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Nonlinear Wave and Solitons
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
There are many important topics related to the KdV and solitons are left out in the present short chapter. For example, nearly all textbooks cover the inverse scattering transform (IST) and its application in deriving soliton solutions. In the present chapter, we prefer to discuss the more direct approach by Hirota. The concept and solution procedure of IST is less straightforward compared to Hirota’s approach. It was said that when Gardner, Greene, Kruskal and Miura thought that the Cole-Hopf transform used in the linearized Burgers equation can be generalized as for the KdV u=-6ψxxψ $$ u = - 6\frac{{\psi _{{xx}} }}{\psi } $$
New Exact Solutions of Fractional Differential Equations by (G′/G)-Expansion Method and Improved (G′/G)-Expansion Method
Published in Santanu Saha Ray, Subhadarshan Sahoo, Generalized Fractional Order Differential Equations Arising in Physical Models, 2018
Santanu Saha Ray, Subhadarshan Sahoo
In the recent past, a great deal of attention has been intended by the researchers on the study of non-linear evolution equations [66,176,208,209] that appeared in mathematical physics. It is widely known that the mKdV equation played an important role in constructing infinitely many conservation laws [210] and the Lax pair for the KdV equation. The Lax pair led to the improvement of the inverse scattering transform (IST) [211] and then the soliton theory. The modified KdV equation is derived by perturbation expansions based on the assumption that the soliton width is small compared with the scale length of the plasma inhomogeneity. In this assumption, soliton maintains all of its identity and its amplitude, width, and speed. The mKdV equation is well known for its special soliton behavior, breathers. Actually, the mKdV equation or its Galilean transformed version, which is usually referred to as the mixed KdV–mKdV equation, appeared in many physical phenomena, such as an harmonic lattices [212], Alfven waves [171], ion acoustic solitons [213], traffic jam [214], Schottky barrier transmission lines [215], and so on. It acquires many remarkable properties, such as Painlevé integrability, Bäcklund transformation, inverse scattering transformation, breather solutions, conservation laws, bilinear transformation, N-solitons, Darboux transformation, and Miura transformation [216–220]. The mKdV equation appears in applications, such as multi-component plasmas and electric circuits, electro-magnetic waves in size-quantized films, electrodynamics, elastic media, and traffic flow [70].
Digital coherent technology-based eigenvalue modulated optical fiber transmission system
Published in Kuppuswamy Porsezian, Ramanathan Ganapathy, Odyssey of Light in Nonlinear Optical Fibers, 2017
Akihiro Maruta, Yuki Matsuda, Hiroki Terauchi, Akifumi Toyota
The nonlinear Schrödinger equation (NLSE), which describes the behavior of the complex envelope of an electric field propagating in a nonlinear dispersive fiber, can be solved analytically by using the inverse scattering transform (IST) [1]. In the framework of IST, the eigenvalues of the Dirac-type eigenvalue equation associated with the NLSE are invariables even though the temporal waveforms and frequency spectra dynamically change during propagation in the fiber. Therefore the eigenvalue is a more ideal information carrier than the pulse’s amplitude, frequency, and/or phase, which are modulated in conventional formats. Hasegawa and Nyu have proposed eigenvalue communication based on the above mentioned concept [2]. They applied cross-phase modulation induced higher order soliton fission to demodulate embedded multiple eigenvalues, and its feasibility has been experimentally demonstrated [3]. Since the receiver configuration is complicated in this method, its practical implementation is difficult. On the other hand, digital coherent technologies have been recently introduced to optical fiber communication systems [4]. For example, a digital back propagation scheme in which the NLSE is iteratively integrated toward the inverse direction has been proposed to compensate for transmission impairments due to fiber dispersion and nonlinearity [5]. It is a time consuming scheme and seems difficult to implement in a real-time system. To overcome the difficulty, we have proposed an eigenvalue demodulation method based on digital coherent technology [6]. In this chapter, we introduce the eigenvalue modulated optical fiber transmission system based on digital coherent technology. After a detailed explanation of the demodulation method [6], we numerically and experimentally demonstrate the eigenvalue modulated transmission system. We also investigate the noise tolerance of the eigenvalues [7]. In general, noise is added to a signal and the noise induces signal distortion. For eigenvalue modulation, constellation points are located in a plane in which orthogonal axes are real and imaginary parts of eigenvalues. We experimentally add amplified spontaneous emission (ASE) noise to a pulse sequence and study the noise tolerance of the demodulated eigenvalues.
