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The nonlinear Schrodinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrodinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results may be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as optical fibers, oceanography and so on.

In this work, we present a numerical study of two-soliton collisions in a system described by a cubic (Kerr-type) nonlinear Schrodinger equation whose nonlinearity has small chaotic imperfections. We use a logistic map in order to obtain a chaotic perturbation, where by defining the values of its seed and the interaction parameter, one can observe a disorder in the nonlinearity of the system. This disorder was varied by changing the parameter values and controlled via the Lyapunov exponent, however, always maintaining a fixed amplitude. We verified a direct relationship between the value of the Lyapunov coefficient and the formation of two-soliton bonded/unbonded states.

We prove that the flow of the discrete nonlinear Schrödinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose–Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative $$L^p$$ L p -estimate, for any $$p \ge 1$$ p ≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.

This study establishes the extended classical optical solitons for a nonlinear Schrodinger equation describing resonant nonlinear light propagation through isolated flaws in optical wave guides. We use the modified Sardar sub-equation approach to get such innovative results. The innovative optical solitons solutions have been investigated to explain unique physical obstacles, and they entail an extended classical M-truncated derivative, which affects the physical properties of the findings greatly. These advancements have been shown to be beneficial in the transmission of long-wave and high-power communications networks. Furthermore, the figures for the acquired solutions are graphed through the depiction of the 3D and contour plots in order to throw additional light on the peculiarities of the obtained solutions.

Optical fiber communication has developed rapidly because of the needs of the information age. Here, the variable coefficients fifth-order nonlinear Schrodinger equation (NLS), which can be used to describe the transmission of femtosecond pulse in the optical fiber, is studied. By virtue of the Hirota method, we get the one-soliton and two-soliton solutions. Interactions between solitons are presented, and the soliton stability is discussed through adjusting the values of dispersion and nonlinear effects. Results may potentially be useful for optical communications such as all-optical switches or the study of soliton control.

In this paper, a novel reduction approach is reported for a physically meaningful nonlocal nonlinear Schrodinger (NLS) equation of reverse-time type to obtain its N-soliton solutions. Firstly, single-soliton solutions of the nonlocal NLS equation are obtained by reducing those of the local NLS equation. Secondly, inspired by the form of single-soliton solutions, N-soliton representations of the nonlocal NLS equation are conjectured and then verified via an algebraic proof. Thirdly, to demonstrate the features of the soliton solutions, some special soliton dynamics are theoretically explored and graphically illustrated. The reduction approach proposed in this paper has the merit that it is purely algebraic which does not need to perform complicated spectral analysis of the corresponding Lax pair.

We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.

In this paper, by using Fokas method, we study the initial-boundary value problems (IBVPs) of the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation on the half-line, which can simulate the nonlinear transmission and interaction of ultrashort pulses in the high-speed optical fiber transmission system, and can also describe the nonlinear spin excitation phenomenon of one-dimensional Heisenberg ferromagnetic chain with eight poles and dipole interaction. By discussing the eigenfunctions of Lax pair of FODNLS equation and analyzing symmetry of the scattering matrix, we get a matrix Riemann–Hilbert (RH) problem from for the IBVPs of FODNLS equation. Moreover, we get the potential function solution u(x, t) of the FODNLS equation by solving this matrix RH problem. In addition, we also obtain that some spectral functions satisfy an important global relation.

In this paper, the generalized exponential rational function method is applied to obtain analytical solutions for the conformable (2+1)-dimensional chiral nonlinear Schrodinger equation. We obtain novel soliton, traveling waves and kink-type solutions with complex structures. We also present the two- and three-dimensional shapes for the real and imaginary part of the obtained solutions. It is illustrated that generalized exponential rational function method is simple and efficient method to reach the various types of the soliton solutions.

The response of light in a nonlinear dispersive medium is governed by the nonlinear Schrodinger equation (NLSE). In general, for higher order group velocity dispersion (GVD) with frequency-dependent gain or absorption, NLSE is numerically solved using the split-step Fourier method. Extended NLSEs including the effects of higher order GVD, frequency-dependent gain and frequency-dependent absorption comprise higher order derivative terms. In this paper, an algorithm to solve partial differential equations with any higher order derivative term is described. A program based on this algorithm was used to solve optical pulse propagation equations in a nonlinear medium with higher order GVD, frequency-dependent gain and saturable absorption.

