# Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation (AM-154)

Spyridon Kamvissis
Kenneth D. T-R McLaughlin
Peter D. Miller
Pages: 304
https://www.jstor.org/stable/j.ctt2jc946

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. List of Figures and Tables
(pp. ix-x)
4. Preface
(pp. xi-xiv)
5. Chapter One Introduction and Overview
(pp. 1-12)

The initial-value problem for the focusing nonlinear Schrödinger equation is

$i\hbar\partial_{t}\psi+\frac{\hbar^{2}}{2}\partial_{x}^{2}\psi+|\psi|^{2} \psi=0,\quad \psi(x,0)=\psi_{0}(x).\quad\caption{(1.1)}$

We are interested in studying the behavior of solutions of this initial-value problem in the so-called semiclassical limit. To make this precise, the initial data is given in the form

$\psi_{0}(x)=A(x)\,\exp(iS(x)/\hbar),\quad\caption{(1.2)}$

whereA(x) is a positive real amplitude function that is rapidly decreasing for large |x| and whereS(x) is a real phase function that decays rapidly to constant values for large |x|. Studying the semiclassical limit means: fix once and for all the functionsA(x) andS(x), and then for each sufficiently small value of$\hbar > 0$, solve...

6. Chapter Two Holomorphic Riemann-Hilbert Problems for Solitons
(pp. 13-22)

The initial-value problem (1.1) is solvable for arbitrary$\hbar$because the focusing nonlinear Schrödinger equation can be represented as the compatibility condition for two systems of linear ordinary differential equations:

$\hbar\partial_{x}\begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix}=\begin{bmatrix}-i\lambda & \psi\\ -\psi^{\ast} & i\lambda\end{bmatrix}\begin{bmatrix}u_{1} \\ u_{2} \end{bmatrix},\quad\caption{(2.1)}$

$i\hbar\partial_{t}\begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix}=\begin{bmatrix}\lambda^{2}-|\psi|^{2}/2 & i\lambda\psi-\hbar\partial_{x}\psi/2\\ -i\lambda\psi^{\ast}-\hbar\partial_{x}\psi^{\ast}/2 & -\lambda^{2}+|\psi|^{2}/2\end{bmatrix}\begin{bmatrix}u_{1} \\ u_{2} \end{bmatrix},\quad\caption{(2.2)}$

whereλis an arbitrary complex parameter. The compatibility condition for (2.1) and (2.2) does not depend on the value ofλand is equivalent to the nonlinear Schrödinger equation.

TheN-soliton solutions of the nonlinear Schrödinger equation are those complex functionsψ(x,t) for which there exist simultaneous column vector solutions of (2.1) and (2.2) of the particularly simple form

$\begin{matrix} \textrm{u}^{+}(x,t,\lambda)=\begin{bmatrix}\sum_{p=0}^{N-1}A_{p}(x,t)\lambda^{p} \\ \lambda^{N}+\sum_{p=0}^{N-1}B_{p}(x,t)\lambda^{p}\end{bmatrix}\exp(i(\lambda x+\lambda^{2}t)/\hbar), \\ \textrm{u}^{-}(x,t,\lambda)=\begin{bmatrix}\lambda^{N}+\sum_{p=0}^{N-1}C_{p}(x,t)\lambda^{p} \\ \sum_{p=0}^{N-1}D_{p}(x,t)\lambda^{p}\end{bmatrix}\exp(-i(\lambda x+\lambda^{2}t)/\hbar), \end{matrix}\quad \caption{(2.3)}$

satisfying the relations

$\begin{matrix}\textrm{u}^{+}(x,t,\lambda_{k})=\gamma_{k}\textrm{u}^{-}(x,t,\lambda_{k}), \\ -\gamma_{k}^{\ast}\textrm{u}^{+} (x,t,\lambda_{k}^{\ast})=\textrm{u}^{-}(x,t,\lambda_{k}^{\ast}),\quad k=1,\ldots,N, \\ \end{matrix}\quad\caption{(2.4)}$

for...

7. Chapter Three Semiclassical Soliton Ensembles
(pp. 23-36)

In this book, we present a technique for studying the behavior, near fixedxandt, of multisoliton solutions for which the number of solitonsNis large but for which the solitons are highlyphase-correlated. This means that for largeNthe discrete measureconverges and at the same time the interpolantsXK(λ) converge and take on a simple limiting form.

The highly correlated situation mentioned above arises naturally when one considers the semiclassical limit of the sequence of initial-value problems (1.1) for initial data of the form (1.2). Becauseħis present explicitly in the scattering problem...

8. Chapter Four Asymptotic Analysis of the Inverse Problem
(pp. 37-120)

In this chapter, we study the asymptotic behavior, in the limit ofNtending to infinity, of semiclassical soliton ensembles corresponding to analytic, even, bell-shaped functionsA(x) with nonzero curvature at the peak and sufficient decay for largex, so that the densityρ⁰(η) defined by (3.1) has an analytic extension from the imaginary interval (0,iA). This means that we are going to study Riemann-Hilbert Problem 2.0.2 for the matrix M(λ) posed with discrete data$\{\lambda_{\hbar_{N},0}^{\textrm{WKB}},\ldots,\lambda_{\hbar_{N},N-1}^{\textrm{WKB}}\}$and$\{\gamma_{\hbar_{N},0}^{\textrm{WKB}},\ldots,\gamma_{\hbar_{N},N-1}^{\textrm{WKB}}\}$defined in terms ofA(x) by Definition 3.1.1, in the limit asNtends to infinity.

