Nonlinear Wave Solutions Research Articles

Space–time L2 estimates for elastic waves and applications

We establish a kind of weighted space–time L2 estimate, which belongs to Keel–Smith–Sogge type estimates, for perturbed linear elastic wave equations in Rn(n≥3). We also apply this estimate to give an alternative and short proof for John's almost global existence theorem for 3-D nonlinear elastic waves (John 1988, [10]), and show the global existence of small smooth solutions for nonlinear elastic waves in high space dimensions n≥4.

Gauge invariant gluon spin operator for spinless nonlinear wave solutions

We consider nonlinear wave type solutions with intrinsic mass scale parameter and zero spin in a pure SU(2) quantum chromodynamics (QCD). A new stationary solution which can be treated as a system of static Wu–Yang monopole dressed in off-diagonal gluon field is proposed. A remarkable feature of such a solution is that it possesses a finite energy density everywhere. All considered nonlinear wave type solutions have common features: presence of the mass scale parameter, nonvanishing projection of the color fields along the propagation direction and zero spin. The last property requires revision of the gauge invariant definition of the spin density operator which is supposed to produce spin one states for the massless vector gluon field. We construct a gauge invariant definition of the classical gluon spin density operator which is unique and Lorentz frame independent.

Steady solutions of high-level Irrotational Green–Naghdi equations for strongly nonlinear periodic waves

The Irrotational Green–Naghdi (IGN) equations are categorized into different levels. The low-level IGN equations can be used in the propagation of weakly dispersive and strongly nonlinear waves. On the other hand, high-level IGN equations can deal with strongly dispersive and strongly nonlinear waves. We focus here on the simulations of the steady solutions of nonlinear periodic waves by a low-level IGN (IGN-2) equations and high-level IGN (IGN-4 and IGN-8) equations. In numerical tests, results of wave speed, wave profile and velocity distribution are given for finite-depth water waves and for four different wave lengths as well as for large amplitude deep water waves. By comparing the simulation results, high-level equations are shown to be in better agreement with an accurate theory, namely the stream function wave theory.

Localized discrete breather modes in neuronal microtubules

We made an attempt to provide a realistic picture of the localization of energy in microtubules (MTs), and we intend to model the nonlinear dynamics of MTs using the “double-well” $$\phi ^4$$ form of the potential describing the dipole–dipole interactions. We investigate the modulational instability (MI) of the nonlinear plane wave solutions by considering both the wave vector (q) of the basic states and the wave vector (Q) of the perturbations as free parameters. A set of explicit criteria of MI is derived, and under the plane-wave perturbation, the constant amplitude solution becomes unstable and localized discrete breathers (DBs) solutions appear. We show numerically that MI is also an indicator of the presence of discrete breathers. We suggest that an electric field favourably leads the DB excitations towards the properly aligned end triggering a dissembly of the protofilament due to the energy release. These DBs could catalyse MT-associated proteins attachment/detachment and promote or inhibit the kinesin walk. We establish that the electromechanical vibrations in MTs can generate an electromagnetic field in the form of an electric pulse (breathers) which propagates along MT serving as signalling pathway in neuronal cells. The DBs in MT can be viewed as a bit of information whose propagation can be controlled by an electric filed. They might perform the role of elementary logic gates, thus implementing a subneuronal mode of computation. The generated DBs present us with novel possibilities for the direct interaction between the local electromagnetic field and the cytoskeletal structures in neurons. Thus, we emphasize that the effect of discreteness and electric field plays a significant role in MTs.

Comparison of Analytical and Numerical Simulations of Long Nonlinear Internal Solitary Waves in Shallow Water

ABSTRACT Mohapatra, S.C.; Fonseca, R.B., and Guedes Soares, C., 2018. Comparison of analytical and numerical simulations of long nonlinear internal solitary waves in shallow water. This study compares the results of long nonlinear internal waves between analytical and numerical wave models in shallow water. To present the analytical solutions, the (G′/G) expansion method is applied to obtain the exact and explicit nonlinear solitary wave solutions of the generalized Gardner equation of any order in the presence of dispersive coefficient. The internal waves are demonstrated by analysing the solutions of the classical Gardner equation, modified Korteweg-de Vries equation, and Korteweg-de Vries equations from the generalized solution. The shape and characteristics of the three-dimensional solitary waves are studied by analysing different numerical results of the nonlinear analytical solutions. The Gardner equation basically behaves with the same properties as the Boussinesq equations of travelling wave solut...

Control of coupled localized nonlinear wave solutions

A method of forced localization of non-linear wave by a feedback control is developed for coupled equations accounting for non-linear dynamic processes in complex lattices. It is shown, that the control of the shape and velocity of the wave function of macro-strain allows to achieve localization of the shape of the function describing variations of defects in the lattice. Moreover, change of the sign of the amplitude of the last wave may be achieved by variation of the parameters of the control function but independent of the initial conditions.

