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Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrodinger equation
Zi-Xiang Zhou
TL;DR
The paper addresses the construction of solutions for a nonlocal derivative nonlinear Schrödinger equation, whose Darboux-generated solutions may be singular. It develops Darboux transformations of degrees one, two, and 2n, and obtains globally defined bounded solutions from the zero seed through suitable eigenvalues and parameters. The result is subject to parameter conditions, and the paper notes the absence of bounded exponential seed solutions for this equation.
Problem
Darboux-generated solutions of the nonlocal derivative nonlinear Schrödinger equation may have singularities, motivating conditions for obtaining global bounded solutions.
Method
The paper constructs degree-one, degree-two, and degree-2n Darboux transformations and applies them to the zero seed using eigenvalues and eigenfunction-ratio parameters.
Results
For μj = aj e^(πi/4) with distinct positive aj and sufficiently small |cj|, the degree-2n solution from q = 0 is globally defined and bounded on R2.
Takeaways & Limitations
Suitable spectral arguments and ratio parameters provide a route to global bounded solutions from the zero seed for arbitrary degree 2n.
Takeaways & Limitations
The paper states that the nonlocal equation has no bounded exponential seed solution of the type available for related equations and that derived solutions may generally be singular.
Abstract
from arXiv · showhide
A nonlocal derivative nonlinear Schrodinger equation is introduced. By constructing its basic Darboux transformations of degrees one and two, the explicit expressions of new solutions are derived from seed solutions by Darboux transformation of degree 2n. Usually the derived solutions of this nonlocal equation may have singularities. However, by suitable choice of eigenvalues and the parameters describing the ratio of the two entries of the solutions of the Lax pair, global bounded solutions of the nonlocal derivative nonlinear Schrodinger equation are obtained from zero seed solution by a Darboux transformation of degree 2n.
1 Introduction
The paper introduces a nonlocal derivative nonlinear Schrödinger equation obtained by replacing the local conjugate interaction with a spatially reversed one. It situates this equation within prior work on nonlocal integrable systems and outlines Darboux-based constructions of explicit and globally bounded solutions.
- The derivative nonlinear Schrödinger equation describes Alfvén waves in plasma physics and is an important integrable equation.
- The proposed nonlocal equation replaces q*(x,t) in the nonlinear term with q*(-x,t), while retaining ε = ±1.
- Unlike the usual derivative nonlinear Schrödinger equation, the proposed equation has a real coefficient for its nonlinear term.
- The paper develops Lax-pair symmetries and Darboux transformations of degrees one, two, and 2n to derive explicit new solutions.
- Because derived solutions can be singular, the paper seeks globally defined and bounded solutions through suitable eigenvalues and parameters.
2 Lax pair and its symmetries
The section formulates the Lax representation for the nonlocal derivative nonlinear Schrödinger equation and derives the reductions and symmetries governing its spectral solutions.
- The Lax compatibility condition Ut − Vx + [U,V] = 0 produces the evolution equations for q and r.
- The notation f̄(x,t) = f(−x,t) encodes spatial reflection, with differentiation introducing the corresponding sign change.
- Imposing r = −εq̄* reduces the two-field Lax system to the nonlocal derivative nonlinear Schrödinger equation.
- Under this reduction, the Lax coefficients obey involutive transformations involving λ → −λ and complex-conjugate spatial reversal.
- These coefficient symmetries map a Lax-pair solution at λ = μ to solutions at −μ, μ*, and −μ*.
3 Darboux transformation of degree one
The degree-one Darboux construction begins with the unreduced Lax system and is then constrained by the nonlocal reduction. The resulting transformation produces a new solution when the spectral parameter and eigenfunction ratio satisfy the stated conditions.
- 3.1 Darboux transformation for unreduced system: A degree-one Darboux matrix has the form G = R(λ − S), with R diagonal and RS constant.
- 3.1 Darboux transformation for unreduced system: The unreduced transformation uses σ = η/ξ, the ratio of the two eigenfunction components, to construct the transformed potential.
- 3.2 Darboux transformation for nonlocal derivative nonlinear Schrödinger equation: After imposing r = −εq̄*, a degree-one transformation requires μ real for ε = 1 or purely imaginary for ε = −1.
- 3.2 Darboux transformation for nonlocal derivative nonlinear Schrödinger equation: The ratio σ evolves through coupled x- and t-equations, and σ̄σ* = 1 is preserved when it holds at one point.
