Nonlinear Schr Research Articles

THE NUMERICAL APPROXIMATION OF STATIONARY WAVE SOLUTIONS FOR TWO-COMPONENT SYSTEM OF NONLINEAR SCHRÖDINGER EQUATIONS BY USING GENERALIZATION PETVIASHVILI METHOD

The Petviashvili method is a numerical method for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: , where is a positive definite and self-adjoint operator and is constant. Due to the case being a system of solitary nonlinear wave equations, we generalize the Petviashvili method. We apply this generalized method for a two-component system of Nonlinear Schr dinger Equations (NLSE) for 2-D.

Perturbation Solution to Modified Nonlinear Schr¨odinger Equation Based on Slowly Varying Envelope Approximation

Perturbation Solution to Modified Nonlinear Schr¨odinger Equation Based on Slowly Varying Envelope Approximation

The periodic soliton solutions for a nonlocal nonlinear Schrödinger equation with higher-order dispersion

A nonlocal nonlinear Schr¨odinger (NNLS) equation with fourth-order dispersion and cubic-quintic nonlinearities has been studied analytically and numeri- cally. Under the constraint conditions, auxiliary functions are introduced, and explicit one- and two-soliton solutions are obtained by the Hirota bilinear method. Accord- ing to the solutions, the propagation dynamics of soliton pulses are investigated. The influences of different parameters on the dynamics of one- and two-soliton solutions have been analyzed. The results show that the two-soliton solution exhibits diverse dy- namic characteristics under the suitable parameter selections. In addition, the stability of one- and two-soliton solutions against the constraint conditions deviations and under the initial perturbations are also studied numerically.

Numerical investigation on nonautonomous optical rogue waves and Modulation Instability analysis for a nonautonomous system

In this paper, we report existence of optical rogue waves in the focussing non—autonomous nonlinear Schrödinger equation (NLSE) through numerical studies of modulation instability (MI). The dynamics of non-autonomous rogue waves discussed and its associated modulation instability through linear stability analysis taken place followed by pulse splitting behaviour due to non—autonomous coefficient. We prove that the excitation of rogue waves with certain conditions in the base band modulation instability regime. The above analysis of complex dynamics in terms of MI processes has allowed to experiments to excite the nonlinear superposition of rogue wave solutions using a modulated plane wave optical field injected into optical fiber which offer the evidence for excitation of nonautonomous rogue waves in an inhomogeneous nonlinear medium. It is identified from the results frequency modulation on a wavefield induces modulation instability as a result of rogue waves. We analyze the dependence of parameters coefficient of group velocity dispersion(GVD) and nonlinearity (α(z)) and non—autonomous coefficient (β(z)) and the instability of rogue waves. Our work suggests that the presence of non-autonomous coefficients can have a significant impact on the emergence of extreme events, particularly in relation to their self—steepening nature.

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

Numerical approximations and asymptotic limits of some nonlinear problems

In the present work, a numerical approach is dedicated to the approximation to the solutions of a time-independent nonlinear Schr¨odinger equation in a mixed case provided with numerical tests on the asymptotic limits of the solution according to some parameters. A finite difference discretization with calibrations is applied leading to a quasi-linear algebraic system. where the its solvability is investigated as well as its stability and convergence via Von Neumann method. Some numerical experiments are developed to validate the result, and to test the effect of some parameters on the asymptotic limit of the problem.

Wave Amplification Outside of the Modulation Instability Band

"We show that linear stability analysis not only describes the effect of modulation instability of a plane wave in nonlinear media but it also predicts significant wave amplification outside of the standard instability band. As an example, we consider the classic MI in the case of the nonlinear Schr¨odinger equation. However, similar amplification may take place in many other nonlinear media that admit modulation instability."

A Liouville integrable hierarchy with four potentials and its bi-Hamiltonian structure

"We aim to construct a Liouville integrable Hamiltonian hierarchy from a specific matrix spectral problem with four potentials through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is exhibited by a bi-Hamiltonian structure explored by using the trace identity. Illustrative examples of novel four-component coupled Liouville integrable nonlinear Schr¨odinger equations and modified Korteweg-de Vries equations are presented."

Dynamical Discussion and Diverse Soliton Solutions via Complete Discrimination System Approach Along with Bifurcation Analysis for the Third Order NLSE

The purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schr\"{o}dinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents.

The solvability of the optimal control problem for a nonlinear Schrödinger equation

In this paper, we analyze the solvability of the optimal control problem for a nonlinear Schr\"{o}dinger equation. A Lions-type functional is considered as the objective functional. First, it is shown that the optimal control problem has at least one solution. Later, the Frechet differentiability of the objective functional is proved and a formula is obtained for its gradient. Finally, a necessary optimality condition is derived.

The long-time asymptotics of the derivative nonlinear Schr[formula omitted]dinger equation with step-like initial value

Consideration in this present paper is the long-time asymptotics of solutions to the derivative nonlinear Schrödinger (DNLS) equation with the step-like initial value q(x,0)=q0(x)=A1eiϕe2iBx,x<0,A2e−2iBx,x>0,by Deift–Zhou method. The step-like initial problem is described by a matrix Riemann–Hilbert problem. A crucial ingredient used in this paper is to introduce the g-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as t→∞. It is shown that the leading order term of the long-time asymptotics solution of the DNLS equation is expressed by the Theta function Θ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.

