In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a *q*-difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (*q*P_{I}), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain *q*-domain. The method, while demonstrated for *q*P_{I}, is also applicable to other *q*-difference Painlevé equations.

We examine the variable-coefficient Kortweg-de Vries equation for the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here, we examine the same case but for a modulated periodic wave train. Using an asymptotic analysis, we show that in contrast a periodic wave is preserved with a finite amplitude as it passes through the critical point, but a phase change is generated causing the wave to reverse its polarity.

]]>We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long-wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small-amplitude parameter. We extend the result to related, nonlinear dispersive equations.

]]>This work is concerned with the initial-boundary value problem for the Boussinesq equation. By employing the unified transform method of Fokas, novel solution formulae for the linearized “good” Boussinesq equation on the half-line with various initial and boundary conditions are obtained. Moreover, these solution formulae are numerically illustrated in the case of concrete data. Finally, Boussinesq's original physical derivation of the so-called “bad” Boussinesq equation is provided.

]]>In this paper, based on matrix and curve integration theory, we theoretically show the existence of Cartesian vector solutions for the general *N*-dimensional compressible Euler equations. Such solutions are global and can be explicitly expressed by an appropriate formulae. One merit of this approach is to transform analytically solving the Euler equations into algebraically constructing an appropriate matrix . Once the required matrix is chosen, the solution is directly obtained. Especially, we find an important solvable relation between the dimension of equations and pressure parameter, which avoid additional independent constraints on the dimension *N* in existing literatures. Special cases of our results also include some interesting conclusions: (1) If the velocity field is a linear transformation on , then the pressure *p* is a relevant quadratic form. (2) The compressible Euler equations admit the Cartesian solutions if is an antisymmetric matrix. (3) The pressure *p* possesses radial symmetric form if is an antisymmetrically orthogonal matrix.

The Landau–Lifshitz equation is analyzed via the inverse scattering method. First, we give the well-posedness theory for Landau–Lifshitz equation with the frame of inverse scattering method. The generalized Darboux transformation is rigorous considered in the frame of inverse scattering transformation. Finally, we give the high-order soliton solution formula of Landau–Lifshitz equation and vortex filament equation.

]]>We consider the Caudrey–Beals–Coifman linear problem (CBC problem) and the theory of the Recursion Operators (Generating Operators) related to it in the presence of -reductions of Mikhailov type defined by Coxeter automorphism of the underlying algebra. We discuss mainly the spectral aspects of the theory of the Recursion Operators related to the expansions over the adjoint solutions of the CBC problem but pay attention also to the algebraic issues related to the recursion relations through which the hierarchies of the nonlinear evolution equations associated with the CBC auxiliary problem are obtained.

]]>In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo-cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first-order derivatives of the new singular solitary wave and periodic waves exist, their second-order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo-cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo-cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.

]]>The linear stability of the solitary waves for the one-dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.

]]>This paper presents a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters. A regular perturbation procedure typically provides asymptotic solutions valid for bounded *t* values as a positive parameter ε tends to zero. One can eliminate secular terms by introducing a slowly-varying amplitude obtained as a solution of an amplitude equation on intervals where is bounded. With sufficient stability hypotheses, the results may even hold for all . These ideas are illustrated for a number of nontrivial problems involving ordinary differential equations.

We study optical spectral singularities of a weakly nonlinear -symmetric bilinear planar slab of optically active material. In particular, we derive the lasing threshold condition and calculate the laser output intensity. These reveal the following unexpected features of the system: (1) for the case that the real part of the refractive index η of the layers are equal to unity, the presence of the lossy layer decreases the threshold gain; (2) for the more commonly encountered situations when is much larger than the magnitude of the imaginary part of the refractive index, the threshold gain coefficient is a function of η that has a local minimum. The latter is in sharp contrast to the threshold gain coefficient of a homogeneous slab of gain material which is a decreasing function of η. We use these results to comment on the effect of nonlinearity on the prospects of using this system as a coherence perfect absorption-laser.

]]>Solitons in one-dimensional parity-time ()-symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex-valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch-band edge for either self-focusing or self-defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of -symmetric multisoliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.

]]>For the stationary Gross–Pitaevskii equation with harmonic real and linear imaginary potentials in the space of one dimension, we study the ground state in the limit of large densities (large chemical potentials), where the solution degenerates into a compact Thomas–Fermi approximation. We prove that the Thomas–Fermi approximation can be constructed by using the unstable manifold theorem for a planar dynamical system. To justify the Thomas–Fermi approximation, the existence problem can be reduced to the Painlevé-II equation, which admits a unique global Hastings–McLeod solution. We illustrate numerically that an iterative approach to solving the existence problem converges but give no analytical proof of this result. Generalizations are discussed for the stationary Gross–Pitaevskii equation with harmonic real and localized imaginary potentials.

]]>We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e., the ones describing self-action and nonlinear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.

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