We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where $$N\ge 1$$ N ≥ 1 , $$p,q>1$$ p , q > 1 , $$\lambda ,\mu \in {\mathbb {C}}$$ λ , μ ∈ C , $$\lambda \ne 0$$ λ ≠ 0 , and $$u(0,\cdot ), w\in L^1_{\mathrm{loc}}({\mathbb {R}}^N,{\mathbb {C}})$$ u ( 0 , · ) , w ∈ L loc 1 ( R N , C ) . We consider both the cases where $$\mu =0$$ μ = 0 and $$\mu \ne 0$$ μ ≠ 0 , respectively. We establish existence/nonexistence of global weak solutions. In each studied case, we compute the critical exponents in the sense of Fujita, and Lee and Ni. When $$\mu \ne 0$$ μ ≠ 0 , we show that the nonlinearity $$|\nabla u|^q$$ | ∇ u | q induces an interesting phenomenon of discontinuity of the Fujita critical exponent.

In the present study, we investigated a numerical simulation for employing fused silica for self-compression of femtosecond laser pulses. Self-compression, resulting from the interaction between self-phase modulation and negative group velocity dispersion, can be solved numerically by the nonlinear Schrodinger equation. Once a laser pulse of 200 fs pulse duration and intensity of 1 TW/cm2 at 2 µm wavelength is applied to a 3.5 mm of fused silica, the pulse is compressed down to nearly 7 fs, corresponding to a compression factor of ~ 27. A high compression factor is acquired by either increasing the input pulse intensity or increasing the input pulse duration. On the other hand, the shortest propagation length for achieving a high compression takes place for either high input intensities or short input pulse duration. Therefore, an optimization of the operational parameters has been studied in the present work based on the B-integral as well as the depression of small-scale self-focusing effect.

The modulational instability of dust acoustic (DA) waves propagating in a four-component dusty plasma system comprising negatively and positively charged dust grains as well as kappa-distributed electrons and ions has been studied in the presence of a generalized polarization force. The derivative expansion technique is used to derive a nonlinear Schrodinger equation (NLSE), which describes the nature of the modulational instability of DA waves. The conditions for the existence of the modulational instability of DA waves has been discussed. The results exhibit that the plasma system supports both a modulationally unstable domain and a stable domain, which are significantly modified by the related plasma parameters (viz., polarization force parameter $$R$$ , dust charge number ratio $$Z$$ , and superthermal parameters of electrons and ions). Moreover, the instability growth rate is found to be effectively modified by the presence of these parameters. Our results could be applicable for different space and astrophysical plasma environments.

In this paper, we prove the existence of small amplitude solutions which are almost-periodic in time for a quasi-periodically forced nonlinear Schrodinger equation $$\begin{aligned} \sqrt{-1}u_{t}-u_{xx}+M_{\xi }u+f(\tilde{\omega }t)|u|^2u=0 \end{aligned}$$ with periodic boundary conditions $$\begin{aligned} u(t,x+2\pi )=u(t,x),~~t\in \mathbb {R}, \end{aligned}$$ where $$M_{\xi }$$ is a real Fourier multiplier, $$f(\tilde{\theta })(\tilde{\theta }=\tilde{\omega }t)$$ is real analytic and the forced frequency vector $$\tilde{\omega }\in \mathbb {R}^{b}$$ is fixed and Diophantine.

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.

In Chen and Wang (Calc Var Part Differ Equ 56:1–26, 2017), we show that, if $$\epsilon >0$$ is small enough, then there exists a sequence of semiclassical states of higher topological type localized at a local minimum set of the potential V for the semiclassical nonlinear Schrodinger equation $$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1\left( {\mathbb{R}}^N\right) . \end{aligned}$$ In this paper, we consider a situation where V has multiple isolated local minimum sets. We show that as $$\epsilon \rightarrow 0,$$ there exist multi-bump solutions of this equation being concentrated at those given local minimum sets while at the same time each bump behaves as a higher-topological-type solution in one local minimum set as aforementioned. Thus, the multi-bump solutions given here are constructed by gluing a sum of higher-topological-type solutions localized in separated local minimal sets of the potential.

