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Dressing for a novel integrable generalization of the nonlinear Schr\"odinger equation
Jonatan Lenells
TL;DR
The paper addresses how to construct soliton solutions for an integrable generalization of NLS. It implements the dressing method to obtain explicit N-soliton formulas, and also simplifies the DNLS N-soliton formulas of Huang and Chen.
Problem
The paper studies an integrable generalization of NLS whose analysis differs from NLS and whose spectral analysis is closely related to DNLS.
Source: §1 · p. 1Method
The authors implement the dressing method for the generalized NLS equation, starting from a seed solution and its Lax-pair eigenfunction.
Source: §1 · pp. 1–2Results
The method gives explicit N-soliton formulas for the generalized NLS equation and simplifies Huang and Chen’s DNLS N-soliton formulas.
Source: §1 · p. 2, §4 · p. 10, §5 · p. 11Takeaways & Limitations
The link between the generalized NLS equation and DNLS allows a simplification of the DNLS soliton formulas.
Source: §6 · p. 11The construction assumes sufficient smoothness and decay of u, and normalizes nonzero γ, ν, and σ to 1 by a change of variables.
Source: §2 · p. 2
Abstract
from arXiv · showhide
We implement the dressing method for a novel integrable generalization of the nonlinear Schr\"odinger equation. As an application, explicit formulas for the $N$-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the $N$-solitons of the derivative nonlinear Schr\"odinger equation given by Huang and Chen.
1. Introduction
The paper studies an integrable generalization of NLS whose analysis differs from NLS in important respects and is related to the DNLS hierarchy. It applies the dressing method to derive explicit N-soliton formulas and simplify formulas for DNLS solitons.
1. Introduction
- Equation (1.1) models nonlinear pulse propagation in optical fibers when certain next-order terms beyond those needed for NLS are retained.Its ν → 0 limit is singular in many respects, and its spectral analysis is closely related to DNLS.Source: §1 · p. 1
- The equation is gauge-related to the first negative member of the DNLS hierarchy, and its x-part Lax pair matches that of equation (1.2) when u_x is identified with q.The equation also admits an inverse scattering analysis for the initial-value problem on the line.Source: §1 · pp. 1–2
- The dressing method constructs a new eigenfunction by multiplying a seed eigenfunction by a 2 × 2 matrix with an even number of spectral poles, producing a new potential that solves (1.1).Lax-pair compatibility ensures the new potential is a solution.Source: §1 · pp. 1–2
- Starting from the zero seed, the method recovers the known one-soliton solution and gives an explicit N-soliton formula for any N.The earlier one-soliton expression was obtained by solving the associated Riemann-Hilbert problem with two poles and no jump.Source: §1 · p. 2
- The paper also simplifies Huang and Chen’s formula for DNLS N-solitons.It introduces a convenient Lax pair, details the dressing implementation, and derives explicit N-soliton expressions.Source: §1 · p. 2
2. A Lax pair
The paper normalizes the nonzero parameters of equation (1.1) to one and constructs a Lax-pair eigenfunction with specified symmetries. Its analytic construction uses Volterra integral equations, while its jump and endpoint singularities do not obstruct the arguments that follow.
2. A Lax pair
- A change of variables permits the nonzero parameters γ, ν, and σ to be set to 1 for the analysis.The transformation u → e^ixu is then applied to equation (1.1).Source: §2 · p. 2
- The matrix-valued eigenfunction ψ has symmetries under ζ → −ζ and complex conjugation, and is normalized to exp(−iθσ3) at infinity when u has sufficient smoothness and decay.The spectral parameter ζ ranges over the extended complex plane.Source: §2 · p. 2
- Two solutions Ψ1 and Ψ2 are defined through linear Volterra integral equations, whose columns have analyticity in complementary regions determined by ζ².The relevant columns extend continuously to the real and imaginary axes, excluding zero and infinity.Source: §2 · p. 3
- Although ψ has singularities at ζ = 0 and ∞ and a jump across R ∪ iR, these features are inconsequential for arguments using ψxψ−1 and ψtψ−1.The jump matrix is independent of x and t; Liouville’s theorem prevents ψ from generally being analytic over the whole Riemann sphere.Source: §2 · p. 3
3. The dressing method
The dressing method transforms a seed solution and symmetry-compatible eigenfunction into a new solution by constructing a dressed eigenfunction. Pole analysis establishes the required Lax pair, while successive dressing steps determine the residue data and preserve analyticity.
