New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.     

Collisionless Boltzmann equations and integrable moment equations

A collisionless Boltzmann equation, describing long waves in a dense gas of particles interacting via short-range forces, is shown to be equivalent to the Benney equations, which describe long waves in a perfect two-dimensional fluid with a free surface. These equations also describe, in a random phase approximation, the evolution, on long space and time scales, of multiply periodic solutions of the nonlinear Schrödinger equation. The derivative nonlinear Schrödinger equation is likewise shown to be related to an integrable system of moment equations.     

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.     

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrödinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrödinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrödinger equation. The stability analysis is discussed both analytically and numerically.     

A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation

In this Letter we introduce a systematic perturbation method for analyzing the effect of small perturbations on critical self-focusing by reducing the perturbed critical nonlinear Schrödinger equation (PNLS) to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simplified depending on whether PNLS is power conserving or not. An important and somewhat surprising result is that various small defocusing perturbations lead to a canonical form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations.     

The Hamiltonian dynamics of the soliton of the discrete nonlinear Schrödinger equation

Hamiltonian equations are formulated in terms of collective variables describing the dynamics of the soliton of an integrable nonlinear Schrödinger equation on a 1D lattice. Earlier, similar equations of motion were suggested for the soliton of the nonlinear Schrödinger equation in partial derivatives. The operator of soliton momentum in a discrete chain is defined; this operator is unambiguously related to the velocity of the center of gravity of the soliton. The resulting Hamiltonian equations are similar to those for the continuous nonlinear Schrödinger equation, but the role of the field momentum is played by the summed quasi-momentum of virtual elementary system excitations related to the soliton.     

Exact solutions of some nonlinear partial differential equations using functional variable method

The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2?+?1)-dimensional Camassa–Holm Kadomtsev–Petviashvili equation and the higher-order nonlinear Schrödinger equation. By using this useful method, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. It is shown that the proposed method is effective and general.     

Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method

In this article, a new version of the trial equation method is suggested. With this method, it is possible to find the new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear Schrödinger equation. New exact solutions are expressed with Jacobi elliptic function solutions, 1-soliton solutions and rational function solutions. When the obtained results are examined, we can say the unstable nonlinear Schrödinger equation shows different dynamic behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.     

On symmetries,conservation laws and exact solutions of the nonlinear Schrödinger–Hirota equation

In this paper, conservation laws and exact solution are found for nonlinear Schrödinger–Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger–Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.     

New optical soliton solutions for nonlinear complex fractional Schrödinger equation via new auxiliary equation method and novel $$({G'}/{G})$$-expansion method

In this research, we apply two different techniques on nonlinear complex fractional nonlinear Schrödinger equation which is a very important model in fractional quantum mechanics. Nonlinear Schrödinger equation is one of the basic models in fibre optics and many other branches of science. We use the conformable fractional derivative to transfer the nonlinear real integer-order nonlinear Schrödinger equation to nonlinear complex fractional nonlinear Schrödinger equation. We apply new auxiliary equation method and novel \(\left( {G'}/{G}\right) \)-expansion method on nonlinear complex fractional Schrödinger equation to obtain new optical forms of solitary travelling wave solutions. We find many new optical solitary travelling wave solutions for this model. These solutions are obtained precisely and efficiency of the method can be demonstrated.     

Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.     

The Nonlinear Schrödinger Equation with a Self-consistent Source in the Class of Periodic Functions

In this work the method of inverse spectral problem is applied to the integration of the nonlinear Schrödinger equation with a self-consistent source in the class of periodic functions.     

Differential equations with supersymmetric time

Partial differential equations with supersymmetric (1, 1) time are investigated by means of superspace Cauchy-Kowalewsky and Cartan-Kähler techniques. Theorems for the existence and uniqueness of solutions are found for a particular class of superanalytic functions. The (1, 1) time evolution equations are very important in applications to supersymmetric quantum mechanics and quantum field theory: the square roots of Schrödinger and heat equations. We considered nonlinear analogs of these equations which can be interpreted as square roots of Maslov's nonlinear Schrödinger and heat equations.     

Subwavelength spatial solitons in inhomogeneous Kerr media

Propagation of a narrow beam in a periodic Kerr medium is analyzed for an arbitrary relation between the beam width, modulation period, and optical wavelength. Maxwell’s equations used to obtain solutions describing spatial optical solitons, and their basic properties are determined. The results are compared with those obtained by solving the paraxial nonlinear Schödinger equation. Conditions are found under which these models are mutually inconsistent.     

On the characterization of vector rogue waves in two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients

We construct vector rogue wave solutions of the two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients, namely diffraction, nonlinearity and gain parameters through similarity transformation technique. We transform the two-dimensional two coupled variable coefficients nonlinear Schrödinger equations into Manakov equation with a constraint that connects diffraction and gain parameters with nonlinearity parameter. We investigate the characteristics of the constructed vector rogue wave solutions with four different forms of diffraction parameters. We report some interesting patterns that occur in the rogue wave structures. Further, we construct vector dark rogue wave solutions of the two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients and report some novel characteristics that we observe in the vector dark rogue wave solutions.     

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients

The research of rogue waves is an advanced field which has important practical and theoretical significances in mathematics, physics, biological fluid mechanics, oceanography, etc. Using the reductive perturbation theory and long wave approximation, the equations governing the movement of blood vessel walls and the flow of blood are transformed into high-order nonlinear Schrödinger (NLS) equations with variable coefficients. The third-order nonlinear Schrödinger equation is degenerated into a completely integrable Sasa-Satsuma equation (SSE) whose solutions can be used to approximately simulate the real rogue waves in the vessels. For the first time, we discuss the conditions for generating rogue waves in the blood vessels and effects of some physiological parameters on the rogue waves. Based on the traveling wave solutions of the fourth-order nonlinear Schrödinger equation, we analyze the effects of the higher order terms and the initial deformations of the blood vessel on the wave propagation and the displacement of the tube wall. Our results reveal that the amplitude of the rogue waves are proportional to the initial stretching ratio of the tube. The high-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave steepness.