We consider the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Schrodinger equation with critical cubic nonlinearity $$\begin{aligned} \left\{ \begin{array}[c]{c} i\partial _{t}u+\frac{1}{2}\partial _{x}^{2}u-\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha }u=\lambda \left| u\right| ^{2}u, t>0, x\in \mathbb {R}\mathbf {,}\\ u\left( 0,x\right) =u_{0}\left( x\right) , x\in \mathbb {R} \mathbf {,} \end{array} \right. \end{aligned}$$ where $$\lambda \in \mathbb {R}$$ and $$\alpha =4$$ or $$\alpha \ge 5,$$ since while estimating pseudodifferential operators below we need that the condition such that the symbol $$\Lambda \left( \xi \right) =\frac{1}{2}\xi ^{2}+\frac{1}{\alpha }\left| \xi \right| ^{\alpha }\in \mathbf {C}^{5}\left( \mathbb {R}\right) .$$ We show that the modified scattering occurs in the uniform norm. We continue to develop the factorization techniques which was started in papers (Ozawa in Commun Math Phys 139(3):479–493, 1991; Hayashi and Ozawa in Ann IHP (Phys Theor) 48:17–37, 1988; Hayashi and Naumkin in Z Angew Math Phys 59(6):1002–1028, 2008; Hayashi and Kaikina in Math Methods Appl Sci 40(5):1573–1597, 2017; Hayashi and Naumkin in J Math Phys 56(9):093502, 2015). The crucial points of our approach presented here are the $$\mathbf {L}^{2}$$ -boundedness of the pseudodifferential operators which are used to obtain estimates of nonlinear terms in weighted Sobolev space.

We consider the one-dimensional (1D) cubic-quintic Gross--Pitaevskii (GP)nequation, which governs the dynamics of Bose--Einstein condensate (BEC) matter waves with time-varying scattering length and loss/gain of atoms in a harmonic trapping potential. We derive the integrability conditions and the compensation condition for the 1D GP equation and obtain, with the help of a cubic-quintic nonlinear Schr\"{o}dinger (NLS) equation with self-steepening and self-frequency shift, exact analytical solitonlike solutions with the corresponding frequency chirp which describe the dynamics of femtosecond solitons and double-kink solitons propagating on a vanishing background. Our investigation shows that under the compensation condition, the matter wave solitons maintain a constant amplitude, the amplitude of the frequency chirp depends on the scattering length, while the motion of both the matter wave solitons and the corresponding chirp depend on the external trapping potential. More interesting, the frequency chirps are localized and their feature depends on the sign of the self-steepening parameter. Our study also shows that our exact solutions can be used to describe the compression of matter wave solitons when the absolute value of the s-wave scattering length increases with time.

The general N-solitons of nonlocal generalized nonlinear Schrodinger equations with third-order, fourth-order and fifth-order dispersion terms and nonlinear terms (NGNLS) are studied. Firstly, the Riemann–Hilbert problem and the general N-soliton solutions of NGNLS equations were given. Then, we study the symmetry relations of the eigenvalues and eigenvectors related to the scattering data which involve the reverse-space, reverse-time and reverse-space-time reductions. Thirdly, some novel solitons and the dynamic behaviors which corresponded to novel eigenvalue configurations and the coefficients of higher-order terms are given. In all the three NGNLS equations, their solutions often collapse periodically, but can remain bounded or nonsingular for wide ranges of soliton parameters as well. In addition, it is found that the higher-order terms of the NGNLS equations not only affect the amplitude variation of the soliton, but also influence the singularity and the motion of the soliton.

Under the framework of the complex column-vector loop algebra Cp, we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme. As applications of the scheme, we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model. The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems, including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell. Specially, we obtain the famous Fokker-Plank equation and its generalized form, which has extensive applications in the stochastic dynamic systems. Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).

A nonlinear iteration scheme for nonlinear Schrodinger equation with 2-step backward differential formula (BDF) finite element method (FEM) is proposed. Energy stability is testified for the constructed scheme, which leads to the boundedness of $\|{U_{h}^{n}}\|_{0}$ and $\sqrt {\tau }\|\nabla {U_{h}^{n}}\|_{0}$ . Auxiliary equation known as a time-discrete system is constructed to get rid of the restriction of τ. The solutions of the time-discrete equation in H1-norm is deduced. On the one hand, $\sqrt {\tau }\|\nabla U^{n}\|_{0}$ reduces to the temporal error in H2-norm. On the other hand, with the help of the boundedness about $\sqrt {\tau }\|\nabla {U_{h}^{n}}\|_{0}$ , the unconditional optimal estimate for spatial error is derived. Without any restriction of the time step, $\|{U_{h}^{n}}\|_{0,\infty }$ is bounded through surmounting the difficulty of nonlinear term. By taking difference between two time levels n and n − 1 of the error equation, an unconditional superconvergence estimate is derived. At last, global superconvergence result is achieved through the known interpolated postprocessing technique. Here, τ is the time step, and Un and ${U_{h}^{n}}$ denote the solutions of the time-discrete system and the finite element approximation equation, respectively. Furthermore, by introducing modified mass and energy functions, the numerical scheme is proved to preserve the total mass and energy in the discrete senses. Finally, numerical results are given to support the theoretical analysis.

