Transformations between Nonlocal and Local Integrable Equations

Recently, a number of nonlocal integrable equations, such as the ‐symmetric nonlinear Schrödinger (NLS) equation and ‐symmetric Davey–Stewartson equations, were proposed and studied. Here, we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey–Stewartson equations, a nonlocal derivative NLS equation, the reverse space‐time complex‐modified Korteweg–de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their Lax pairs and analytical solutions from those of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially ‐symmetric Davey–Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multisoliton and quasi‐periodic solutions in the reverse space‐time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa–Satsuma equations.     

General soliton and (semi-)rational solutions to the nonlocal Mel'nikov equation on the periodic background

In this paper, the Hirota's bilinear method and Kadomtsev-Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi-)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram-type determinants. When N is even, soliton, line breather and (semi-)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi-rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.     

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

In this paper, the partially party‐time () symmetric nonlocal Davey–Stewartson (DS) equations with respect to x is called x‐nonlocal DS equations, while a fully symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x‐nonlocal DS equations, the usual (2 + 1)‐dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)‐dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.     

Solutions to ABS Lattice Equations via Generalized Cauchy Matrix Approach

The usual Cauchy matrix approach starts from a known plain wave factor vector and known dressed Cauchy matrix . In this paper, we start from a determining matrix equation set with undetermined and . From the determining equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The determining equation set admits more choices for and and in the paper we give explicit formulae for all possible and . As applications, we get more solutions than usual multisoliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation, and some lattice equations in ABS list.     

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

We study here the initial value problem for a two‐dimensional Korteweg–de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp‐discontinuity. Finally, we give conditions under which the evolution is reduced to the classical KdV.     

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

In this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg‐de Vries hierarchy, and nonlocal versions of the sine‐Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two‐soliton solutions of the reverse‐t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.     

Inverse scattering transforms and soliton solutions of nonlocal reverse-space nonlinear Schrödinger hierarchies

The aim of the paper is to construct nonlocal reverse-space nonlinear Schrödinger (NLS) hierarchies through nonlocal group reductions of eigenvalue problems and generate their inverse scattering transforms and soliton solutions. The inverse scattering problems are formulated by Riemann-Hilbert problems which determine generalized matrix Jost eigenfunctions. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A solution formulation to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless transforms, is presented and applied to -soliton solutions of the nonlocal NLS hierarchies.     

Using Symmetries à Rebours

We consider nonclassical symmetries of partial differential equations (PDEs) in dimensions. Given a th‐order ordinary differential equation in the unknown we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to .     

A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems

The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem , where is skew‐symmetric and is self‐adjoint. If has a bounded inverse the index is well established, and it is given by the number of negative eigenvalues of the operator constrained to act on some finite‐codimensional subspace. There is an important class of problems—namely, those of KdV‐type—for which does not have a bounded inverse. In this paper, we overcome this difficulty and derive the index for eigenvalue problems of KdV‐type. We use the index to discuss the spectral stability of homoclinic traveling waves for KdV‐like problems and Benjamin—Bona—Mahony‐type problems.     

Stable blow-up dynamics in the -critical and -supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities.     

Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background satisfies a “small norm” condition . On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally stable for , becomes unstable if , and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with , analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view. We formulate the direct and inverse problems by using a suitable uniformization variable, and pose the inverse problem as an RHP across a simple contour in the complex plane of the uniform variable. As a by‐product of the IST, we also obtain explicit soliton solutions, which are the discrete analog of the celebrated Kuznetsov–Ma, Akhmediev, Peregrine solutions, and which mimic the corresponding solutions for the focusing AL equation. Soliton solutions that are the analog of the dark soliton solutions of the defocusing AL equation in the case are also presented.     

Solutions with concentration for conservation laws with discontinuous flux and its applications to numerical schemes for hyperbolic systems

Measure-valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux-based numerical schemes for the class of hyperbolic systems that admit nonclassical -shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient. The article also discusses the concentration phenomenon of solutions along the line of discontinuity, for scalar transport equations with a discontinuous coefficient. The existence of the solutions for transport equation is shown using the vanishing viscosity approach and the asymptotic behavior of the solutions is also established. The performance of the numerical schemes for both scalar conservation laws and systems to capture the -shocks effectively is displayed through various numerical experiments.     

Invariant discrete flows

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.     

Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as are found. These asymptotics are parameterized by monodromy data of the associated linear ODE, The parameterization allows one to derive connection formulae for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulae are also considered.     

The short pulse equation: Bäcklund transformations and applications

A Bäcklund transformation (BT), which involves both independent and dependent variables, is established and studied for the short pulse (SP) equation. Based it, the nonlinear superposition formulae for 2-, 3-, and 4-BT are presented. The general result for the composition of -BTs is achieved and given in terms of determinants. As applications, various solutions including loop solitons, breather solutions, and their interaction solutions are worked out.     

Localized Analytical Solutions and Parameters Analysis in the Nonlinear Dispersive Gross–Pitaevskii Mean‐Field GP (m,n) Model with Space‐Modulated Nonlinearity and Potential

The novel nonlinear dispersive Gross–Pitaevskii (GP) mean‐field model with the space‐modulated nonlinearity and potential (called GP equation) is investigated in this paper. By using self‐similar transformations and some powerful methods, we obtain some families of novel envelope compacton‐like solutions spikon‐like solutions to the GP equation. These solutions possess abundant localized structures because of infinite choices of the self‐similar function . In particular, we choose as the Jacobi amplitude function and the combination of linear and trigonometric functions of space x so that the novel localized structures of the GP(2, 2) equation are illustrated, which are much different from the usual compacton and spikon solutions reported. Moreover, it is shown that GP(m, 1) equation with linear dispersion also admits the compacton‐like solutions for the case and spikon‐like solutions for the case .     

A Boundary Value Problem for Conjugate Conductivity Equations

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half‐plane with conductivity , . The representations are obtained via the so‐called unified transform method or Fokas method, involving a Riemann–Hilbert problem on the complex plane when p is even and on a two‐sheeted Riemann surface when p is odd. They are given in terms of the Dirichlet and Neumann data on the boundary of the domain. For even exponent p, we also show how to make the conversion from one type of conditions to the other by using the global relation that follows from the closedness of some differential form. The method used to derive our integral representations could be applied in any bounded simply connected domain of the right half‐plane with a smooth boundary.     

Well‐posedness,positivity, and time asymptotics properties for a reaction–diffusion model of plankton communities,involving a rational nonlinearity with singularity

In this work, we consider a reaction–diffusion system, modeling the interaction between nutrients, phytoplankton, and zooplankton. Using a semigroup approach in , we prove global existence, uniqueness, and positivity of the solutions. The nonlinearity is handled by providing estimates in , allowing to deal with most of the functional responses that describe predator/prey interactions (Holling I, II, III, Ivlev) in ecology. The paper finally exhibits some time asymptotic properties of the solutions.