The effect of nonlinear wave-structure and soil-structure interactions in the design of an offshore structure

This work highlights the importance of the consideration of nonlinearities in the analysis and design of deep water platforms. The nonlinearities concern both the calculation of the wave loading and the structural analysis of the platform. More specifically, numerical calculations have been undertaken using a fully nonlinear three-dimensional wave model capable of accurately representing a realistic distribution of wave energy in both frequency and direction. The purpose of these calculations is to provide what is essentially a benchmark against which to assess the accuracy of the commonly applied design wave solutions; linear and 2nd order random waves and linear and nonlinear steady wave solutions. On the other hand, in the field of structural analysis, three cases are considered for comparison, linear analysis, nonlinear analysis with the nonlinearity limited to the modelling of the soil and finally, geometrically nonlinear analysis with nonlinear modelling of the soil. The calculations are performed using the well-known structural analysis software SAP2000, enhanced with an interface that was developed in order to calculate the forces due to wave loading and apply them on the structural members. The example treated in the context of the present paper is a compliant tower, set-up in deep water. The results show that for the specific environmental conditions and for the structure examined herein, the consideration of the nonlinear particle kinematics in the wave loading and the consideration of the structural nonlinearities leads to significant differences with respect to the common design methods in both the displacements and stresses of the structure. Moreover, irrespectively of the adopted wave theory, the structural nonlinear analyses lead to significant discrepancies from the linear ones.

A laboratory investigation concerning the superharmonic free wave suppression in shallow and intermediate water conditions

This paper concerns laboratory wavemaking in shallow and intermediate water conditions. A comparison is made between two wave generation techniques, a first based on controlling the wavemaker displacement, and a second based on controlling the wavemaker force. Nonlinear wave generation in position control is well understood, and many laboratories rely on established second-order or Stream-function inputs. In deep water, using flap-type wavemakers, a force-control approach based on a linear demand signal was recently shown to offer benefits in terms of wave quality. The shallow water operation of such force-control strategies is less certain, which motivates the present study.To investigate the influence of the water depth on this type of control, a range of generation scenarios is considered, including small amplitude and large amplitude regular waves. Adopting both supporting calculations and experimental evidence, the work demonstrates that first-order force-based wave generation in shallow water suffers from similar limitations as first-order position control. This principally concerns the contamination of the testing area due to unwanted free waves, where the present focus is placed on the superharmonic range.The main advance of the work lies in the solutions it offers to overcome this free wave contamination. A number of nonlinear wave solutions upon which force-based generation can be based are discussed, and a suitable methodology is proposed and validated for each case. The developed methodology allows for high quality wave generation, whilst maintaining the benefit of active wave absorption. The work is timely in the sense that is responds to two recent developments. First, the majority of wavemaking facilities commissioned over the past two decades are computer controlled, and active absorption has become commonplace. The work presented offers solutions highly relevant to such installations. Second, developments particularly in offshore wind, have seen many new structures placed in relatively shallow-water depth. It is essential that the model testing of such structures adequately accounts for the issues and solutions presented herein.

Symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime.

We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a single pulse formed by a nonlinear superposition of a periodic wave and a single-peak (W-shaped or antidark) soliton. In particular, a phase diagram for different types of nonlinear excitations on a continuous wave background, including the unusual multipeak soliton, the W-shaped soliton, the antidark soliton, the periodic wave, and the known breather rogue wave, is established based on the explicit link between exact solution and modulation instability analysis. Numerical simulations are performed to confirm the propagation stability of the multipeak solitons with symmetric and asymmetric structures. Further, we unveil a remarkable shape-changing feature of asymmetric multipeak solitons. It is interesting that these shape-changing interactions occur not only in the intraspecific collision (soliton mutual collision) but also in the interspecific interaction (soliton-breather interaction). Our results demonstrate that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.

A non-abelian quasi-particle model for gluon plasma

We propose a quasi-particle model for the thermodynamic description of the gluon plasma which takes into account non-abelian characteristics of the gluonic field. This is accomplished utilizing massive non-linear plane wave solutions of the classical equations of motion with a variable mass parameter, reflecting the scale invariance of the Yang-Mills Lagrangian. For the statistical description of the gluon plasma we interpret these non-linear waves as quasi-particles with a temperature dependent mass distribution. Quasi-Gaussian distributions with a common variance but different temperature dependent mean masses for the longitudinal and transverse modes are employed. We use recent Lattice results to fix the mean transverse and longitudinal masses while the variance is fitted to the equation of state of pure $SU(3)$ on the Lattice. Thus, our model succeeds to obtain both a consistent description of the gluon plasma energy density as well as a correct behaviour of the mass parameters near the critical point.