Ground state energy threshold and blow-up for NLS with competing nonlinearities

We consider the nonlinear Schr\odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subcritical perturbations, we show the existence of a critical prescribed mass, precisely the mass of the unique, static, positive solution to the associated elliptic equation, such that the ground state energy is achieved for any mass equal or smaller than the critical one. Moreover, the ground state energy is not achieved for mass larger than the critical one. As a byproduct of the variational characterization of the ground state energy, we prove the existence of blowing-up solutions in finite time, for any energy below the ground state energy threshold.

Darboux Transformation and Soliton Solutions of the Generalized Sasa–Satsuma Equation

The Sasa-Satsuma equation, a higher-order nonlinear Schr\"{o}dinger equation, is an important integrable equation, which displays the propagation of femtosecond pulses in optical fibers. In this paper, we investigate a generalized Sasa-Satsuma(gSS) equation. The Darboux transformation(DT) for the focusing and defocusing gSS equation is constructed. By using the DT, various of soliton solutions for the generalized Sasa-Satsuma equation are derived, including hump-type, breather-type and periodic soliton. Dynamics properties and asymptotic behavior of these soliton solutions are analyzed. Infinite number conservation laws and conserved quantities for the gSS equation are obtained.

Semiclassical analysis of quantum asymptotic fields in the Yukawa theory

In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics, in the semiclassical regime $\hslash\to 0$, with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schr\odinger-Klein-Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schr\odinger-Klein-Gordon equation. Moreover, we prove that the asymptotic vacuum states of the Yukawa model have a phase space concentration property around classical radiationless solutions. Under further assumptions, we show that the S-KG energy admits a unique minimizer modulo symmetries and identify exactly the semiclassical measure of Yukawa ground states. Some additional consequences of asymptotic completeness are also discussed, and some further open questions are raised.

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

In this article multi-soliton solutions of reverse space-time nonlocal nonlinear Schr ödinger (NLS) equation have been constructed. Darboux transformation is applied to the associated linear eigenvalue problem for the generalized NLS equation and we obtain a determinant formula for multi-soliton solutions. Under suitable reduction conditions and appropriate choice of spectral parameters, the generalized expression of first-order nontrivial solution gives some novel solutions such as double-hump and flat-top soliton solutions for reverse space-time nonlocal NLS equation. The dynamics and interaction of double-hump soliton solutions are studied in detail and it is indicated that these solutions undergo collisions without any energy redistribution. For higher-order double-hump solutions, the relative velocities of solitons play a crucial role to have humps and also induce nonlinear interference in the collision zone. The dynamics of individual decaying and growing unstable and stable double-humps as well as their interactions are explained and illustrated.

Higher-order rogue-wave fission in the presence of self-steepening and Raman self-frequency shift

Using the generalized nonlinear Schr\"odinger equation, we investigate how the effect of self-steepening and Raman-induced self-frequency shift impact higher-order rogue-wave solutions. We observe that each effect breaks apart the higher-order rogue wave, reducing it to its constituent fundamental parts, in a similar manner to how a higher-order soliton undergoes fission. Applying a local inverse-scattering technique, we show that under the effect of self-frequency shift, the emergence of a rogue wave significantly influences the surrounding wave background, triggering solitons, breathers, and new rogue waves. We demonstrate that under the combined effect of third-order dispersion, self-steepening, and Raman-induced self-frequency shift, the disintegrated elements of higher-order rogue waves become fundamental solitons, creating an asymmetrical spectral profile that generates both red- and blue-shifted frequency components. We also show the intermediary processes of the fission steps prior to soliton transformation which can only be observed in the presence of weak perturbations. These observations reveal the mechanisms that create a large number of solitons in the process of modulation instability-induced supercontinuum generation from a continuous-wave background in optical fibers.

Computing the Action Ground State for the Rotating Nonlinear Schrödinger Equation

We consider the computations of the action ground state for a rotating nonlinear Schr\"odinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. Based on it, we propose a normalized gradient flow method with asymptotic Lagrange multiplier and establish the energy-decaying property. Popular optimization methods are also applied to gain more efficiency. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization. Then the direct gradient flow method and unconstrained optimization methods are applied. Numerical experiments show the convergence and accuracy of the proposed methods in both cases, and comparisons on the efficiency are discussed. Finally, the relation between the action and the energy ground states are numerically investigated.

Resonant collisions of high-order localized waves in the Maccari system

Exploring new nonlinear wave solutions to integrable systems has always been an open issue in physics, applied mathematics, and engineering. In this paper, the Maccari system, a two-dimensional analog of nonlinear Schrödinger equation, is investigated. The system is derived from the Kadomtsev–Petviashvili (KP) equation and is widely used in nonlinear optics, plasma physics, and water waves. A large family of semi-rational solutions of the Maccari system are proposed with the KP hierarchy reduction method and Hirota bilinear method. These semi-rational solutions reduce to the breathers of elastic collision and resonant collision under special parameters. In case of resonant collisions between breathers and rational waves, these semi-rational solutions describe lumps fusion into breathers, or lumps fission from breathers, or a mixture of these fusion and fission. The resonant collisions of semi-rational solutions are semi-localized in time (i.e., lumps exist only when t → +∞ or t → −∞), and we also discuss their dynamics and asymptotic behaviors.

Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates.

The condensation of a trapped dilute Bose gas with three-body interactions

We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting via a three-body interaction potential of the form $N\, V(N^{1/2}(x-y,x-z))$. In the limit $N\to \infty$, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schr\"odinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the $N$-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schr\"odinger minimizer is unique up to a complex phase.