REVIEW 4 major objections 6 minor 44 references
Selected aspects of the Korteweg-de Vries equation
T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The Lax equation for the quaternionic Cauchy-Riemann-Fueter flow holds only for real-valued solutions, so noncommutativity blocks integrability in this ansatz.
desk verdict A clear set of KdV lecture notes with one small quaternionic Lax-pair aside whose stated moral about noncommutativity goes beyond what the computation proves. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the Lax pair $(L,M)$ and the Lax equation $L_t+[L,M]=0$, with $L=\not\partial+u$ and $M=\not\partial$. Here $\not\partial=I_1\partial_1+I_2\partial_2+I_3\partial_3$ is the Dirac-type Cauchy-Riemann-Fueter operator, which squares to the three-dimensional Laplacian. The calculation works because every second-derivative term in the commutator cancels, leaving only first-order and zero-order terms; the first-order terms vanish exactly when $u$ commutes with $I_1,I_2,I_3$, and the zero-order term vanishes exactly when $u$ obeys $\partial_t u=-\not\partial(u)$.
What would settle it
Exhibit a Lax pair for the negative Cauchy-Riemann-Fueter equation whose $L$ is $\not\partial+u$ with a genuinely quaternionic (non-real) $u$ that nonetheless satisfies $\partial_t u=-\not\partial(u)$, or find any pair $(L,M)$ satisfying $L_t+[L,M]=0$ for such a $u$; either would overturn the note's conclusion. Concretely, allowing $M$ to carry quaternionic coefficient functions of order zero or one and solving the resulting coefficient equations is a finite algebraic check.
Extended reading notes
Core claim
The paper's own computation shows that expanding $(L_t+[L,M])(v)$ for $L=\not\partial+u$ and $M=\not\partial$ leaves only the terms $(I_m u-u I_m)\partial_m v$ and $(u_t+\not\partial u)v$; all mixed second-order terms cancel by symmetry of partial derivatives. Hence the Lax equation is equivalent to the evolution equation plus the three commutativity conditions $I_m u = u I_m$. For the standard quaternionic structure, these conditions force $u$ to be real, so the Lax formalism, within this ansatz, cannot describe quaternionic solutions of the Cauchy-Riemann-Fueter evolution equation.
Load-bearing premise
The computation only treats the fixed ansatz $L=\not\partial+u$, $M=\not\partial$; the notes do not prove that this ansatz is the most general Lax pair, so if a different pair admitted non-real quaternionic solutions, the claim that noncommutativity is the obstacle would be weaker.
Editorial extensions
If this is right
- For the negative CR-Fueter flow, the Lax pair $(L,M)=(\not\partial+u,\not\partial)$ exists only for real-valued $u$, so the note's integrability machinery does not reach quaternionic solutions.
- The same coefficient comparison yields the two standard KdV Lax pairs: $M=(\gamma+u_x)-(4\lambda+2u)\partial_x$ and $M=-4\partial_x^3-6u\partial_x-3u_x$, each of which reproduces $u_t+6uu_x+u_{xxx}=0$.
- The KdV hierarchy gives infinitely many Poisson-commuting Hamiltonians $F_j$, so every $F_k$ is an integral of the standard KdV equation.
- Inverse scattering linearizes the time evolution of the scattering data: $\rho(k,t)=\rho(k,0)e^{8ik^3t}$ and $B(k,t)=B(k,0)$, which is how the initial value problem is solved.
Reading between the lines
- The obstructions are purely algebraic, so the same commutativity condition should appear for any noncommutative target algebra (for instance matrix-valued $u$) paired with a Dirac-type operator; this is an extension, not a claim of the paper.
- If a different $M$, say with quaternionic coefficients of lower order, could satisfy the Lax equation for a non-real $u$, the note's conclusion about noncommutativity would weaken; searching for such an $M$ is a direct testable extension.
