Search

The generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities is considered. A series of direct transformations is performed for the traveling wave reduction of the original equation. The implicit function method is used for nonlinear ordinary differential equation with some constraints on the parameters. Some analytical solutions are found in the form of periodic and solitary waves. Three conservation laws, corresponding to the generalized nonlinear Schrödinger equation are obtained and conserved quantities corresponding to the solution in implicit form are found [ABSTRACT FROM AUTHOR]

In the current paper, a generalized nonlinear Schrödinger (gNLS) equation with logarithmic nonlinearity is studied as a model for the propagation of optical pulses. More precisely, after applying a specific hypothesis for the solution of the governing equation, its Gaussian solitary wave is retrieved using the ansatz method. Some numerical simulations in two- and three-dimensional postures are presented to investigate the impact of different physical parameters on Gaussian solitary wave' dynamics. Results confirm that the physical parameters of the gNLS equation have a key role in controlling the dynamics of the Gaussian solitary wave. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several important properties of the generalized nonlinear Schrödinger equation with P T -symmetric δ -sech optical potentials, such as the phase breaking of P T -symmetric δ -sech potentials, and existence, dynamics and excitations of new soliton solutions. Specifically, we identify the fully real spectral region of the non-Hermitian Hamiltonian and observe the phase-breaking phenomenon. Additionally, we discover a new soliton solution that represents a peakon solution, a smooth soliton solution, and a double-hump soliton solution under different potential parameters. We analyze the stability of these three types of solutions and determine their stability domains. Furthermore, we study the numerical peakon solution and its stability. In particular, we investigate the interaction of peakon solutions and observe semi-elastic interactions. Finally, we explore the stable adiabatic excitations of peakons. These results contribute to a deeper understanding of P T -symmetric optical fields and provide insights for related experimental works. [ABSTRACT FROM AUTHOR]

In optical fibers, the generalized nonlinear Schrödinger equations with self-steepening (SS), self-frequency shift (SFS), intermodal dispersion (IMD), and third-order dispersion (TOD) play an important role. Our investigation covers a variety of physical parameters based on how optical solitons change their structure as they move through an optical medium. Our study shows that modifying the coefficients for SS, SFS, IMD, and TOD can affect optical solitons' profiles either by altering their nature or without doing so. We used the extended rational sinh–cosh method, which works with various types of soliton profiles. These profiles include dark, kink-dark, kink, and anti-kink solitons. By selecting appropriate physical parameter values, the behavior of various optical solitons is graphically depicted. As a result, we utilize the eigenvalue spectrum to investigate linear stability analysis. [ABSTRACT FROM AUTHOR]

Today, nonlinear Schrödinger-type equations are the focus of explorers and scientists. Hereby, we look into a generalized nonlinear Schrödinger equation by constructing the modified generalized Darboux transformation and analysing the type-I, type-II and type-III degenerate solitons for generalized nonlinear Schrödinger equation via some semirational solutions. Type-I degenerate solitons refer to the degenerate solitons, type-II degenerate solitons mean the interactions between the solitons and the degenerate solitons, and type-III degenerate solitons denote the bound states among a series of the degenerate solitons. We hope that the mathematical research method used in this paper could provide some theoretical assistance for future research on the nonlinear Schrödinger-type equations. [ABSTRACT FROM AUTHOR]

Research on the shear horizontal (SH) waves propagating in a nonlinear elastic half-space coated with a nonlinear elastic layer having slow variation in the boundary surfaces is presented. It is assumed that both the free surface and interface change as a function of the distance in the direction of propagation of the waves. By employing a perturbation method, nonlinear self-modulation of the surface SH waves for a general geometry is characterized by a generalized nonlinear Schrödinger (GNLS) equation with variable coefficients depending on functions representing the irregularities of boundary surfaces in addition to linear and nonlinear material parameters of the medium. The solitary wavelike solutions of this GNLS equation are obtained for certain special cases of the irregular boundary surfaces. It has been demonstrated by graphs that the propagation characteristics of SH waves are significantly affected by slowly varying layer and nonlinear properties of the media. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the generalized nonlinear Schrödinger (g-NLS) equation which has extensive applications in various physical fields. Firstly, we prove Liouville integrability of this equation by deriving its bi-Hamiltonian structures applying the variational identity. Nextly, we calculate the modulational instability for the possible reason of the formation of the rogue wave. Moreover, based on the generalized (2 , N - 2) -fold Darboux transformation (DT), we can derive several mixed localized wave solutions such as breathers, rogue waves and semi-rational solitons for this equation, and accurately analyze a lot of important physical quantities. Finally, we present these solutions graphically by choosing appropriate parameters and discuss their dynamic behavior. It is worth noting that all of these solutions can change from a strong interaction to a weak interaction by choosing the parameters. This may also be one of the reasons why relevant wave structures presenting diversity, and useful to explain some physical phenomena in nonlinear optics. [ABSTRACT FROM AUTHOR]