The (2+1)-dimensional generalized coupled nonlinear Schrodinger equation with the four-wave mixing (FWM) term is studied in this paper, which describes the optical solitons in a birefringent fiber. By virtue of the Hirota method, the one- and two-soliton solutions are derived. On the basis of solutions obtained, we discuss how the values of the FWM and some free parameters affect the solitons’ peformance. The FWM parameter can help to control the amplitude of the solitons. Meanwhile, by setting the values of certain free parameters, we can control the solitons’ propagation direction and speed, and reduce the interactions between them as well. In addition, the energy transfer of solitons during elastic collision and separation is also discussed. The conclusions here may be useful in improving the communication quality in multi-mode fibers.

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrodinger equation $$\left\{ {\matrix{ { - {\rm{\Delta }}u + V\left( x \right)u = {\mu _q}u + a{{\left| u \right|}^q}u\,\,\,{\rm{in}}\,{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{{\left| u \right|}^2}dx = 1,} } \hfill \cr } } \right.$$ where μq is the Lagrange multiplier. We show that for q > 2 close to 2, the problem admits two solutions: one is the local minimal solution uq and the other one is the mountain pass solution υq. Furthermore, we study the limiting behavior of uq and υq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state υq.

Propagation of terahertz (THz) waves in silicon waveguides is of special interest for THz applications such as optical integrated circuits. In this paper, a photonic crystal-based rib silicon waveguide with a specific structure is designed for the first time which can be utilized for applications in THz region (including wavelength range of 30-35 μm). The dispersion curve and effective index are simulated over 30–35 μm for different rib heights. In addition, the dispersion coefficients up to the third order as well as the nonlinear parameter are calculated at the center wavelength of 33 μm from which the pulse evolution along the waveguide is simulated via solving the nonlinear Schrodinger equation (NLSE). The split-step Fourier (SSF) method is used to numerically solve the NLSE and then the pulse spectra at the input and output of the waveguide are simulated and compared to each other. Also, the effect of different parameters including rib height, pulse width, chirp and peak power on the THz pulse propagation is investigated. The results show that for lengths less than 1 cm, the pulse propagates almost without distortion, while for longer lengths, the shape and spectrum of the pulse are changed.

We investigate the dromion-like excitations corresponding to intramolecular chain-like proteins. In the present work, the dromion-like excitations are described by using cubic-quintic nonlinear Schrodinger equation (CQNSE) governing the dynamics of proteins and we analytically analyze the velocity (v) of dromion-like structure compared with velocity ( $$v_a$$ ) of acoustical sound waves corresponding to the longitudinal vibrations of protein molecules. Our work is motivated by the effectiveness and powerful mathematical techniques such as modified extended tanh function method and sine–cosine function method for solving CQNSE to obtain dromion-like structures.

we investigate the existences of KAM tori for the infinite dimensional Hamiltonian system with finite number of zeros among normal frequencies. By constructing a constant vector we show that, for “most” tangent frequencies in the sense of Lebesgue measure, either if the vector is zero, there is a KAM torus or if the vector is not zero, there is no KAM torus in some domain. As an application, we show that the nonlinear Schrodinger equation with a zero among normal frequencies possesses many quasi-periodic solutions.

A generalized nonlinear Schrodinger 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.

We study the focusing stochastic nonlinear Schrodinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the $$L^2$$ -critical and supercritical cases. The mass ( $$L^2$$ -norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate both theoretically and numerically how the energy is affected by various spatially correlated random perturbations and its dependence on the discretization parameters and the schemes. We then perform numerical investigation of the noise influence on the global dynamics measuring the probability of blow-up versus scattering behavior depending on parameters of correlation kernels. Finally, we study numerically the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed for a space-time white driving noise in Millet et al. (Numerical study of solutions behavior to the 1d stochastic $$L^2$$ -critical and supercritical nonlinear Schrodinger equation, 2020. arXiv:2006.10695 ).