- One might test numerically whether solutions of $\partial_t u=-\not\partial(u)$ with quaternionic initial data exhibit any conserved quantities at all; the paper's Lax obstruction suggests none of the usual integrable type, but that is an inference from the note's framework.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript is a set of lecture notes on the Korteweg-de Vries equation. It covers the physical motivation and basic properties, a finite-dimensional integrable-systems review, symmetries, the Hamiltonian formulation and KdV hierarchy, Lax pairs, direct and inverse scattering, and a quaternionic example intended to illustrate limitations of the Lax formalism. Most of the material is standard and is presented with explicit computations and references; the only essentially new-looking item is Example 2.31, which characterizes a Lax pair for the negative Cauchy-Riemann-Fueter evolution under the fixed ansatz L = /∂ + u and M = /∂. The paper does not state a single overarching theorem; its intended value is expository.
Significance. If corrected, the notes could be a useful teaching resource: the Hamiltonian computation in §2.1, the uniqueness proof in §1.6, and the quaternionic computation in Example 2.31 are explicit and largely checkable. Example 2.31 is the strongest point of the manuscript, and the computation is internally consistent for the fixed ansatz. However, the significance is limited by the interpretation in §2.5, which overstates what has actually been proved, and by errors in the symmetry and scattering chapters that currently make parts of the text unreliable for classroom use. The paper's contribution is therefore more modest than the text suggests.
major comments (4)
- [§2.5, Example 2.31, Remark 2.32] The section concludes that noncommutativity of the quaternionic target space is 'the real obstacle' to rich solutions, but this is demonstrated only for the fixed ansatz L = /∂ + u, M = /∂ (with M understood as the conjugate operator in the computation). No argument rules out a broader class, for example a first-order M = Σ_{m=1}^3 a_m ∂_m + c with quaternionic coefficients and a zero-order term. Such a pair could in principle admit non-real solutions, in which case the conclusion in §2.5 would be an artifact of the ansatz. The text should either prove a no-go statement for a natural class of M or explicitly weaken the conclusion to 'for the ansatz considered here'.
- [§3, opening paragraph and §3.4] The Lax equation sign changes without comment: Definition 2.18 defines the Lax equation as L_t + [L,M] = 0, but the opening of Chapter 3 states that both Lax pairs satisfy L_t = [L,M]. In §3.4, after setting v_k(t,x) = M_k(t,x)e^{-ikx}, the displayed equation for ∂_t M_k contains the terms γ − 4ik^3 and +(4k^2 − 2u)∂_x M_k. For the first-order M = (γ+u_x) − (4λ+2u)∂_x with λ = k^2, a direct computation gives (γ+u_x+4ik^3+2iku)M_k − (4k^2+2u)∂_x M_k; for the third-order M of Example 2.23 the transformed equation would be different again. Thus equation (3.34) and the resulting time evolution A(k,t) = A(k,0)e^{8ik^3t} are not justified by either M. The asymptotic statements for M_k also list 'x→−∞' twice, where the second should be the other limit, and the statement that one may impose γ − 4ik^3 = 0 is problematic because γ is a constant and k varies. This section needs a consistent sign convention and a correct derivation.
- [§1.5, Corollary 1.21] The corollary claims that the KdV equation has precisely four symmetries that transform solutions into solutions. This is contradicted later in the same manuscript: §2.2, Example 2.16 exhibits a nontrivial degree-seven symmetry and Theorem 2.6 presents the whole KdV hierarchy, whose flows are solution-preserving transformations. The cited reference [Olver, Example 2.44] concerns point symmetries, not all symmetries in the generalized sense used in §2.2. The corollary should be restated as a classification of point symmetries, or the discussion in §2.2 should be reconciled with it.
- [§3.3, Proposition 3.24] The proof of simplicity of the zeros of B(k) is not coherent as written. After 'If B(k) has a zero in iσ', the text writes φ_{iσ}(x) = B(iσ)ψ_{iσ}(x), even though B(iσ) = 0 at a zero; later A(iσ) = 0 is used without justification. In Proposition 3.20, the claimed limit 'a(x) → 1' refers to a quantity a(x) that has not been defined. If these propositions are to be proved rather than cited, the proof must be corrected; otherwise the statements should be attributed clearly to a reference with only a sketch.
minor comments (6)
- [§2.4, proof of Example 2.23] The term 'a3uv' should read 'c3uv'.
