Huang Nian-Ning

Inverse Scattering Transform for the Derivative Nonlinear Schrödinger Equation

Since the Jost solutions of the derivative nonlinear Schrodinger equation do not tend to the free Jost solutions, when the spectral parameter tends to infinity(|λ| → ∞), the usual inverse scattering transform (IST) must be revised. If we take the parameter κ = λ−1 as the basic parameter, the Jost solutions in the limit of |κ| → ∞ do tend to the free Jost solutions, hence the usual procedure to construct the equations of IST in κ-plane remains effective. After we derive the equation of IST in terms of κ, we can obtain the equation of IST in λ-plane by the simple change of parameters λ = κ−1. The procedure to derive the equation of IST is reasonable, and attention is never paid to the function W(x) introduced by the revisions of Kaup and Newell. Therefore, the revision of Kaup and Newell can be avoided.

Marchenko Equation for the Derivative Nonlinear Schrödinger Equation

A simple derivation of the Marchenko equation is given for the derivative nonlinear Schrodinger equation. The kernel of the Marchenko equation is demanded to satisfy the conditions given by the compatibility equations. The soliton solutions to the Marchenko equation are verified. The derivation is not concerned with the revisions of Kaup and Newell.

A SOLITON PERTURBATION THEORY FOR THE UNSTABLE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

Based on the Zakharov-Shabat equation of the inverse scattering transform for the unstable nonlinear Schr?dinger equation, for which a perturbation theory with corrections is developed in this paper. All necessary formulae for calculating the scattering data are derived. Based upon these formulae, the effect due to the corrections can be studied. As an example, the correction due to the damping is calculated.

On Kaup and Newell's Method for Solving DNLS Equation

Kaup and Newells revised inverse scattering transform for the derivative nonlinear Schrodinger (DNLS) equation is investigated. We compared it with a more reasonable approach proposed recently, which is rigorously proven by the Liouville theorem. It is concluded that Kaup and Newells revision is only suitable for giving single-soliton solution and can not be generalized to multi-soliton case.

Demonstration of Inverse Scattering Transform for DNLS Equation

Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as | | ! 1, the usual inverse scattering transform (IST) must be revised. Beside the Kaup and Newells approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying 2 or 1 . To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.

Gauge Transformation and Conservation Laws of Landau–Lifschitz Equation for a Spin Chain

Introducing an explicit expression of gauge transformation turning the spin in linear order of spectral parameter in the first Lax pair into the third axis in spin space, we derive the required Hamiltonian densities for the spin chain with the easy axis, easy plane, and full anisotropy, respectively.

Conservation Laws in sine-Gordon Equation

There are two conservation quantities needed to derive conservation laws of sine-Gordon equation in a frame of laboratory reference. One of them could be derived easily with a standard process, but the other could not. After the gauge transformation e-iθσ2/2 is introduced, we simply obtain the other one without any questionable assumption.

Green's Function Method for Perturbed Korteweg-de Vries Equation

The x-derivatives of squared Jost solution are the eigenfunctions with the zero eigenvalue of the linearized equation derived from the perturbed Korteweg-de Vries equation. A method similar to Greens function formalism is introduced to show the completeness of the squared Jost solutions in multi-soliton cases. It is not related to Lax equations directly, and thus it is beneficial to deal with the nonlinear equations with complicated Lax pair.

Darboux transformation method for solving the sine-gordon equation in a laboratory reference

The sine-Gordon equation is solved in a laboratory reference using the method of Darboux transformation. Using the Liouville theorem, explicit expressions of the single soliton solution and the breather solution are derived from the Darboux matrix in the case of a null spectral parameter.

A direct perturbation theory of the nonlinear Schrödinger equation with corrections

An exact direct perturbation theory of nonlinear Schrodinger equation with corrections is developed under the condition that the initial value of the perturbed solution is equal to the value of an exact multisoliton solution at a particular time. After showing the squared Jost functions are the eigenfunctions of the linearized operator with a vanishing eigenvalue, suitable definitions of adjoint functions and inner product are introduced. Orthogonal relations are derived and the expansion of the unity in terms of the squared Jost functions is naturally implied. The completeness of the squared Jost functions is shown by the generalized Marchenko equation. As an example, the evolution of a Raman loss compensated soliton in an optical fiber is treated.