In this paper, higher-order localized waves of three-component coupled nonlinear Schrodinger equation are investigated using generalized Darboux transformation. By constructing a novel seed solution as the function of evolution time and normalized distance, a series of new phenomena for higher-order localized waves are displayed with numerical simulations. Specially, the angles between propagation directions of dark-bright solitons(breathers) and the positive direction of x-axis are analysed on the basis of free parameters. The results obtained will contribute to reveal dynamic features of localized waves for the coupled system.

The long-time asymptotic behavior of solutions to the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the $xt$-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations., Comment: 76 pages, 35 figures

We consider the Cauchy problem for the focusing nonlinear Schr\"odinger equation with initial data approaching two different plane waves $A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. Using Riemann-Hilbert techniques and Deift-Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases $B_1B_2$, and $B_1=B_2$ deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors., Comment: 50 pages, 52 figures

In this paper, a variable-coefficient nonlinear Schrodinger equation that describes the optical soliton propagation in dispersion management fiber systems is studied. Two- and three-soliton solutions are obtained by using the Hirota bilinear method. Based on those solutions, the effects of related parameters on optical soliton propagation are discussed. By choosing different values of the third-order dispersion, the amplification of optical solitons can be realized. In addition, the interactions among the solitons can be reduced by setting a proper value of the group velocity dispersion. The results of this paper may be helpful to design optical amplifiers or to improve the quality of optical communications.

Variable coefficients nonlinear evolution equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counter constant coefficients in some real-world problems. Under consideration is a nonlinear variable coefficients Schrödinger’s equation with spatio-temporal dispersion in the Kerr law media. We are aimed at constructing novel solutions to the equation under consideration. Bright and combined dark– bright optical solitons are successfully revealed with aid of the complex amplitude ansatz scheme. Using two test functions, two nonautonomous complex wave solutions in dark and bright optical solitons forms are successfully revealed. The effect of the variable coefficients on the reported results can be clearly seen on the 3-dimensional and contour graphs.

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrodinger equation $$\text {i}\partial _t u + \Delta u + |u|^{p-1}u=0$$ on $$\mathbb {R}^d$$ , close to the mass critical case and for any space dimension $$d\ge 1$$ . These profiles bifurcate from the ground-state solitary wave. The argument relies on the classical matched asymptotics method suggested in Sulem and Sulem (The nonlinear Schrodinger equation. Self-focusing and wave collapse. Applied mathematical sciences, 139, Springer, New York, 1999) which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log–log blow-up law observed in the mass critical case.

We describe in detail the construction of numerical fourth- and sixth-order schemes using the Magnus expansion for solving the Zakharov–Shabat system, which makes it possible to solve accurately the direct scattering problem for the nonlinear Schrodinger equation. To avoid numerical instabilities inherent in the procedure of solving the direct scattering problem, high-precision arithmetic is used. At present, application of the proposed schemes in combination with the highprecision arithmetic is a unique tool that can be used for analysis of complex wave fields, which contain a great number of solitons, and allows one to determine the complete discrete spectrum, including both eigenvalues and normalization constants. In this work, we study the errors in the proposed scheme using an example of the potential in the form of a hyperbolic secant. It is found that the time of calculation of the scattering matrix using the sixth-order algorithm is almost twice as long compared with that for the standard second-order Boffetta–Osborne algorithm, whereas the time gain resulting from the reduction of number of the wave-field discretization points with retaining the desired precision can reach an order of magnitude or more. Exact solution of the direct scattering problem requires that the discretization increment of high-amplitude wave fields should be comparable with the characteristic width of the largest solitons contained in such fields. In this case, the discretization increment can be sufficiently smaller than that required for reconstruction of the full Fourier spectrum of the wave field. Application of the proposed high-order approximation schemes can be of fundamental importance for successful operation with a great number of complex nonlinear wave fields, such as, e.g., that in the process of the statistical study of the scattering data.

Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge–Kutta methods, however, relies on a convergence result that requires weakening many of the order conditions, resulting in schemes whose stages must be implemented in a sequential way. In this work, after showing a stronger convergence result, we are able to derive two new families of fourth- and fifth-order exponential Runge–Kutta methods, which, in contrast to the existing methods, have multiple stages that are independent of one another and share the same format, thereby allowing them to be implemented in parallel or simultaneously, and making the methods to behave like using with much less stages. Moreover, all of their stages involve only one linear combination of the product of $$\varphi $$ -functions (using the same argument) with vectors. Overall, these features make these new methods to be much more efficient to implement when compared to the existing methods of the same orders. Numerical experiments on a one-dimensional semilinear parabolic problem, a nonlinear Schrodinger equation, and a two-dimensional Gray–Scott model are given to confirm the accuracy and efficiency of the two newly constructed methods.

