Initial-Boundary Value Problem for Two-Component Gerdjikov–Ivanov Equation with 3 × 3 Lax Pair on Half-Line

The Fokas unified method is used to analyze the initial-boundary value problem of two-component Gerdjikov–Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation.     

An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa–Holm equation is integrable by the inverse spectral transform in the multi-peakon case.     

Estimates for the KdV-Limit of the Camassa–Holm Equation

In a number of scaling limits, we prove estimates relating the solutions of the Camassa–Holm equation to the solutions of the associated KdV equation. As a consequence, suitable solutions of the water wave problem and solutions of the Camassa–Holm equation stay close together for long times.     

The Solutions for the Boundary Layer Problem of Boltzmann Equation in a Half-Space

We study the half space boundary layer problem for Boltzmann equation with cut-off potentials in all the cases −3<γ≤1, while the boundary condition is imposed on the incoming particles of Dirichlet type, and the solution is assumed to approach to a global Maxwellian at the far field. The same as for cut-off hard sphere model, there is an implicit solvability condition on the boundary data which gives the co-dimensions of the boundary data in terms of positive characteristic speeds.     

Conservation Laws for the Derivative Nonlinear Schroedinger Equation with Non-vanishing Boundary Conditions

Conservation laws for the derivative nonlinear Schr6dinger equation with non-vanishing boundary conditions are derived, based on the recently developed inverse scattering transform using the affine parameter technique.     

Local Well-Posedness of Vlasov–Poisson–Boltzmann Equation with Generalized Diffuse Boundary Condition

Journal of Statistical Physics - The Vlasov–Poisson–Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We...     

Suitable Solutions for the Navier–Stokes Problem with an Homogeneous Initial Value

This paper is devoted to the study of strong or weak solutions of the Navier–Stokes equations in the case of an homogeneous initial data. The case of small initial data is discussed. For large initial data, an approximation is developed, in the spirit of a paper of Vishik and Fursikov. Qualitative convergence is obtained by use of the theory of Muckenhoupt weights.     

Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition

This paper is contributed to explore all possible single peakon solutions for the Degasperis–Procesi (DP) equation m _t  + m _x u + 3mu _x  = 0, m = u − u _xx . Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the inhomogeneous boundary condition lim_|x|→ ∞ u =A ≠0, or possesses the regular peakon solutions ce ^{− |x − ct|} ∈ H ¹ (c is the wave speed) only when lim_|x|→ ∞ u = 0 (see Theorem 4.1). In particular, we first time obtain the stationary cuspon solution of the DP equation. Moreover we present new cusp solitons (in the space of ) and smooth soliton solutions in an explicit form. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation.      

Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value

We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation i q t + q xx ? i q 2 q ? x + 1 2 | q | 4 q = 0 $$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0 $$ with step-like initial data q ( x , 0 ) = 0 $q(x,0)=0$ for x ≤ 0 $x \leqslant 0$ and q ( x , 0 ) = A e ? 2 iBx $q(x,0)=A\mathrm {e}^{-2iBx}$ for x > 0 $x>0$ , where A > 0 $A>0$ and B ∈ ? $B\in \mathbb R$ are constants. We show that there are three regions in the half-plane { ( x , t ) | ? ∞ < x < ∞ , t > 0 } $\{(x,t) | -\infty <x<\infty , t>0\}$ , on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x > ? 4 tB $x>-4tB$ , a plane wave region: x > ? 4 t B + 2 A 2 B + A 2 4 $x<-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )$ , an elliptic region: ? 4 t B + 2 A 2 B + A 2 4 > x > ? 4 tB $-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )<x<-4tB$ . Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method.     

Elliptic Linear Problem for Painlevé VI Equation with Spectral Parameter

We introduce a linear problem with a spectral parameter for the elliptic form of the Painlevé VI equation. The corresponding nonautonomous version produces the Lax pair with spectral parameter for the Calogero-Inozemtsev model with a single degree of freedom.     

Geometric Integrability of the Camassa–Holm Equation

It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a Miura transform and a modified Camassa–Holm equation are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.     

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.     

Wave-Breaking and Peakons for a Modified Camassa–Holm Equation

In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.     

Dark Multi-Soliton Solution of the Nonlinear Schrödinger Equation with Non-Vanishing Boundary

The inverse scattering transform for the nonlinear Schrödinger equation in normal dispersion with non-vanishing boundary values is re-examined using an affine parameter to avoid double-valued functions. An operable algebraic procedure is developed to evaluate dark multi-soliton solutions. The dark two-soliton solution is given explicitly as an example, and is verified by direct substitution. The additional motion of the soliton center is given by its asymptotic behavior.     

Condition of Prequantization to Two—Dimensional Manifolds with Boundary

The Weil‘s integrality condition of prequantization is generalized to two-dimensional phase space with boundaries.It is shown that in the prequantization condition a term related to the symplectic potential on the boundary appears.The necessity of the generalized condition is proved by analyzing the isolated singularities of the Hermitian bundle while the sufficiency of the condition is proved via geometric construction on the space of equivalence class.     

Hamiltonian Formalism of mKdV Equation with Non-vanishing Boundary Values

Hamiltonian formalism of the mKdV equation with non-vanishing boundary value is re-examined by a revised form of the standard procedure. It is known that the previous papers did not give the final results and involved some questionable points [T.C. Au Yeung and P.C.W. Fung, J. Phys. A 21 （1988） 3575]. In this note, simple results are obtained in terms of an affine parameter and a Galileo transformation is introduced to ensure the results compatible with those derived from the inverse scattering transform.     

Bethe-Salpeter Equation with Self-Energy Graphs Ⅲ-Numerical Solution and the Abnormal State Problem

Parameter imbedding method for nonlinear parameter integral equation developed in paper I is applied to the BS equation with self-energy graphs discussed in paper 11. Numerical algorithm and computer code are presented. An example problem, which can be treated analytically, is run on the computer VAX-8700 in the USTC, China. The numerical results are excellent. The continuous spectrum and the discrete eigenvalues of the model problem are investigated attentively and calculated on nearly the whole closed complex plane by using the numerical code. Several sets of parameters p (the mass of the exchanged bosons) and μ(the bound energy of the bound state) are given. The numerical results exhibit some strong evidences, which make us come to the suggestion that there are no discrete abnormal states in all the cases we considered when the self-energy is included.     

An Integro-Differential Equation for Bose�CEinstein Condensates

We present an integro-differential equation describing systems with large number of bosons. The new equation includes the two-body correlations exactly into account and the kernel has a simple analytic form. The equation has been employed to obtain results for ${A\in\{10,100\}}$ ⁸⁷Rb atoms confined by an externally applied trapping potential V _trap(r). Our results are in excellent agreement with those of the Potential Harmonic Expansion Method (PHEM) and the Diffusion Monte Carlo (DMC) method.