Types Of Nonlinear Waves Research Articles

New exact solutions of nonlinear wave type PDEs with delay

We consider nonlinear wave type PDEs with delay of the form utt+H1(u)ut=[G(u)ux]x+H2(u)ux+F(u,w),where u=u(x,t) is the unknown function, w=u(x,t−τ), and τ is the delay time. The source function F(u,w) depends on one or several arbitrary functions of one argument. Using the modified method of functional constraints, we obtain a number of new exact solutions with generalized and functional separation of variables, as well as traveling-wave solutions. Most solutions are expressed in terms of elementary functions and contain free parameters. These solutions can be used to formulate test problems intended to evaluate the accuracy of numerical methods for solving nonlinear delay PDEs.

Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity

In this work, we consider a nonlinear Schrodinger equation with stimulated Raman scattering and quintic nonlinearity, which works as a model for the propagation of ultrashort pulse in optical fiber and also corresponds to one-dimensional anisotropic Heisenberg ferromagnetic spin chain. Introducing a Galilean transformation, we transform the model into a fifth-order complex modified KdV equation, the integrability of which has been considered in the AKNS technique framework. The N-fold Darboux transformation for the model is derived in terms of the gauge transformation of the associated $$3 \times 3$$ matrix spectral problem. As applications, abundant intriguing types of nonlinear waves are obtained in zero and nonzero boundary conditions. The dynamical evolution of these nonlinear waves can be well controlled under stimulated Raman scattering and quintic nonlinearity management. Beside that, we also find some novel and interesting conversion phenomena, where a high-order term plays a pivotal role.

Dynamics of the boundary layer created by the explosion of a dense object in an ambient dilute gas triggered by a high power laser

The dynamics of the boundary layer in between two distinct collisionless plasmas created by the interaction between a dense object modeling a cluster and a short laser pulse in the presence of an ambient gas is studied with two dimensional relativistic particle-in-cell simulations, which are found to be described by three successive processes. In the first phase, a collisionless electrostatic shock wave, launched near the cluster expansion front, reflects the ambient gas ions at a contact surface as a moving wall, which allows a particle acceleration with a narrower energy spread. In the second phase, the contact surface disappears and the compressed surface of the ambient gas ions passes over the shock potential, forming an overlapping region between the cluster expansion front and the compressed surface of the ambient gas. Here, another type of nonlinear wave is found to be evolved, leading to a relaxation of the shock structure, while continuing to reflect the ambient gas ions. The nonlinear wave exhibits a bipolar electric field structure that is sustained for a long timescale coupled with slowly evolving ion dynamics, suggesting that a quasistationary kinetic equilibrium dominated by electron vortices in the phase space is established. In the third phase, a rarefaction wave is triggered and evolves at the compressed surface of ambient gas. This is because some of the ambient gas ions tend to pass over the potential of the bipolar electric field. Simultaneously, a staircase structure, i.e., a kind of internal shock, is formed in the cluster due to the deceleration of cluster ions. Such structure formations and successive dynamics accompanied by the transitions from the shock wave phase through the overlapping phase to the rarefaction wave phase are considered to be a unique nature at the boundary layer created by an explosion of a dense plasma object in an ambient dilute plasma.

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.

Modulation theory solution for nonlinearly resonant, fifth‐order Korteweg–de Vries, nonclassical, traveling dispersive shock waves

AbstractA new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.

Analytic solutions for the (2+1)-dimensional generalized sine-Gordon equations in nonlinear optics

The (2+1)-dimensional generalized sine-Gordon equations can be used to describe the propagation of femtosecond laser pulse in a systems of two-level atoms. In this paper, bilinear forms of the equations have been constructed through variable transformation and Bell polynomial. Based on the bilinear method and truncated Painlevé expansion, multi-soliton and quasi-periodic peakon solutions for such a system are derived, respectively. Overtaking elastic interactions between the two solitons and among the three solitons have been found. It is presented that the soliton with the smaller amplitude can travel faster than the larger one. Apart from the soliton, other type of nonlinear wave propagating in the systems of two-level atoms has been displayed, namely the quasi-periodic peakon. The peakon oscillates in both space and time, while its left and right derivatives at peak point are relatively large, which means the electric-field intensity of the femtosecond laser pulse is rather strong.

Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation

We study the linear stability analysis and nonlinear excitations on a continuous wave background for the fifth-order nonlinear Schrödinger equation. We find that there are a modulation stability (MS) quasi-elliptic ring and a MS curve in the modulation instability (MI) regime. The concrete expressions of different types of nonlinear waves are obtained by using Darboux transformation. And we present the relations and phase diagram between MI and these waves. We discover that the antidark (AD) soliton and nonrational W-shaped soliton can only exist on the resonance line of outside of the MS quasi-elliptic ring and the right of the MS curve. We perform numerical simulation to test the stability of the AD soliton. By introducing the perturbation energy, we further show that solitons in the MI regime are caused by both the fourth- and fifth-order effects.