- [§2.5, Example 2.31] The text sets M := /∂ but then uses the conjugate operator (negative of /∂) in the computation; the notation should be made consistent, for example by writing M := /∂ with an explicit overbar or by using a different letter.
- [§1.5, Lemma 1.16] After defining h(x,t) = (x−x0, t−t0), the inverse is (ξ+x0, τ+t0); the proof writes ξ0 and τ0, which have not been introduced.
- [§3.2] The transmitted beam is written as e^{−ikx}, which is inconsistent with the convention stated in Remark 3.8 that a right-moving beam has the form e^{+ikx}; the conventions should be aligned.
- [§3.3 and §3.4] The name 'Faddeev' is spelled 'Fadeev' in §3.4, and the bibliography entry [KdVFaddev] is formatted in reverse author order; the citation style should be normalized.
- [General] The manuscript contains several YouTube links; for a published version, these should be replaced or supplemented by archival references where possible.
Circularity Check
No significant circularity: the paper is an expository derivation of standard KdV theory, and the quaternionic Lax-pair example is a direct algebraic computation with an explicit ansatz.
full rationale
The paper is self-contained lecture notes. Its central derivations—D'Alembert's solution, the KdV equation from the wave equation, the four symmetries, the Hamiltonian hierarchy, the Lax pairs, and the inverse scattering procedure—are carried out by explicit computation or are cited to standard external sources such as Olver and Ablowitz–Clarkson. There are no fitted parameters presented as predictions and no quantity derived from itself by definition. In Example 2.31, the ansatz L = /∂ + u, M = /∂ is stated explicitly; the Lax equation is then solved algebraically, yielding both the negative Cauchy–Riemann–Fueter evolution equation and the commutation conditions I_m u = u I_m. The resulting equivalence is a genuine characterization of that ansatz, not an assumption smuggled in as a conclusion. The interpretive statement in §2.5 that noncommutativity 'poses here the real obstacle' is broader than what the example proves, because only one family of M-operators is considered; however, this is a scope limitation or overreach, not circularity. The self-citations in §2.5 ([Hohloch & Noetzel & Salamon], [Hohloch], [Brilleslijper & Fabert]) are used for background and physical motivation on hyperkähler Floer theory, not as load-bearing evidence for the algebraic claim. A proof left to the reader in Example 2.16 is referred to the external textbook [Miwa & Jimbo & Date], which is independent support rather than circular self-citation. Accordingly, no circular step meeting the required standard can be exhibited.
Assumptions & free parameters
assumptions (5)
- standard math The standard definitions and theorems of symplectic geometry (Darboux, Hamiltonian vector fields, Poisson brackets) are assumed.
- domain assumption The function space E in §2.1 admits at least three weak derivatives and vanishing boundary terms; the symplectic form is treated as a genuine symplectic form despite being weak.
- domain assumption The Faddeev condition u ∈ P1 = {g∈C^2(R): ∫(1+|x|)|g(x)|dx < ∞} ensures the scattering analysis is valid.
- domain assumption In the quaternionic example, the multiplication on H^n is associative and linear in both factors, and I1,I2,I3 form a quaternionic structure.
- standard math The completeness of the symmetry classification (exactly four symmetries) follows from Peter Olver's classification of Lie symmetries.
Cite this review
Pith. "Pith review of Selected aspects of the Korteweg-de Vries equation." pith.science (2026). https://pith.science/paper/XL5PMMX7
@misc{pith2026241118504,
author = {Pith},
title = {Pith review of: Selected aspects of the Korteweg-de Vries equation},
year = {2026},
howpublished = {\url{https://pith.science/paper/XL5PMMX7}},
note = {Machine review of arXiv:2411.18504}
}
read the original abstract
These lecture notes grew out of notes for courses around Integrable PDEs and the KdV equation given by the authors during the past five years at the University of Antwerp (Belgium). Comments and suggestions are welcome.
