Nonvanishing Boundary Conditions Research Articles

Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.

Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions

Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions

On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work.

Quaternion algebra approach to nonlinear Schrödinger equations with nonvanishing boundary conditions

In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices.

Thermomechanically coupled theory in the context of the multiphase-field method

The modeling and simulation of microstructure evolution is usually subject to the multiphase-field method. In this context, thermomechanical coupling is often neglected, even when non-isothermal phase transformations are considered. Using a simplified example, the present work shows that this assumption is not justified for small strains and small strain rates with respect to a non-vanishing coefficient of thermal expansion. To this end, both a thermomechanically coupled and a thermomechanically weakly coupled theory are briefly revisited. The difference between the coupled and the weakly coupled theory regarding the growth of an inclusion is discussed. The considered elastoplastic inclusion, subjected to eigenstrains, is embedded in an elastoplastic matrix under load. It is shown, that the weakly coupled theory overestimates the growth of the inclusion, and, thus, the volume concentration, compared to the coupled theory. Moreover, only the application of the coupled theory reflects a load-induced anisotropic growth of the inclusion, which exhibits an isotropic material behavior, due to non-vanishing uniaxial Neumann boundary conditions. In addition, it is shown that the smaller the heat conduction coefficient, the more pronounced the anisotropic growth of the inclusion.

Riemann–Hilbert problem for the Fokas–Lenells equation in the presence of high-order discrete spectrum with non-vanishing boundary conditions

We extend the Riemann–Hilbert (RH) method to study the Fokas–Lenells (FL) equation with nonzero boundary conditions at infinity and successfully find its multiple soliton solutions with one high-order pole and N high-order poles. The mathematical structures of the FL equation are constructed, including global conservation laws and local conservation laws. Then, the conditions (analytic, symmetric, and asymptotic properties) needed to construct the RH problem are obtained by analyzing the spectral problem. The reflection coefficient r(z) with two cases appearing in the RH problem is considered, including one high-order pole and N high-order poles. In order to overcome the difficulty of establishing the residue expressions corresponding to high-order poles, we introduce the generalized residue formula. Finally, the expression of exact soliton solutions with reflectionless potential is further derived by a closed algebraic system.

Global Well-Posedness of the 4-D Energy-Critical Stochastic Nonlinear Schrödinger Equations with Non-Vanishing Boundary Condition

We consider the energy-critical stochastic cubic nonlinear Schrödinger equation on R4 with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrödinger equation on R4, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.

Riemann-Hilbert approach for the Kundu equation with non-vanishing boundary conditions: Simple and double poles

The inverse scattering transform for the Kundu equation with non-vanishing boundary conditions at infinity is studied via the Riemann-Hilbert approach. A detailed spectral analysis for analyticity, symmetry properties, and asymptotic behavior of the modified Jost eigenfunctions and scattering matrix is discussed. And an appropriate matrix Riemann-Hilbert problem is formulated. A closed system is solved for the explicit N-soliton solutions to the Kundu equation in the case of the reflectionless potential. Besides, the trace formula for the analytic scattering coefficients and the so-called theta condition for the phase difference between the boundary values are derived. A consistent framework is constructed to obtain solutions associated with double zeros to the analytic scattering coefficients. The solitons in the simple and double poles case of the Chen-Lee-Liu equation as a reduced example are studied, including any combination of dark, bright and breather solitons, and the dynamical properties of these solutions are displayed and analyzed.

Tau‐function formulation for bright, dark soliton and breather solutions to the massive Thirring model

AbstractIn the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two‐component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single‐component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one‐ and two‐soliton for bright, dark soliton and breather solutions are analyzed in details.

A Matrix Schrödinger Approach to Focusing Nonlinear Schrödinger Equations with Nonvanishing Boundary Conditions

We relate the scattering theory of the focusing AKNS system with equally sized nonvanishing boundary conditions to that of the matrix Schrödinger equation. This (shifted) Miura transformation converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method of solving the Marchenko integral equations by separation of variables to derive the multisoliton solutions of this nonlocal equation, thus proposing a method to solve the reflectionless matrix NLS equation.

Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions

We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The nonlinear hydrodynamic equations of the WNT are solved analytically through a connection to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a nonvanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.

Darboux transformation and solitonic solution to the coupled complex short pulse equation

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright-soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis. Breather and rogue wave solutions are constructed under non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution.

Multiple-high-order pole solutions for the NLS equation with quartic terms

The aim of this article is to investigate the multiple-high-order pole solutions to the focusing NLS equation with quartic terms (QNLS) under the non-vanishing boundary conditions (NVBC) via the Riemann–Hilbert(RH) method. The determinant formula of multiple-high-order pole soliton solutions for NVBC is given. Further the double 1nd-order, mixed 2nd- and 1nd-order pole solutions are obtained.