Quantum stability of nonlinear wave type solutions with intrinsic mass parameter in QCD

The problem of existence of a stable vacuum field in a pure quantum chromodynamics (QCD) is revised. Our approach is based on using classical stationary non-linear wave type solutions with intrinsic mass scale parameter. Such solutions can be treated as quantum mechanical wave functions describing massive spinless states in quantum theory. We verify whether non-linear wave type solutions can form a stable vacuum field background within the framework of effective action formalism. We demonstrate that there is a special class of stationary generalized Wu-Yang monopole solutions which are stable against quantum gluon fluctuations.

Super-soliton dust-acoustic waves in four-component dusty plasma using non-extensive electrons and ions distributions

Based on Sagdeev pseudo-potential and phase-portrait, the dynamics of four-component dust plasma with non-extensively distributed electrons and ions are investigated. Three distinct types of nonlinear waves, namely, soliton, double layer, and super-soliton, have been found. The basic features of such waves are high sensitivity to Mach number, non-extensive parameter, and dust temperature ratio. It is found that the multi-component plasma is a necessary condition for super-soliton's existence, having a wider amplitude and a larger width than the regular soliton. Super-solitons may also exist when the Sagdeev pseudo-potential curves admit at least four extrema and two roots. In our multi-component plasma system, the super-solitons can be found by increasing the Mach number and the non-extensive parameter beyond those of double-layers. On the contrary, the super-soliton can be produced by decreasing the dust temperature ratio. The conditions of the onset of such nonlinear waves and its merging to regular solitons have been studied. This work shows that the obtained nonlinear waves are found to exist only in the super-sonic Mach number regime. The obtained results may be of wide relevance in the field of space plasma and may also be helpful to better understand the nonlinear fluctuations in the Auroral-zone of the Earth's magnetosphere.

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.

A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method

ABSTRACTIn this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions. This work extends the idea of Tamsir et al. [An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation, Appl. Math. Comput. 290 (2016), pp. 111–124] for 3D nonlinear wave type problems. Expo-MCB-DQM transforms the 3D nonlinear wave equation into a system of ordinary differential equations (ODEs). To solve the resulting system of ODEs, an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) scheme is used. Stability analysis of the proposed method is also discussed and found that the method is conditionally stable. Four test problems are considered in order to demonstrate the accuracy and efficiency of the algorithm.

The bifurcations of solitary and kink waves described by the Gardner equation

In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenonis that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.

2D in-band solitons in PT-symmetric waveguide arrays

We address two types of two-dimensional (2D) localized solitons in Kerr media with an imprinted quasi-one-dimensional lattice featuring a parity-time (PT) symmetry. Solitary waves originating from the edges of the Bloch bands are stable in their entire existence domains. Purely nonlinear multipeaked states propagate stably in wide parameter windows. Both types of nonlinear waves exist in the finite bandgaps of the corresponding linear system and, cross-continuously, the Bloch band (continuous spectrum) sandwiched between (or neighboring) them. To the best of our knowledge, our findings thus provide the first example of "embedded solitons" in 2D PT periodic systems.

Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects.

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.

Asymmetric Rogue Waves, Breather-to-Soliton Conversion, and Nonlinear Wave Interactions in the Hirota–Maxwell–Bloch System

We study the nonlinear localized waves on constant backgrounds of the Hirota–Maxwell–Bloch (HMB) system arising from the erbium doped fibers. We derive the asymmetric breather, rogue wave (RW) and semirational solutions of the HMB system. We show that the breather and RW solutions can be converted into various soliton solutions. Under different conditions of parameters, we calculate the locus of the eigenvalues on the complex plane which converts the breathers or RWs into solitons. Based on the second-order solutions, we investigate the interactions among different types of nonlinear waves including the breathers, RWs and solitons.

Nonlocal symmetries and interaction solutions of the (2+1)-dimensional higher order Broer-Kaup system