Reference graph
Works this paper leans on
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[1]
˜uτ + 6˜u˜uξ + ˜uξξξ = 1 bκut + 6 aκ2 uux + 1 a3κuxxx ◦ h−1
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SYMMETRIES OF THE KDV EQUATION 15
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If ˜u is a solution of KdV and b aκ = 1 = b a3 then u is a solution of KdV
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If u is a solution of KdV and b aκ = 1 = b a3 then ˜u is a solution of KdV
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The solutions (a, b,κ ) of b aκ = 1 = b a3 are given by (a, b,κ ) = (λ,λ 3,λ 2) forλ∈ R. Proof. 1) Lemma 1.15 and Lemma 1.17 together yield (1.19) ˜ uτ + 6˜u˜uξ + ˜uξξξ = 1 bκut + 6 aκ2 uux + 1 a3κuxxx ! ◦ h−1
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If ˜u is a solution of KdV then the left hand side of (1.19) vanishes and multiplication with bκ leads to 0 = ut + 6 b aκuux + b a3 uxxx ! ◦ h−1 Thus, if b aκ = 1 = b a3 then u solves KdV
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Using this, b aκ = 1 leads toκ = a2
From b a3 = 1 we get b = a3. Using this, b aκ = 1 leads toκ = a2. □ Furthermore Lemma 1.20 (Galilean boost). Let h : R2 → R2, h(x, t) = (x−λt, t) and let λ ∈ R. Then u∈ C3(R2, R) is a solution of KdV if and only if v := (u◦ h−1)− λ 6 is a solution of KdV . Proof. We calculate Dh = 1−λ 0 1 . Finding h−1(ξ,τ ) = (ξ +λτ,τ), we get D(h−1) = 1 λ 0 1 and thus f...
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INITIAL V ALUE PROBLEM FOR KDV 17 1.6 Initial value problem for KdV When dealing with ODEs, there are the theorems of Peano and Picard-Lindel ¨of that establish ex- istence resp. existence & uniqueness of solutions of initial value problems in a very general way: If the function underlying an ODE is continuous, we have existence of solutions of initial va...
Show all 44 references
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u is said to have initial data u0 : R→ R in t0∈ R if u(x, t0) = u0(x) for all x∈ R
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u decays sufficiently rapidly if limx→±∞∂k xu(x, t) = 0 for all t∈ R and all k∈ N0
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The following statement stems in fact from ODE theory but is often very useful when estimating a growth rate of solutions
The energy of u is given by Eu : R→ R≥0, Eu(t) := ∞Z −∞ |u(x, t)|2 dx. The following statement stems in fact from ODE theory but is often very useful when estimating a growth rate of solutions. It is due to the (originally Swedish) mathematician Thomas Hakon Gr¨onwall (1877 – ...
1932
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INITIAL V ALUE PROBLEM FOR KDV 19 Now we look for a PDE of which z is a solution. To this aim, plug z into the KdV equation and compute zt + 6zzx + zxxx = ut− vt + 6(u− v)(ux− vx) + uxxx− vxxx = ut− vt + 6uux− 6uvx− 6uxv + 6vvx + uxxx− vxxx− 6vvx + 6vvx =−6uvx− 6uxv + 6vvx + 6...
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KdV equation: Hamiltonian PDE and integrability In this chapter, we study how the notions of Hamiltonian vector field, Hamiltonian equation, and integrability extend to the infinite dimensional setting. 2.1 Hamiltonian formalism for the KdV equation and the KdV hierarchy Inspi...
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HAMILTONIAN FORMALISM FOR THE KDV EQUATION AND THE KDV HIERARCHY 23 = ∞Z −∞ 3g2(x)v(x) + gx(x)vx(x) dx and finally conclude by integration by parts = ∞Z −∞ 3g2(x)− gxx(x) v(x) dx. Now abbreviate (XH(g))(z) =: XH(z)∈ Tg(z)R≃ R and consider ωg(XH(g), v) = 1 2 ∞Z −∞ xZ −∞ v(x)XH(...
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HAMILTONIAN FORMALISM FOR THE KDV EQUATION AND THE KDV HIERARCHY 25 We will see that the KdV equation is in fact only one item within a whole family of similarly generated partial di fferential equations. This was discovered and studied in a se- ries of papers by [ Miura], [ M...