On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

<p style='text-indent:20px;'>In this paper we consider the Schrödinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces <inline-formula><tex-math id="M1">\begin{document}$ X^k( \mathbb{R}) $\end{document}</tex-math></inline-formula> and in <inline-formula><tex-math id="M2">\begin{document}$ \phi+H^k( \mathbb{R}) $\end{document}</tex-math></inline-formula>. Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.</p>

Riemann–Hilbert approach to the modified nonlinear Schrödinger equation with non-vanishing asymptotic boundary conditions

The modified nonlinear Schrödinger (NLS) equation was proposed to describe the nonlinear propagation of the Alfven waves and the femtosecond optical pulses in a nonlinear single-mode optical fiber. In this paper, we present the inverse scattering transform for the modified NLS equation iut+uxx+2|u|2u+i1α(|u|2u)x=0,α>0,with nonvanishing boundary values at infinity u(x,t)∼u±e−4iα2t+2iαx,x→±∞.An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. The direct scattering problem is shown to be well posed for potentials u such that u−u±∈L1,2(R±), for which existence and analyticity properties of eigenfunctions and scattering data are established. Their asymptotic behaviors and the symmetries are analyzed in details based on the Lax pair for the modified NLS equation. Then the inverse scattering problem is formulated as a Riemann–Hilbert (RH) problem associated with the problem of nonzero boundary conditions. The existence and uniqueness of solution for the mixed RH problem for t>0 are strictly proved by decomposing it into a pure RH problem and a scattering RH problem. The N-soliton solutions for the modified NLS equation are obtained via a reconstruction formulae between solution of the modified NLS equation and the solution of above mixed RH problem. As an illustrate example of N-soliton formula, for N=1 and N=2, two kinds of one-soliton solutions and three kinds of two-soliton solutions are explicitly presented, respectively according to different distribution of the spectrum. The dynamical feature of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions. Finally, we analyze the differences between our results and those on zero boundary case.

A (3,2) reduced degree-of-freedom unified zigzag laminated beam theory

A (3,2) unified zigzag beam theory is developed with a reduced number of degree-of-freedom. Comparing to previous methods in the field of zigzag beam theory, the main novelty in this paper's method is that a more general non-vanishing top/bottom surface's shear stress boundary conditions are satisfied automatically in strong form. The bottom surface shear stress condition and the interface shear stress continuity conditions are used to uniquely determine the coefficients of zigzag functions. For the top surface shear stress condition, it is used to eliminate one degree-of-freedom, changing the 7°-of-freedom (3,2) zigzag beam to a 6°-of-freedom (3,2) zigzag beam. The zigzag coefficients are derived with an explicit formulation. Since the proposed method's formula is based on the unified beam theory, the formulation can be applied to any specific beam theory. The corresponding zigzag coefficients are also dependent on the specific beam theory's thickness basis function.In the numerical test section, several benchmark problems are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for both thick and thin beams. The shear stress accuracy is also good for both vanishing and non-vanishing shear stress boundary conditions on top/bottom surfaces.

Multiple Soliton Solutions of the Sawada-Kotera Equation with a Nonvanishing Boundary Condition and the Perturbed Korteweg de Vries Equation by Using the Multiple Exp-Function Scheme

The Sawada-Kotera equation with a nonvanishing boundary condition, which models the evolution of steeper waves of shorter wavelength than those depicted by the Korteweg de Vries equation, is analyzed and also the perturbed Korteweg de Vries (pKdV) equation. For this goal, a capable method known as the multiple exp-function scheme (MEFS) is formally utilized to derive the multiple soliton solutions of the models. The MEFS as a generalization of Hirota’s perturbation method actually suggests a systematic technique to handle nonlinear evolution equations (NLEEs).

Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation

We consider a coupled nonlinear Schrödinger (NLS) equation, which can be reduced to the generalized NLS equation by constituting a certain constraint. We first construct a generalized Darboux transformation (DT) for the coupled NLS equation. Then, by using the resulting DT, we analyse the solutions with vanishing boundary condition and non-vanishing boundary condition, respectively, including positon wave, breather wave and higher-order rogue wave solutions for the coupled NLS equation. Moreover, in order to better understand the dynamic behavior, the characteristics of these solutions are discussed through some diverting graphics under different parameters choices.

Busch-Anglin effect for matter-wave and optical dark solitons in external potentials

Discovered by Busch and Anglin, the most important and puzzling nonlinear effect for solitons in external potentials lies in the fact that in a harmonic oscillator potential, a dark soliton oscillates with a frequency that is smaller by a factor of 2 in comparison with the ground state oscillations frequency. We review the main features of the Busch-Anglin effect and show that the physical nature of the self-defocusing nonlinearity can radically alter the Busch-Anglin dynamics. To bridge the gap between the matter-wave and optical dark solitons, we consider both the impact of linear and nonlinear absorption and the crucial role of nonlocality (or nonstationarity) of the nonlinear response in the triggering of the Busch-Anglin effect. We conclude that nonlocality (or nonstationarity) brings into existence the self-induced Busch-Anglin effect. Both linear and nonlinear losses cannot radically affect the period of dark solitons oscillations. We establish that from the physical point of view, it is best to redefine the specific dark soliton angle between the asymptotic phases of non-vanishing boundary conditions in accordance with Faddeev and Takhtajan's mathematical arrangement based on the phase shift across a dark soliton “core” controlling the soliton “blackness”. Among other things, we show that the effects analogous to the self-induced Busch-Anglin effect exist for bright solitons as well. Namely, triggered by nonlocal (or time-delayed) self-focusing nonlinearity, the bright solitons oscillate in the parabolic trap with a shifted equilibrium position and a frequency that exactly coincides with the oscillator frequency.