In our analysis, we would...

9. Chapter Five Direct Construction of the Complex Phase
(pp. 121-162)

In this chapter, we turn to the question of reducing Riemann-Hilbert Problem 2.0.2 for the matrix M(λ) ultimately to the simple form of the outer model Riemann-Hilbert Problem 4.2.3 for the matrix Õ(λ), which was solved exactly in §4.3. Achieving the reduction required finding an appropriate complex phase function$g^{\sigma}(\lambda)$on an appropriate contourCC*. We describe here a construction of good “candidate” complex phase functions. It is then often possible to prove directly that an appropriate candidate actually satisfies all of the criteria (cf. Definition 4.2.5) required for reducing the Riemann-Hilbert problem to an analytically tractable form,...

10. Chapter Six The Genus-Zero Ansatz
(pp. 163-194)

ForG= 0, there is only one complex endpoint to determine,λ₀. This endpoint is constrained by one moment condition and one measure reality condition. Both conditions are real and, taken together, are expected to determine the endpoint up to a discrete multiplicity of solutions. The equations that constrain the endpoint forG= 0 are

$M_{0}=J\int_{I_{0}}\frac{2ix+4i\eta t}{R_{+}(\eta)}d\eta+\int_{\Gamma_{1}\cap C_{1}}\frac{\pi i\rho^{0}(\eta)}{R(\eta)}d\eta+\int_{\Gamma_{1}\cap C_{1}^{\ast}}\frac{\pi i\rho^{0}(n^{\ast})^{\ast}}{R(\eta)}d\eta=0\quad\caption{(6.1)}$

and

$R_{0}=\Im \left ( \int_{0}^{\lambda_{0}}\rho^{\sigma}(\eta)d\eta \right )=0.\quad\caption{(6.2)}$

In the measure reality conditionR₀ = 0, we use the formula (5.34) for the candidate measure$\rho^{\sigma}(\eta)$valid in the band$I_{0}^{+}$:

$\rho^{\sigma}(\lambda)=\rho^{0}(\lambda)-\frac{4Jt}{\pi}R_{+}(\lambda)+\frac{R_{+}(\lambda)}{\pi i}\int_{\Gamma_{I}\cap C_{I}}\frac{\rho^{0}(\eta)d\eta}{(\lambda-\eta)R(\eta)}+\frac{R_{+}(\lambda)}{\pi i}\int_{\Gamma_{I}\cap C_{I}^{\ast}}\frac{\rho^{0}(\eta^{\ast})^{\ast}d\eta}{(\lambda-\eta)R(\eta)}.\quad\caption{(6.3)}$

In these formulas,$I_{0}=I_{0}^{+} \cup I_{0}^{-}$is the unknown band connecting$\lambda_{0}^{\ast}$in the lower half-plane to...

11. Chapter Seven The Transition to Genus Two
(pp. 195-214)

Recall that in §6.2.2 it was shown that for each fixedx≠ 0 there exists some choice of the parametersσandJsuch that theG= 0 ansatz holds for |t| sufficiently small. Furthermore, it was shown in §5.2 that if the pair (x₀,t₀) is such that theG= 0 ansatz holds and the endpoint functions are differentiable, then there is a small neighborhood of (x₀,t₀) on which theG= 0 ansatz holds as well, and this allows us to define a region of the (x,t)-plane containing (x₀,t₀) throughout which the...

12. Chapter Eight Variational Theory of the Complex Phase
(pp. 215-222)

Apart from relying heavily on the analyticity of the functionρ⁰(η) characterizing the asymptotic density of eigenvalues on the imaginary interval [0,iA] by the WKB formula (3.2), in the direct construction of the complex phase functiongσ(λ) presented in chapter 5 there was no way to determine a priori the value of the genusGfor which a successful ansatz could be constructed for given values ofxandt, nor indeed whether such a finiteGexists at all. To begin to address these issues, we need to reformulate the conditions for an admissible density functionρσ(η)...

13. Chapter Nine Conclusion and Outlook
(pp. 223-228)

The generalized steepest-descent scheme we have described in detail for analyzing the semiclassical limit of the initial-value problem (1.1) for the focusing nonlinear Schrödinger equation provides what we believe to be the first rigorous result of its kind: that solutions of a sequence of well-posed problems (i.e., the initial-value problem (1.1) with WKB-modified initial data corresponding to true data of the formψ₀(x) =A(x) for the sequenceħ=ħN) converge to an object whose macroscopic properties (weak limits of conserved local densities) are described by a system of elliptic modulation equations, whose initial-value problem is significantly less well...

14. Appendix A. Hölder Theory of Local Riemann-Hilbert Problems
(pp. 229-252)
15. Appendix B. Near-Identity Riemann-Hilbert Problems in L²
(pp. 253-254)
16. Bibliography
(pp. 255-258)
17. Index
(pp. 259-265)