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Then the family of equations ut = X j(u), for j≥ 1 is called KdV hierarchy or higher order KdV equations and recovers for a = 2, b =−1 and j = 2 the standard KdV equation ut− 6uux + uxxx = 0
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The higher order KdV equations u t = X j(u) are Hamiltonian with Hamiltonian func- tions F j(g) := R∞ −∞ f j(g(x)) dx and Hamiltonian vector fields X F j = X j
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The higher order KdV equations are integrable in the sense that {F j, Fk} = 0 for all j, k≥ 1
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Thus all F k are integrals of the standard KdV equation
The Hamiltonian H from the standard KdV equation satisfies H = F2 and therefore {H, Fk} = 0 holds true for all k ≥ 1. Thus all F k are integrals of the standard KdV equation. Proof. 1) X1(u) := ux leads to ut = X1(u) = ux. Now keep in mind that ux = Du and calculate with a = 2...
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Using the identity X j(g) = ∂x δ f j δg and applying Remark 2.5 to the Hamiltonian F j(g) =R∞ −∞ f j(g(x)) dx, we obtain XF j = X j
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In order to show integrability we first prove a recurrence relation for the Poisson bracket. We use the short notation X j(z) := X j(g)(z) and compute {F j, Fk} =ω(XF j, XFk) =ω(X j, Xk) = 1 2 ∞Z −∞ xZ −∞ Xk(x)X j(y)− X j(x)Xk(y) dy dx = 1 2 ∞Z −∞ Xk(x) xZ −∞ X j(y)...
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Now we show that in fact{F j, Fk} = 0 holds true for all j, k≥ 1
HAMILTONIAN FORMALISM FOR THE KDV EQUATION AND THE KDV HIERARCHY 27 thus obtaining the relation {F j, Fk} ={F j+1, Fk−1}. Now we show that in fact{F j, Fk} = 0 holds true for all j, k≥ 1. First, consider the case| j− k| even and assume w.l.o.g. that j< k. By increasing j7→ j +...
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This follows immediately from 3). □ 28 2. KDV EQUATION: HAMILTONIAN PDE AND INTEGRABILITY 2.2 Evolution equations and their symmetries Intuitively, an integrable system is a system of differential equations whose behavior is determined by initial conditions and which can be so...
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Let K : C∞(R× R, R)→ C∞(R× R, R) with K (u) : =−(6uux + uxxx) and u depending on (x, t)∈ R2
EVOLUTION EQUATIONS AND THEIR SYMMETRIES 29 Example 2.10. Let K : C∞(R× R, R)→ C∞(R× R, R) with K (u) : =−(6uux + uxxx) and u depending on (x, t)∈ R2. Then the associated evolution equation ∂tu = K(u) is equivalent to the KdV equation ut + 6uux + uxxx = 0. Intuitively, a ‘symm...
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x adds +1 to the degree, i.e., deg( u) = 2, deg(ux) = 3, deg(uxx) = 4 etc., and multiplication leads to addition of the degrees, i.e., deg( uux) = 2 + 3 = 5 etc
Each differentiation w.r.t. x adds +1 to the degree, i.e., deg( u) = 2, deg(ux) = 3, deg(uxx) = 4 etc., and multiplication leads to addition of the degrees, i.e., deg( uux) = 2 + 3 = 5 etc. This gives the following monomials of odd degree: • degree 3: ux • degree 5: uux, uxxx ...
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• degree 7: c1u2ux + c2uuxxx + c3uxuxx + c4uxxxxx with (constant) coefficients c1, c2, c3, c4∈ R
EVOLUTION EQUATIONS AND THEIR SYMMETRIES 31 • degree 5: c1uux + c2uxxx with (constant) coefficients c1, c2∈ R. • degree 7: c1u2ux + c2uuxxx + c3uxuxx + c4uxxxxx with (constant) coefficients c1, c2, c3, c4∈ R. For homogeneous polynomials of degree 3, we obtain: Example 2.14. Co...
