Vortex solitons of the (3+1)-dimensional spatially modulated cubic–quintic nonlinear Schrödinger equation with the transverse modulation

Vortex solitons in the spatially modulated cubic–quintic nonlinear media are governed by a (3+1)-dimensional cubic–quintic nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation. Via the variable separation principle with the similarity transformation, we derive two families of vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media. For the disappearing and parabolic transverse modulation, vortex solitons with different configurations are constructed. The similar configurations of vortex solitons exist for the same value of \(l-k\) with the topological charge k and degree number l. Moreover, the number of the inner layer structure of vortex solitons getting rid of the package covering layer is related to \((n-1)/2+1\) with the soliton order number n. For the disappearing transverse modulation, there exist phase azimuthal jumps around their cores of vortex solitons with \(2\pi \) phase change in every jump, and any two jumps one after another realize the change in \(\pi \). For the parabolic transverse modulation, all phases of vortex soliton exist k-jump, and every jump realizes the change in \(2\pi /k\); thus, k-jumps totally realize the azimuthal change in \(2\pi \) around their cores.     

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.     

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.     

Fundamental solitons and dynamical analysis in the defocusing Kerr medium and $$\varvec{\mathcal {}}$$-symmetric rational potential

We find that a class of parity-time- (\(\mathcal {PT}\)-) symmetric rational potentials can support stable solitons in the defocusing Kerr-nonlinear media, though they may not enjoy entirely real linear spectra. Analytical expressions of spatial solitons are elicited at lots of isolated propagation-constant points, around which several families of numerical fundamental solitons can be found to be stable, which is validated by linear stability analysis and nonlinear wave propagation. Many other intriguing properties of nonlinear localized modes are also discussed in detail, including the interactions, excitations, and transverse power flows. The idea of the \(\mathcal {PT}\)-symmetric rational potentials can also be extended to other types of nonlinear wave models.     

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.     

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.     

Resonance solitons produced by azimuthal modulation in self-focusing and self-defocusing materials

We demonstrate azimuthally modulated resonance scalar and vector solitons in self-focusing and self-defocusing materials. They are constructed by selecting appropriately self-consistency and resonance conditions in a coupled system of multicomponent nonlinear Schrödinger equations. In the case with zero modulation depth, it was found that the larger the topological charge, the smaller the intensity of the soliton in the self-focusing material, while in the self-defocusing material the opposite holds. For the solitons with the same parameters, the ones in the self-focusing material possess larger optical intensity than the ones in the self-defocusing material. The stability of resonance solitons is examined by direct numerical simulation, which demonstrated that a new class of stable scalar fundamental soliton states with m=0 and low-order vector vortex soliton states with m=1 can be supported by self-focusing and self-defocusing materials. Higher-order solitons are found unstable, however, displaying quasi-stable propagation over prolonged distances.     

Soliton solutions for the reduced Maxwell–Bloch system in nonlinear optics via the N-fold Darboux transformation

Under investigation in this paper is the reduced Maxwell?CBloch system, which describes the propagation of the intense ultra-short optical pulses through a two-level dielectric medium. Through symbolic computation, conservation laws are derived and N-fold Darboux transformation (DT) is constructed for that system. By virtue of the DT obtained, multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Elastic interactions between two bright one-peak solitons, between two bight two-peak solitons and between two dark two-peak solitons; (2) Parallel propagations between two bright one-peak solitons, between two bright two-peak solitons and between two dark two-peak solitons; (3) Periodic propagations of hump solitons, of a pair of bound hump solitons with the same amplitude and of dark solitons.     

Interaction soliton–sable dans un canal en eau peu profonde

Interaction between solitons and a sandy bed in shallow water is investigated. In our experiments, solitons are generated on the background of a harmonic wave, in a wave flume used in resonant mode. It is found that the sand ripples formed by the solitons propagation induce a significant decrease of solitons amplitude and of the phase shift between the soliton and the harmonic wave. However, the amplitude of the harmonic wave is approximately constant. The possible physical processes of such behaviour for the soliton amplitude and for the harmonic wave amplitude are discussed. To cite this article: F. Marin et al., C. R. Mecanique 333 (2005).     