A Riemann–Hilbert approach for the modified short pulse equation
Published in Applicable Analysis, 2019
In the context of the inverse scattering transform method, non-real zeros of the scattering coefficient a(k) correspond to eigenvalues of the ‘x-equation’ of the Lax pair and consequently to multi-soliton solutions of the underlying non-linear equation. In the framework of the Riemann–Hilbert method, pure soliton solutions corresponds to solving a meromorphic Riemann–Hilbert problem with trivial jump conditions () and with nontrivial residue conditions at zeros of a(k), thus the solution of the Riemann–Hilbert problem, being a rational function of the spectral parameter, reduces to solving a system of linear algebraic equations only. Hence, this construction leads to explicit formulas for multi-solitons.
Bifurcation, chaotic and multistability analysis of the $(2+1)$-dimensional elliptic nonlinear Schrödinger equation with external perturbation
Published in Waves in Random and Complex Media, 2022
Samina Samina, Adil Jhangeer, Zili Chen
The author [15] investigated the stability of two inviscid, incompressible, properly conducting fluids moving parallel to their interface with uniform velocities and parallel magnetic fields along the direction of flowing. He demonstrated that the magnetic fields have a stabilizing impact on the configuration within the context of the linear theory. In Ref. [16], the author established the linear theory that governs the Kelvin–Helmholtz instability in magneto-hydrodynamics (MHD) when two superposed fluids flow at the same uniform speeds. In ideal fluids, the nonlinear characteristics of Kelvin–Helmholtz instability were studied [17]. In Ref. [18], the authors have shown the nonlinear propagation of Kelvin–Helmholtz instability of stable packet of waves in MHD. They studied magneto-hydrodynamics flows that sub-critical with velocity differences are less than critical velocity, the nonlinear Schrödinger equation is the dynamical equation that gives the amplitudes. According to Ref. [18], even in the existence of a strong magnetic field, the magnetic amplitude of the waves is still unstable under modulation. In Refs. [19, 20], the authors investigated and analyzed the convective flow of a third-grade non-Newtonian fluid caused by a linearly stretched sheet subjected to a magnetic field. They used the analytical technique named a homotopy analysis method for the nonlinear differential equation. In Ref. [21], the authors applied inverse scattering transform to obtain the solutions of the one dimensions nonlinear Schrödinger equation. The results gained are applied for estimating the nonlinear frame of the evolution of modulation instability.
Dynamical solitons for the perturbated Biswas–Milovic equation with Kudryashov's law of refractive index using the first integral method
Published in Journal of Modern Optics, 2022
Lanre Akinyemi, Mohammad Mirzazadeh, Seyed Amin Badri, Kamyar Hosseini
The study of nonlinear dynamical systems is crucial for comprehending the physical world around us since most physical phenomena are nonlinear in real life. The nonlinear partial differential equations (NPDEs) are the most appropriate tool for describing such phenomena. NPDEs are characterized as one of the most fundamental and important branches in applied sciences and engineering, particularly for discovering novel properties of complex phenomena in diverse fields such as physics, biology, atomic physics, plasma physics, nuclear energy, optical physics, shallow water wave dynamics, so on and so forth. Exact solutions to NPDEs have always piqued the interest of many researchers and numerous effective approaches have been suggested the like of extended trial function method [1], the soliton ansatz method [2–6], F-expansion method [7], -expansion method [8–10], modified simple equation [11], new extended direct algebraic method [12,13], generalized auxiliary equation method [14], modified Khater method [15], sine-Gordon expansion method [16–18], Sardar sub-equation method [19–21], model scheme and the traveling wave hypothesis [22], new Kudryashov method [23–25], Jacobi elliptic function method [26], the Wronskian technique [27], F-expansion method [7], homogeneous balance method [28], symmetry algebra method [29], Hirota's bilinear method [30–32], generalized Riccati equation mapping method [33], inverse scattering transform [34], trial solution method [35], -expansion method [36], and so on.