Symmetry reductions and exact solutions for a class of nonlinear PDEs

This paper uses Lie symmetry group method to study a special kind of PDE. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions are obtained.

Superposed nonlinear waves in coherently coupled Bose–Einstein condensates

We study the dynamics of superposed nonlinear waves in coherently coupled Gross–Pitaevskii (CCGP) equations with constant (autonomous system) and time varying (non-autonomous system) nonlinearity coefficients. By employing a linear transformation, the autonomous CCGP system is converted into two separate scalar nonlinear Schrödinger equations and we show that linear superposition of different nonlinear wave solutions of these scalar equations results into several kinds of nonlinear coherent structures namely, coexisting rogue wave-Ma breather, Akhmediev–Ma breathers, collision and bound states of Ma breathers and solitons. Next, the non-autonomous CCGP system is converted into an autonomous CCGP system with a similarity transformation. We show an interesting possibility of soliton compression and appearance of creeping solitons for kink-like and periodically modulated nonlinearity coefficient.

On general form of the Tanh method and its application to nonlinear partial differential equations

The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.

Highly nonlinear wave solutions in a dual to the chiral model

We consider a two-dimensional scalar field theory with a nilpotent current algebra, which is dual to the Principal Chiral Model. The quantum theory is renormalizable and not asymptotically free: the theory is strongly coupled at short distances (encountering a Landau pole). We suggest it can serve as a toy model for $\lambda\phi^{4}$ theory in four dimensions, just as the principal chiral model is a useful toy model for Yang-Mills theory. We find some classical wave solutions that survive the strong coupling limit and quantize them by the collective variable method. They describe excitations with an unusual dispersion relation $\omega\propto|k|^{\frac{2}{3}}$ . Perhaps they are the "preons" at strong coupling, whose bound states form massless particles over long distances.

The explicit periodic wave solutions and their limit forms for a generalized b-equation

In this paper, for b ∈ (−∞,∞) and b ≠ −1,−2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux − uxxt + (1+b)u2ux = buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable α. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for α. When α tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = −1 or b = −2, we guess that the equation does not have any one of above solutions.

Data dependence for the amplitude equation of surface waves

We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves (Hamilton et al. in J Acoust Soc Am 97:891–897, 1995), surface waves on current-vortex sheets in incompressible MHD (Ali and Hunter in Q Appl Math 61(3):451–474, 2003; Ali et al. in Stud Appl Math 108(3):305–321, 2002) and on the incompressible plasma–vacuum interface (Secchi in Q Appl Math 73(4):711–737, 2015). The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in Hunter (J Hyperbolic Differ Equ 3(2):247–267, 2006), Secchi (Q Appl Math 73(4):711–737, 2015). In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.

Semiclassical limit of the focusing NLS: Whitham equations and the Riemann–Hilbert Problem approach

The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N-phase nonlinear wave solutions to the focusing nonlinear Schrödinger (fNLS) equation, and b) the Riemann–Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated N-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.

Travelling wave states in pipe flow

In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$. We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$, though this remains to be confirmed numerically.

Magnetoacoustic solitons and shocks in dense astrophysical plasmas with relativistic degenerate electrons

Two-fluid quantum magnetohydrodynamic (QMHD) equations are employed to investigate linear and nonlinear properties of the magnetosonic waves in a semi-relativistic dense plasma accounting for degenerate relativistic electrons. In the linear analysis, a plane wave solution is used to derive the dispersion relation of magnetosonic waves, which is significantly modified due to relativistic degenerate electrons. However, for a nonlinear investigation of solitary and shock waves, we employ the reductive perturbation technique for the derivation of Korteweg–de Vries (KdV) and Korteweg–de Vries Burger (KdVB) equations, admitting nonlinear wave solutions. Numerically, it is shown that the wave frequency decreases to attain a lowest possible value at a certain critical number density $N_{c}^{(0)}$, and then increases beyond $N_{c}^{(0)}$ as the plasma number density increases. Moreover, the relativistic electrons and associated pressure degeneracy lead to a reduction in the spatial extents of the magnetosonic waves and a strengthening of the shock amplitude. The results might be important for understanding the linear and nonlinear magnetosonic excitations in dense astrophysical plasmas, such as in white dwarfs, magnetars and neutron stars, etc., where relativistic degenerate electrons are present.

Nonlinear Dynamics and Exact Traveling Wave Solutions of the Higher-Order Nonlinear Schrödinger Equation with Derivative Non-Kerr Nonlinear Terms

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.