Finding explicit solutions of nonlinear partial differential equation is one of the most important problems in mathematical physics. And it is very difficult to find interaction solutions among different types of nonlinear excitations except for soliton-soliton interactions. It is known that Painlev analysis is an important method to investigate the integrable property of a given nonlinear evolution equation, and the truncated Painlev expansion method is a straight way to provide auto-Bcklund transformation and analytic solution, furthermore, it can also be used to obtain nonlocal symmetries. Symmetry group theory plays an important role in constructing explicit solutions, whether the equations are integrable or not. By applying the nolocal symmetry method, many new exact group invariant solution can be obtained. This method is greatly valid for constructing various interaction solutions between different types of excitations, for example, solitons, cnoidal waves, Painlev waves, Airy waves, Bessel waves, etc. It has been revealed that many more integrable systems are consistent tanh expansion (CTE) solvable and possess quite similar interaction solutions which can be described by the same determining equation with different constant constraints. In this paper, the (2+1)-dimensional higher-order Broer-Kaup (HBK) system is studied by the nonlocal symmetry method and CTE method. By using the nonlocal symmetry method, the residual symmetries of (2+1)-dimensional higher order Broer-Kaup system can be localized to Lie point symmetries after introducing suitable prolonged systems, and symmetry groups can also be obtained from the Lie point symmetry approach via the localization of the residual symmetries. By developing the truncated Painlev analysis, we use the CTE method to solve the HBK system. It is found that the HBK system is not only integrable under some nonstandard meaning but also CTE solvable. Some interaction solutions among solitons and other types of nonlinear waves which may be explicitly expressed by the Jacobi elliptic functions and the corresponding elliptic integral are constructed. To leave it clear, we give out four types of soliton+cnoidal periodic wave solutions. In order to study their dynamic behaviors, corresponding images are explicitly given.

Solitary wave-based delamination detection in composite plates using a combined granular crystal sensor and actuator

We experimentally and numerically investigate a diagnostic method for detecting hidden delamination in composite panels, using highly nonlinear solitary waves. Solitary waves are a type of nonlinear waves with strong energy intensity and non-distortive nature, which can be controllably generated in one-dimensional granular crystals. In this study, we use granular crystals as a combined sensor and actuator to detect hidden delamination in carbon fiber reinforced polymer (CFRP) composite panels. Specifically, we locally excite a CFRP composite specimen using the granular crystal as an actuator and measure the reflected waves that carry the specimen’s diagnostic information using the same device as a sensor. We first investigate the effect of the panel’s boundary conditions on the response of the reflected solitary waves. We then investigate the interactions of a solitary wave with delamination hidden in the CFRP composite specimen. Lastly, we define a damage index based on the solitary waves’ responses to identify the location of the hidden delamination in the CFRP composite panel. The solitary wave-based diagnostic method can provide unique merits, such as portable and fast sensing of composites’ hidden damage, thereby with the potential of being used for hot spot monitoring of composite-based structures.

Global uniqueness theorems for linear and nonlinear waves

We prove a unique continuation from infinity theorem for regular waves of the form [ □ + V ( t , x ) ] ϕ = 0 . Under the assumption of no incoming and no outgoing radiation on specific halves of past and future null infinities, we show that the solution must vanish everywhere. The “no radiation” assumption is captured in a specific, finite rate of decay which in general depends on the L ∞ -profile of the potential V . We show that the result is optimal in many regards. These results are then extended to certain power-law type nonlinear wave equations, where the order of decay one must assume is independent of the size of the nonlinear term. These results are obtained using a new family of global Carleman-type estimates on the exterior of a null cone. A companion paper to this one explores further applications of these new estimates to such nonlinear waves.

Two kinds of important bifurcation phenomena of nonlinear waves in a generalized Novikov–Veselov equation

This paper considers a generalized Novikov–Veselov equation $$u_{t}+ a u_{xxx}+b u_{yyy}+c (u^{2}\partial _{y}^{-1}u_{x})_{x}+d(u^{2}\partial _{x}^{-1}u_{y})_{y}=0$$ . By utilizing the bifurcation method and qualitative theory of dynamical systems, we show that there exist two kinds of important bifurcation phenomena when the bifurcation parameters vary. One of the bifurcation phenomena is that the fractional solitary waves can be bifurcated from three types of nonlinear waves which are the trigonometric periodic waves, the elliptic periodic waves and the hyperbolic solitary waves. Another bifurcation phenomenon is that the kink waves can be bifurcated from two types of nonlinear waves which are the solitary waves and the singular waves. Moreover, we carry out many numerical simulations of bifurcation processes of these nonlinear waves to verify our main results.

Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions

In this article, the author proposed two differential quadrature methods to find the approximate solution of one and two dimensional hyperbolic partial differential equations with Dirichlet and Neumann’s boundary conditions. The methods are based on Lagrange interpolation and modified cubic B-splines respectively. The proposed methods reduced the hyperbolic problem into a system of second order ordinary differential equations in time variable. Then, the obtained system is changed into a system of first order ordinary differential equations and finally, SSP-RK3 scheme is used to solve the obtained system. The well known hyperbolic equations such as telegraph, Klein–Gordon, sine-Gordon, Dissipative non-linear wave, and Vander Pol type non-linear wave equations are solved to check the accuracy and efficiency of the proposed methods. The numerical results are shown in L∞,RMSandL2 errors form.