- 3.2 Darboux transformation for nonlocal derivative nonlinear Schrödinger equation: Under these conditions, Theorem 1 states that the degree-one Darboux formula yields a new solution of the nonlocal equation.
4 Darboux transformation of degree two
The degree-two construction incorporates the full eigenvalue quartet required by the nonlocal reduction when μ is neither real nor purely imaginary. Its symmetry-compatible Darboux matrix yields a new solution of the equation.
- The degree-two construction applies successive degree-one transformations to eigenfunctions associated with μ² and −μ².
- When μ² is not real, the reduction requires the four eigenvalues μ, −μ, μ*, and −μ*, making a degree-two Darboux matrix necessary.
- The eigenvalue and eigenfunction data are paired as μ1 = μ, μ2 = μ* with σ1 = σ and σ2 = ε/σ̄*.
- The resulting Darboux matrix satisfies reductions compatible with the Lax-pair symmetries.
- For nonzero complex μ that is neither real nor purely imaginary, Theorem 2 gives a new solution using σ = η/ξ.
5 Darboux transformation of degree 2n
The paper constructs degree-2n Darboux transformations that preserve the nonlocal reduction by pairing appropriately chosen eigenvalues and Lax-pair solutions. The resulting polynomial Darboux matrix yields explicit transformed solutions.
- Degree-two Darboux transformations are the lowest-degree transformations preserving all reductions when µ^2 is not real.
- For each µ_j with µ_j^2 nonreal, the construction pairs eigenvalues µ_j and −µ_j with Lax-pair vectors (ξ_j,η_j)^T and (ξ_j,−η_j)^T.
- The resulting Darboux matrix is a polynomial in λ of degree 2n and has a constant scalar value at λ=0.
- The degree-2n matrix can be viewed as a composition of n degree-two Darboux matrices and therefore preserves the reduction r = −ε¯q∗.
- The transformation of q is obtained from the coefficient of λ^(2n+1) in the Darboux compatibility relation.
6 Examples
The examples use degree-two, degree-four, and degree-eight Darboux transformations to construct solutions from the zero seed. Under a suitable parameter choice, the degree-two solution is global and periodic.
- Solutions from zero seed are unbounded or singular for even or noninteger values of 4 arg µ/π, so the examples use odd integer values.
- The degree-two example uses µ = a(1 + i), with a a positive constant, and introduces the ratio parameter c = b/a.
- The degree-two solution is global when |c| ≠ 1, periodic in both x and t, and has a norm depending only on x.
- Figure 1 plots the norm for ε = 1, µ = 1.5(1 + i), and c = 0.5, with the right panel showing its contour plot.
- Norm figures for degree-four and degree-eight Darboux transformations are plotted in Figures 2 and 3 using specified parameter sets.
7 Globalness of the solutions
The paper proves that suitable small parameter choices produce globally defined and bounded solutions from the zero seed, while higher-degree Darboux solutions can otherwise be singular. The proof establishes invertibility of the matrices entering the solution formula and uses this to obtain boundedness.
- Globalness conditions: Higher-degree Darboux solutions may be singular, but they have no singularities when all parameter magnitudes |c_j| are sufficiently small.Degree-two solutions are always global when |c| ≠ 1.
- Global bounded solutions: For distinct positive a_j with μ_j = a_j e^{πi/4}, degree-2n transformations of the zero seed yield globally defined and bounded solutions when |c_j| < δ.A positive constant δ exists, and the bound holds for all (x,t) ∈ R^2.
- Invertibility argument: The proof reduces globalness to showing that the matrices Γ and F remain invertible for all (x,t) when the |c_j| are small.Γ converges uniformly to its zero-parameter limit, while F likewise remains invertible for sufficiently small parameters.
- Boundedness: Boundedness of the transformed solution follows from the solution formula, bounded inverse matrices, and bounded σ_j.The argument concludes after establishing invertibility and boundedness of the relevant factors.
- Parameter regimes: The same global-solution theorem also holds when all |c_j| are sufficiently large, by using σ_j^{-1} instead of σ_j.This is stated as an equivalent parameterization of the preceding result.
- Scope and open direction: Unlike the usual derivative and nonlocal nonlinear Schrödinger equations, this equation lacks bounded exponential seed solutions for generating global solutions.The paper identifies finding other interesting bounded seed solutions as an open direction.