The present paper is concerned with the half-space Dirichlet problem "Equation missing"where $$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$ R + N : = { x ∈ R N : x N > 0 } for some $$N \ge 1$$ N ≥ 1 and $$p > 1$$ p > 1 , $$c > 0$$ c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ). We prove that the existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ) depend in a striking way on the value of $$c > 0$$ c > 0 and also on the dimension N. We find an explicit number $${c_p}\in (1,\sqrt{e})$$ c p ∈ ( 1 , e ) , depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions $$N \ge 2$$ N ≥ 2 , we prove that, for $$0< c < {c_p}$$ 0 < c < c p , problem ($$P_c$$ P c ) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$ c > c p , there are no bounded positive solutions to ($$P_c$$ P c ).

In this paper, the nonlinear ion acoustic rogue waves (IARWs) are investigated in an unmagnetized, collisionless plasma which consist of positive ions, superthermal electrons and positrons, respectively. A nonlinear Schrodinger equation (NLSE) of IAW packets has been obtained by using the reductive perturbation technique. Furthermore, the effects of superthermal electrons and positrons on the nonlinear dispersion relation, the group velocity and the modulational instability of nonlinear ion acoustic wave (IAW) packets have been studied in detail. At the same time, the characteristics of first- and second-order IARW also have been discussed. It seems that there is some significant influence of superthermal electrons and positrons on the nonlinear ion acoustic waves. Meanwhile, it is also found that the first- and second-order IARW consisting of Peregrine breathers can be excited by using the modulational instability in this system.

In this paper, we investigate the propagation of short pulses in dispersion-engineered silicon nanowires via solving nonlinear Schrodinger equation (NLSE) using the split-step Fourier (SSF) method. By assuming secant-hyperbolic input pulse and including the nonlinear parameter, second- and third-order dispersion coefficients as a function of nanowire length, the impact of chirp parameter on the pulse evolution along the nanowire is simulated. The results show that the value and sign of the chirp parameter have significant impacts on the pulse evolution and specially the pulse broadening along the nanowire. Hence, the broadening factor can be reduced by adding a positive chirp to the input pulse. By choosing proper values of pulse parameters, pulse propagation can be controlled such that the minimum broadening during the propagation happens.

The nonlinear propagation of dust acoustic (DA) waves (DAWs) in an unmagnetized electron depleted dusty plasma (containing opposite polarity warm dust grains and non-extensive positive ions) has been theoretically investigated. The nonlinear Schrodinger equation (NLSE) is derived by employing the reductive perturbation method. It is observed that the dusty plasma system under consideration supports two branches of modes, namely, fast and slow DA modes, and that both of these two modes can be modulationally stable or unstable depending on the sign of ratio of the dispersive and nonlinear coefficients of the NLSE. The conditions for the formation of bright and dark envelope solitons as well as the first- and second-order rogue waves have been examined. The numerical analysis has shown that the basic features (viz., stability/instability, amplitude and width of the envelope solitons and rogue structures, etc.) of the DAWs associated with the fast and slow DA modes are significantly modified by dust masses, dust charges, non-extensivity of ions, and other various plasma parameters. The implications of the results to space plasma research (viz., magnetosphere of Jupiter, upper mesosphere, Saturn’s F-ring, cometary tail, etc.) are briefly discussed.

We consider a damped/driven nonlinear Schrodinger equation in $$\mathbb {R}^n$$ , where n is arbitrary, $${\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,$$ under odd periodic boundary conditions. Here $$\eta (t,x)$$ is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $$ \Vert u(t)\Vert _m^2 \le C\nu ^{-m}, $$ uniformly in $$t\ge 0$$ and $$\nu >0$$ . In this work we prove that for small $$\nu >0$$ and any initial data, with large probability the Sobolev norms $$\Vert u(t,\cdot )\Vert _m$$ with $$m>2$$ become large at least to the order of $$\nu ^{-\kappa _{n,m}}$$ with $$\kappa _{n,m}>0$$ , on time intervals of order $$\mathcal {O}(\frac{1}{\nu })$$ . It proves that solutions of the equation develop short space-scale of order $$\nu $$ to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation.