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So it is of particular interest to find equivalent formulations that are hopefully easier to solve
LAX FORMALISM OF AN EVOLUTION EQUATION 33 2.3 Lax formalism of an evolution equation Recall that, intuitively, an integrable system is a system of di fferential equations whose behavior is determined by initial conditions and which can be solved (‘integrated’) from those initi...
1926
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Similarly we can recover the Hamiltonian, i.e., the conserved quantity, by computing the trace of the square of the matrix L
LAX FORMALISM OF AN EVOLUTION EQUATION 35 Computing the eigenvalues of L via 0 = det(L−λI2) =−(P−λ)(P +λ)− Q2 yields λ =±1 2 p Q2 + P2 =± r k 4mq2 + p2 4m2 =± r 1 2m √ H, which means that the eigenvalues are constants of motion, since the Hamiltonian is a constant of motion. S...
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• 0 = 2(v1)x + uv0 + (v0)xx = 2(v1)x + uv0 implies v1(x) =− v0 2 R x u(y) dy
LAX PAIRS OF THE KDV EQUATION 37 = √ λe √ λx X k≥0 vk √ λ k + e √ λx X k≥0 (vk)x √ λ k x + ue √ λx X k≥0 vk √ λ k−λe √ λx X k≥0 vk √ λ k =λe √ λx X k≥0 vk √ λ k + 2 √ λe √ λx X k≥0 (vk)x √ λ k + e √ λx X k≥0 (vk)xx √ λ k + ue √ λx X k≥0 vk √ λ k−λe √ λx X...
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0 = Lt + [L, M] holds true if and only if the coe fficients of vxxxx , vxxx, vxx, vx, and v vanish
LAX PAIRS OF THE KDV EQUATION 39 = utv + 2(c1)xvxxxx + (c1)xxvxxx + 2(c2)xvxx− 3c1uxvxx + (c2)xxvx − 3c1uxxvx + 2(c3)xvx + (c3)xxv− c1uxxxv− c2uxv = 2(c1)xvxxxx + (c1)xxvxxx + 2(c2)x− 3c1ux vxx + (c2)xx− 3c1uxx + 2(c3)x vx + ut + (c3)xx− c1uxxx− c2ux v. 0 = Lt + [L, M] holds t...
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If the underlying field is commutative one may hope for a reasonable number of terms arising from the commutator in the Lax equation to cancel each other
LIMITATIONS OF THE LAX FORMALISM 41 2.5 Limitations of the Lax formalism A look at the proofs of Example 2.22, Example 2.23, and Example 2.24 shows that an ansatz usually leads to several constraints. If the underlying field is commutative one may hope for a reasonable number ...
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/∂ is a Dirac type operator in the sense that/∂◦ /∂ = ∆ where ∆ is the 3-dimensional Laplace operator
LIMITATIONS OF THE LAX FORMALISM 43 Remark 2.30. /∂ is a Dirac type operator in the sense that/∂◦ /∂ = ∆ where ∆ is the 3-dimensional Laplace operator. The Cauchy-Riemann-Fueter equation is of importance for instance in hyperk ¨ahler Floer the- ory where it describes the criti...
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When studying the conditions for L t + [L, M] = 0, then the coefficient of v is given by ut− a∂1u− b∂2u− c∂3u which yields ut + /∂(u) = 0 if and only if a =−i and b =− j and c =−k
Example 2.31 was found with the ansatz M = a∂1 + b∂2 + c∂3 where a, b, c : T3× R→ H are coefficient functions depending on (x, t)∈ T3× R andI1 = i,I2 = j,I3 = k is the standard quaternionic structure on H. When studying the conditions for L t + [L, M] = 0, then the coefficient...
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KdV equation: direct and inverse scat- tering In this chapter, we will use the Lax formalism and the scattering technique to solve the initial value problem for the Korteweg–de Vries (KdV) equation. In the previous chapter, in Examples 2.22 and 2.23, we gave the following two ...
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ANALOGY WITH FOURIER TRANSFORMATION 47 Example 3.3. Let f, g∈ L1(Rn, C) and denote by∇ = Pn j=1∂2 j the Laplace operator on Rn (often called vector Laplacian in Physics) and consider the following partial differential equation: (3.4) −∇ f + f = g Since the Fourier transform tu...