Multi-soliton solutions for the three-coupled KdV equations engendered by the Neumann system

Korteweg–de Vries (KdV)-type equations describe certain nonlinear phenomena in fluids and plasmas. In this paper, three-coupled KdV equations corresponding to the Neumann system of the fourth-order eigenvalue problem is investigated. Through the dependent variable transformations, bilinear forms of such equations are obtained, from which the multi-soliton solutions are derived. Soliton propagation and interaction are analyzed: (1) Bell- and anti-bell-shaped solitons are found; (2) Among the soliton images, one depends on the sign of wave numbers k _i ’s (i=1,2,3), while the others are independent of such a sign; (3) Interaction between two solitons and among three solitons are elastic, i.e., the amplitude and velocity of each soliton remain unvaried after the interaction except for the phase shift.     

Solitons and their collisions in the spinor Bose–Einstein condensates

Under investigation in this paper is a system of three-component Gross?CPitaevskii equations, which can describe the dynamics of F=1 spinor Bose?CEinstein condensates in one dimension. By employing the Hirota method and symbolic computation, we obtain the explicit bright one- and two-soliton solutions for the system in the integrable case, which is associated with the attractive mean-field collision and ferromagnetic spin-exchange collision. According to the spin states, we classify the one-soliton solutions into two types: the ferromagnetic and polar solitons. Ferromagnetic solitons in three components share the same pulse shape. Polar solitons in three components have the one- or two-peak profiles, and the separated distance between two peaks is related to the polarization parameters. Based on the asymptotic analysis, collisions between two solitons are discussed in the polar?Cpolar, polar?Cferromagnetic, and ferromagnetic?Cferromagnetic cases, respectively.     

Stable localized spatial solitons in \mathcal {PT}-symmetric potentials with power-law nonlinearity

We derive analytical spatial soliton solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity in \(\mathcal {PT}\) -symmetric potentials. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulations. Moreover, some dynamical characteristics of these solutions, such as the phase switch, the power, and the transverse power-flow density, are also examined.     

Solution of the heat-transfer equation for equilibrium turbulent boundary layers when the temperature distribution of the streamlined surface is arbitrary

We give an approximate solution of the heat-transfer equation for equilibrium turbulent boundary layers for which the velocity distribution and the coefficient of turbulent viscosity can be described by functions of two parameters. In [1–4] equilibrium turbulent boundary layers characterized by a constant dimensionless pressure gradient were investigated. The $$\beta = \frac{{\delta ^{* \circ } }}{{\tau _w ^ \circ }}\left( {\frac{{dP}}{{dx^ \circ }}} \right)$$ profile of the velocity defect was calculated in [4] for such layers throughout the whole range ?0.5≤β≤∞, while a method was indicated in [5] for combining the defect velocity profiles with the universal profiles of the wall law, and a composite function defining the coefficient of turbulent viscosity was proposed. In this paper we construct the solution of the heat-transfer equation for equilibrium boundary layers under the assumption that the velocity distribution in the layer and the coefficient of turbulent viscosity are described by functions, obtained in [4, 5], of the dimensionless coordinateη=y/Δ, depending on two parametersβ and Re_*, while the turbulent Prandtl number Pr_t is either constant or is also a known function of η and the parametersβ and Re_*. The temperature of the surface T_w(x) is assumed to be an arbitrary function of the longitudinal coordinate and the solution is constructed in the form of series in the form parameters containing the derivatives of T_w(x). These form parameters are similar to those used in [6–9] to construct exact solutions of the equations of the laminar boundary layer.     

Helical Structures in the Near Field of a Turbulent Pipe Jet

We perform a finely resolved Large-eddy simulation to study coherent vortical structures populating the initial (near-nozzle) zone of a pipe jet at the Reynolds number of 5300. In contrast to ‘top-hat’ jets featured by Kelvin-Helmholtz rings with the non-dimensional frequency S t≈0.3?0.6, no high-frequency dominant mode is observed in the near field of a jet issuing from a fully-developed pipe flow. Instead, in shear layers we observe a relatively wide peak in the power spectrum within the low-frequency range (S t≈0.14) corresponding to the propagating helical waves entering with the pipe flow. This is confirmed by the Fourier transform with respect to the azimuthal angle and the Proper Orthogonal Decomposition complemented with the linear stability analysis revealing that this low-frequency motion is not connected to the Kelvin-Helmholtz instability. We demonstrate that the azimuthal wavenumbers m=1?5 contain the most of the turbulent kinetic energy and that a common form of an eigenmode is a helical vortex rotating around the axis of symmetry. Small and large timescales are identified corresponding to “fast” and “slow” rotating modes. While the “fast” modes correspond to background turbulence and stochastically switch from co- to counter-rotation, the “slow” modes are due to coherent helical structures which are long-lived and have low angular velocities, in agreement with the previously described spectral peak at low S t.     