A generalized nonlinear Schrödinger equation with higher-order dispersive and nonlinear effects is presented to govern the pulse propagation in negative index materials. Via the mapping method, Jacobian elliptic function solutions and their corresponding soliton solutions are found. Parameter modulation of higher-order dispersive effect and nonlinearities for periodic waves and solitons is studied. With the addition of the third dispersion parameter, or the quintic nonlinear parameter, the change of real and imaginary parts of periodic waves and solitons is same; namely, their amplitudes increase, and yet their widths decrease. Moreover, phase transitions of solutions with parameter modulation are found. [ABSTRACT FROM AUTHOR]

Modulational collapse of a wave pulse in a complex nonlinear plasma described by a complex nonlinear Schrödinger equation (CNSE) including the effects of viscous heating and nonlinear damping, and the adjusting by an external potential are investigated. Theoretical and numerical results reveal that the original pulse first suffers modulational instability, the growth rate of the modulational instability increases with the increase of viscous heating, but it is not sensitive to the nonlinear damping. The instability induces the self-similar collapsing, then the fields rapidly transform into a turbulent state with short-wavelength modes. The results are explained in terms of the evolution of the total energy, which can become invariant during the evolution even though the system is nonconservative. The external potential can delay the self-defocusing process of the pulse. Those results can apply to the phenomena described by the CNSE. [ABSTRACT FROM AUTHOR]

In this paper, the dispersion coefficients and the nonlinear parameter of a wagon wheel fiber is obtained by finite element method. Efficient broadband near infrared supercontinuum generation is predicted in a 15 cm of the single-mode small core silicon wagon wheel fiber by solving the generalized nonlinear Schrödinger equation when the fiber is pumped by femtosecond pulses in the abnormal dispersion regime. Here, it is shown that the flattened supercontinuum spectrum over 850 nm wide is achieved. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. Firstly, the n-fold Darboux transformation (DT) of the GNLS equation is constructed. Then, the soliton solutions, breather solutions, and rogue wave solutions of the GNLS equation are studied based on the DT by choosing different seed solutions. Furthermore, the dynamic features of these solutions are explicitly delineated through some figures with the help of Maple software. [ABSTRACT FROM AUTHOR]

In this paper, formation of dark plasmon-soliton in a plasmonic photonic crystal fiber (PPCF) structure is predicted and analyzed by the use of the generalized nonlinear Schrödinger equation (GNLSE). Linear analysis of the plasmonic guided modes in the proposed structure offers a lower attenuation and dispersion values compared with the conventional core metallic fiber. Kerr coefficient of the silica background and the thermo-modulational nonlinearity of the gold are considered as the origins of nonlinear behavior. A delicate balance between normal dispersion and nonlinear characteristics of the structure provides the necessary condition for the formation of dark plasmon-soliton. Moreover, the thermo-modulational nonlinearity of metal is accompanied by the delayed propagation of the guided modes which is well justified throughout the simulations. Undistorted and time-delayed propagation of solitary plasmonic modes is a key point for the potential applications in the future all-optical devices, sensing platforms, and frequency conversion. [ABSTRACT FROM AUTHOR]

The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convexity or concavity) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrödinger equation. These include superfluids (Bose-Einstein condensates and ultracold fermions) and 'optical fluids.' Estimates of breaking times for smooth initial data and the long-time behavior of the shock tube problem are presented. Detailed numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory. [ABSTRACT FROM AUTHOR]

A correspondence between the families of generalized nonlinear Schrödinger (NLS) equations and generalized KdV equations was recently found using a Madelung fluid description. We similarly consider a special derivative NLS equation. We find a number of solitary waves and periodic solutions (expressed in terms of elliptic Jacobi functions) for a motion with a stationary profile current velocity. We study the stability of a bright solitary wave (ground state) by conjecturing that the Vakhitov-Kolokolov criterion is applicable. [ABSTRACT FROM AUTHOR]

Soliton solutions of a class of generalized nonlinear evolution equations are discussed analytically and numerically. This is done by using a travelling wave method to formulate one-soliton solution and the finite difference method to the numerical solutions and the interactions between the solitons for the generalized nonlinear Schrodinger equations. The characteristic behavior of the nonlinearity admitted in the system has been investigated and the soliton states of the system in the limit when α → 0 and α → ∞have been studied. The results presented show triat the soliton phenomenon is characteristics associated with the nonlinearities of the dynamical systems. [ABSTRACT FROM AUTHOR]

The propagation patterns of optical waves in medium which are described by the generalized nonlinear Schrödinger equation are studied in details. A varied of exact solutions are obtained, which include singular solution, solitary wave solution, periodic solution and double periodic solution. Under the concrete parameters, we analyze the modulus of solutions, from which we can get new propagation patterns. [ABSTRACT FROM AUTHOR]