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In this sec- tion, we will give an introductory overview of quantum mechanics and scattering theory to clarify some aspects of the physics behind the scattering scheme
MOTIV ATION FROM PHYSICS 49 3.2 Motivation from physics Scattering is a well-studied framework in physics, as a significant portion of modern particle physics relies on the analogy between particle interactions and scattering phenomena. In this sec- tion, we will give an intro...
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We will examine the L-operator associated with the KdV equation and show that its asymptotic analysis corresponds to the previously illustrated physical setting
SCATTERING PROCEDURE 51 3.3 Scattering procedure In this section, we will derive the set of scattering data by approaching the Korteweg–de Vries (KdV) equation as a scattering problem. We will examine the L-operator associated with the KdV equation and show that its asymptotic...
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Let e∈ R and consider a Schr¨odinger equation with a potential u(x)∈ P1, (3.14) −∂2 x f + u(x) f = e f
SCATTERING PROCEDURE 53 Proof. Let e∈ R and consider a Schr¨odinger equation with a potential u(x)∈ P1, (3.14) −∂2 x f + u(x) f = e f. Let f, g be two different solutions of (3.14). Then computing the Wronskian yields ∂xW( f, g) =∂x ( f∂g− g∂ f ) = f∂2 xg− g∂2 x f = f (u(x)− e...
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Integration paths: γ+ andγ− whereδ(x) is the Dirac delta function
SCATTERING PROCEDURE 55 Re(l) Im(l) l = 0 l = 2k γ+ γ− Figure 3.3. Integration paths: γ+ andγ− whereδ(x) is the Dirac delta function. This allows to write the solution of Equation (3.21) as mk(x) = 1 + +∞Z −∞ Gk(x−ξ)u(ξ)mk(ξ)dξ if for|x|→ +∞ the function m(x)→ 1. By means of t...
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SCATTERING PROCEDURE 57 = 1− B(k) + 1 2ik +∞Z −∞ 1− e2ik(x−y) u(y)Mk(y)dy − 1 2ik +∞Z x 1− e2ik(x−y) u(y) Mk(y)− B(k)Nk(y) dy . Now consider the right-hand side of Equation (3.23): A(k)e2ikxN−k(x) = A(k)e2ikx 1− 1 2ik +∞Z x ...
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This implies (3.29) lim x→+∞ W(φk(x),∂ kφk(x))− lim x→−∞ W(φk(x),∂ kφk(x)) =−2k ∞Z −∞ φ2 k(x)dx
SCATTERING PROCEDURE 59 The difference between Equations (3.27) and (3.28) gives φk(x)∂k∂xφk(x)−∂xφk(x)∂kφk(x) + 2k φk(x) 2 = 0. This implies (3.29) lim x→+∞ W(φk(x),∂ kφk(x))− lim x→−∞ W(φk(x),∂ kφk(x)) =−2k ∞Z −∞ φ2 k(x)dx. Now compute (3.30) W (∂kφk(x),ψ k(x)) + W (φk(x),∂ ...
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As in the previous subsection, the asymptotic behavior will take a central role in the procedure
TIME EVOLUTION OF SCATTERING DATA 61 3.4 Time evolution of scattering data Let us consider again the Lax pair, and in particular the time-evolution operator M. As in the previous subsection, the asymptotic behavior will take a central role in the procedure. Let us con- sider t...
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See [ Bro]
INVERSE SCATTERING 63 Then for every constant a∈ R+, then: lim R→+∞ Z CR f (z)eıazdz = 0, Proof. See [ Bro]. □ We can define the two projections: Π+F (κ) = 1 2πi ∞Z −∞ F (ξ) ξ− k + (i0)dξ, Π−F (κ) = 1 2πi ∞Z −∞ F (ξ) ξ− k− (i0)dξ. Now rewrite Equation...
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J.; Clarkson, P
Bibliography [Ablowitz & Clarkson] A blowitz, M. J.; Clarkson, P. A.:Solitons, nonlinear evolution equations and inverse scatter- ing. London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. xii+516 pp. [Ablowitz & Kaup & Newell & Seg...
1974 arXiv
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