Three-dimensional instability on the interaction between a vortex and a stationary sphere

We present direct numerical simulations of the interaction between a vortex ring and a stationary sphere for Re = 2,000. We analyze the vortex dynamics of the ring as it approaches the sphere surface, and the boundary layer formed on the surface of the sphere undergoes separation to form a secondary vortex ring. This secondary vortex ring can develop azimuthal instabilities, which grow rapidly as it interacts with the primary ring. The azimuthal instabilities on both rings are characterized by analysis of the azimuthal component decomposition of the axial vorticity.     

Scalar and vector multipole and vortex solitons in the spatially modulated cubic–quintic nonlinear media

We derive scalar and vector multipole and vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media, which is governed by a (3+1)-dimensional N-coupled cubic–quintic nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation. If the modulation depth \(q=1\), the vortex soliton is constructed, and if \(q=0\), the multipole soliton, including dipole, quadrupole, hexapole, octopole and dodecagon solitons, is constructed, respectively, when the topological charge \(k=1\)–5. If the topological charge \(k=0\), scalar solitons can be obtained. Moreover, the number of layers for the scalar and vector multipole and vortex solitons is decided by the value of the soliton order number n.     

On the N-wave equations and soliton interactions in two and three dimensions

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann-Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave equations with ?₂ × ?₂ reduction group allow breather-like solitons. Finally it is demonstrated that RHP with sewing function depending on three variables t, x and y provides some special solutions of the N-wave equations in three dimensions.     

Freak waves as nonlinear stage of Stokes wave modulation instability

Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done using two-dimensional code, based on conformal mapping of the fluid to the lower half-plane. We have considered two problems: (i) modulation instability of wave train and (ii) evolution of NLSE solitons with different steepness of carrier wave. In both cases we have observed formation of freak waves.     

Estimates of azimuthal numbers associated with elementary elliptic cylinder wave functions

The paper deals with issues related to the construction of solutions, 2 π-periodic in the angular variable, of the Mathieu differential equation for the circular elliptic cylinder harmonics, the associated characteristic values, and the azimuthal numbers needed to form the elementary elliptic cylinder wave functions. A superposition of the latter is one possible form for representing the analytic solution of the thermoelastic wave propagation problem in long waveguides with elliptic cross-section contour. The classical Sturm-Liouville problem for the Mathieu equation is reduced to a spectral problem for a linear self-adjoint operator in the Hilbert space of infinite square summable two-sided sequences. An approach is proposed that permits one to derive rather simple algorithms for computing the characteristic values of the angular Mathieu equation with real parameters and the corresponding eigenfunctions. Priority is given to the application of the most symmetric forms and equations that have not yet been used in the theory of the Mathieu equation. These algorithms amount to constructing a matrix diagonalizing an infinite symmetric pentadiagonal matrix. The problem of generalizing the notion of azimuthal number of a wave propagating in a cylindrical waveguide to the case of elliptic geometry is considered. Two-sided mutually refining estimates are constructed for the spectral values of the Mathieu differential operator with periodic and half-periodic (antiperiodic) boundary conditions.     

Stability of Bragg grating solitons in a semilinear dual-core system with cubic–quintic nonlinearity

The existence and stability of quiescent Bragg grating solitons in a dual-core fiber, where one core contains a Bragg grating with cubic–quintic nonlinearity, and the other is a linear are studied. The model admits two disjoint bandgaps when the relative group velocity in the linear core, c, is zero: one in the upper half and the other in the lower half of the system’s linear spectrum. In the general case (i.e., \(c\ne 0\)), a central gap (which is a genuine gap) is formed, while the lower and upper gaps overlap with one branch of continuous spectrum, and therefore, they are not genuine bandgaps. For quiescent solitons, exact analytical solutions are found in implicit form for \(c=0\). For nonzero c, soliton solutions are obtained numerically. The system supports two disjoint families (referred to as Type 1 and Type 2) of zero-velocity soliton solutions, separated by a border. Both Type 1 and Type 2 soliton solutions exist throughout the upper and lower gaps but not in the central gap. The stability of both soliton families is investigated by means of systematic numerical simulations. It is found that Type 2 solitons are always unstable and are destroyed upon propagation. On the other hand, unstable Type 1 solitons may either decay into radiation or radiate some energy and evolve into a moving Type 1 soliton. Also, in the case of Type 1 solitons, we have identified stable regions in the plane of quintic nonlinearity and frequency. The influence of coupling coefficient and the relative group velocity in the linear core on the stability of solitons are analyzed.