3. The dressing method.
- A rational dressing factor is built from a seed solution and compatible eigenfunction using simple poles and residues chosen to preserve the eigenfunction symmetries.The factor is normalized to approach the identity as ζ tends to infinity and is analytic with its inverse at zero and infinity.Source: §3 · pp. 3–4
3.1. Analysis at ζ = 0.
- Near ζ=0, the dressed x- and t-derivative products have double poles, and the coefficient Q−1 is off-diagonal.The pole expansions identify coefficients at orders O(1/ζ²) and O(1/ζ).Source: §3.1 · pp. 4–5
3.2. Analysis at ζ = ∞.
- As ζ tends to infinity, ψxψ−1 tends to zero while ψtψ−1 has a double pole; the coefficient expressions simplify using the dressing-factor expansions.The expansions identify terms of orders O(ζ^n), for n=0, 1, 2.Source: §3.2 · p. 5
3.3. The generated solution.
- The dressed eigenfunction yields a Lax pair whose compatibility makes u=U12 solve (2.1), with u obtained by adding residue entries to the seed u0.The generated field is given by the (1,2) entries of the residues plus u0.Source: §3.3 · p. 6
3.4. The dressing transformation.
- N successive dressing steps add two poles apiece and produce eigenfunctions whose x- and t-derivative products are analytic at the specified pole points.The analytic regularity is established for the points ±ζj and ±¯ζj.Source: §3.4 · pp. 6–7
4. Solitons
Applying the dressing method to the zero solution produces explicit N-soliton formulas for the generalized NLS equation. The solution is expressed using an N×N matrix and depends on 2N complex parameters.
4. Solitons
- For the zero seed u0=0, adding 2N poles yields the N-soliton solution, given explicitly in Proposition 4.1.The algebraic system simplifies for this seed, enabling explicit soliton formulas.Source: §4 · pp. 9–10
- The N × N matrix K = K(x, t) is defined by (4.3), and fj := fj(x, t) = bjeiθ(x,t,ζj) for j = 1, . . . , N.Source: §4 · pp. 9–10
- The N-soliton solution depends on the 2N complex parameters {bj, ζj}.The seed eigenfunction used to derive the formula is ψ0=exp(−iθσ3).Source: §4 · pp. 9–10
5. DNLS solitons
The DNLS N-soliton solution is expressed explicitly in terms of the matrix K and coefficients used in the generalized NLS construction. The derivation links the two formulas through their Lax pairs and an x-derivative relation.
5. DNLS solitons
- Identifying Q with Ux makes the Lax-pair x-parts coincide; this motivates expecting the DNLS N-soliton qN to have an expression similar to the right-hand side of (5.3), as discussed in.Source: §5 · p. 10
- The DNLS expression is the x-derivative of the generalized NLS expression, a relation that persists with the DNLS-specific coefficients because the time dependence does not affect x-differentiation.Source: §5 · pp. 10–11
- Proposition 5.1 gives an explicit DNLS N-soliton formula using K from (4.3), coefficients (5.4), and 2N complex parameters.Source: §5 · p. 11
- The authors report explicit checks for small N confirming that the generalized NLS and DNLS solitons satisfy their respective equations.Source: §5 · p. 11
6. Conclusions
The dressing method yields an explicit generalized NLS N-soliton solution from the zero seed, while the linked DNLS formulas admit a simplification. The generalized NLS construction is more complicated because of additional singularities in the time part of its Lax pair.
6. Conclusions
- Starting from u0 ≡ 0, the dressing construction gives the explicit generalized NLS N-soliton expression (4.4).Source: §6 · p. 11
- Additional singularities in the time part of the Lax pair make the dressing argument for the generalized NLS equation more complicated.Source: §6 · p. 11
- The link between the generalized NLS and DNLS formulas yields a simplification of the DNLS N-solitons in, whose right-hand side is an x-derivative.Source: §6 · p. 11
- The DNLS hierarchy’s potential formulation is associated with solitons expressible as x-derivatives of elementary functions.Source: §6 · p. 11