\hbox {MgCl}_{2}\) indicated strongly increased specific surface area (from 4.9 \(\hbox {m}^{2}\hbox {/g}\) to within 7–9 \(\hbox {m}^{2}\hbox {/g}\)). Mineral precipitation effects were negligible in chalk samples flooded with NaCl compared to \(\hbox {MgCl}_{2}\). Silicates were the main mineralogical impurity in the studied chalk samples (0.3–6 wt%). The cores with higher \(\hbox {SiO}_{2}\) content showed less deformation when injecting NaCl brine, but more compaction when injecting \(\hbox {MgCl}_{2}\)-brine. The observations were successfully interpreted by mathematical geochemical modelling which suggests that the re-precipitation of Si-bearing minerals leads to enhanced calcite dissolution and mass loss (as seen experimentally) explaining the high compaction seen in \(\hbox {MgCl}_{2}\)-flooded Aalborg chalk. Our work demonstrates that the original mineralogy, together with the newly formed minerals, can control the chemo-mechanical interactions during flooding and should be taken into account when predicting reservoir behaviour from laboratory studies. This study improves the understanding of complex flow reaction mechanisms also relevant for field-scale dynamics seen during brine injection.     

Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

We study the asymptotic behaviour of the resolvents \({(\mathcal{A}^\varepsilon+I)^{-1}}\) of elliptic second-order differential operators \({{\mathcal{A}}^\varepsilon}\) in \({\mathbb{R}^d}\) with periodic rapidly oscillating coefficients, as the period \({\varepsilon}\) goes to zero. The class of operators covered by our analysis includes both the “classical” case of uniformly elliptic families (where the ellipticity constant does not depend on \({\varepsilon}\)) and the “double-porosity” case of coefficients that take contrasting values of order one and of order \({\varepsilon^2}\) in different parts of the period cell. We provide a construction for the leading order term of the “operator asymptotics” of \({(\mathcal{A}^\varepsilon+I)^{-1}}\) in the sense of operator-norm convergence and prove order \({O(\varepsilon)}\) remainder estimates.     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

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.     

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 \(b|u|^{2p}+ ic|u|^{2q}u\).

     

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.