The complete discrimination system was proposed to search traveling wave solutions for a (1 + 1)-dimensional nonlinear Schrodinger equation with parabolic law nonlinearity. As a consequence, we obtained a series of solutions which contain dark soliton, bright soliton, triangular solitary wave solution, kink soliton, periodic solutions and implicit analytical solutions. At the same time, numerical simulations were demonstrated to visualize the mechanics of these solutions. Lastly, linear stability of dark soliton $$u_{25}$$ in Lyapunov’s sense was discussed.

The admissibility of the initial-boundary data, which characterizes the existence of solution for the initial-boundary value problem, is important. Based on the Fokas method and the inverse scattering transformation, an approach is developed to solve the initial-boundary value problem of the nonlinear Schrodinger equation on a finite interval. A necessary and sufficient condition for the admissibility of the initial-boundary data is given, and the reconstruction of the potential is obtained.

The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut = iαuxx + βu2ūx + γ∣u∣2ux + i∣u∣2u on the half-line with inhomogeneous boundary condition. We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces. Moreover, we show that the nonlinear part of the solution on the half-line is smoother than the initial data.

In this work, we study the Kundu-nonlinear Schrodinger (Kundu-NLS) equation (so-called the extended NLS equation), which can describe the propagation of the waves in dispersive media. A Lax spectral problem is used to construct the Riemann–Hilbert problem, via a matrix transformation. Based on the inverse scattering transformation, the general solutions of the Kundu-NLS equation are calculated. In the reflection-less case, the special matrix Riemann–Hilbert problem is carefully proposed to derive the multi-soliton solutions. Finally, some novel dynamics behaviors of the nonlinear system are theoretically and graphically discussed.

We prove local well-posedness for the periodic derivative nonlinear Schrodinger equation, which is $$L^2$$ critical, in Fourier-Lebesgue spaces which scale like $${H}^s({\mathbb {T}})$$ for $$s>0$$ . Our result is optimal in the sense that it covers the full subcritical regime. In particular we close the existing gap in the subcritical theory by improving the result of Grunrock and Herr (SIAM J Math Anal 39(6):1890–1920, 2008), which established local well-posedness in Fourier-Lebesgue spaces which scale like $${H}^s({\mathbb {T}})$$ for $$s >\frac{1}{4}$$ . We achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space. Together these allow us to construct the unique solution to the given subcritical data. This constructive procedure is inspired by the theory of para-controlled distributions developed by Gubinelli et al. (Forum Math Pi, 3:75, 2015) and Catellier and Chouk (Ann Probab 46(5):2621–2679, 2018) in the context of stochastic PDE. Our proof and results however, are purely deterministic.

The amplitude modulation of electron acoustic waves (EAWs) is investigated in an unmagnetized four-component quantum plasma containing dynamic cold electron fluid, Thomas–Fermi distributed hot electrons and positrons and immobile ions in presence of exchange-correlation potentials. For this purpose, basic set of quantum hydrodynamic equations is reduced to a nonlinear Schrodinger equation (NLSE) adopting a reductive perturbation technique. The regions of the stable and unstable wave packets are obtained precisely in terms of the intrinsic plasma parameters. It is shown that the NLSE leads to the modulational instability (MI) of EAWs, and the formation of EA rogue waves, which are due to the effects of nonlinearity and dispersion in the propagation of EAWs. The effects of relevant plasma parameters on the MI criterion are numerically investigated. It is also found that the modulated EAW packets can propagate in the form of bright envelope solitons as well as dark envelope solitons. Within the modulational instability region, the dependence of first- and second-order EA rogue wave profiles on the system parameters is discussed.

We consider the two-dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. In order to describe the evolution of the envelopes of small oscillating wave packet-like solutions to this problem the Nonlinear Schr\"odinger equation can be derived as a formal approximation equation. In recent years, the validity of this approximation has been proven by several authors for the case without surface tension. In this paper, we rigorously justify the Nonlinear Schr\"odinger approximation for the cases with and without surface tension by proving error estimates over a physically relevant timespan in the arc length formulation of the two-dimensional water wave problem. The error estimates are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero., Comment: This paper is a minor revision of the previous submission arXiv:1907.03014v1. In its present form, the paper has been accepted for publication in Archive for Rational Mechanics and Analysis