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Classical Integrable Field Theories with Defects and Near-Integrable Boundaries

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Classical Integrable Field Theories with Defects and Near-Integrable Boundaries Classical Integrable Field Theories with Defects and near-Integrable Boundaries Robert Charles Parini PhD University of York Mathematics March 2018 Abstract In the first part of this thesis algebro-geometric solutions for the sine-Gordon and KdV equations in the presence of a type I integrable defect are found, generalising the previously known soliton solutions. Elliptic (genus one) solutions where the defect induces only a phase shift are obtained via ansätze for the fields on each side of the defect. Algebro- geometric solutions for arbitrary genus and involving soliton emission by the defect are constructed using a Darboux transformation, exploiting the fact that the defect equations have the form of a Bäcklund transformation at a point. All the soliton and phase-shifted elliptic solutions to the defect equations are recovered as limits of the algebro-geometric solutions constructed in this way. Certain energy and momentum conserving defects for the Kadomtsev-Petviashvili equa- tion are then presented as a first step towards the construction of integrable defects in higher dimensions. Algebro-geometric solutions to the sine-Gordon equation on the half-line with an in- tegrable two parameter boundary condition are obtained by imposing a corresponding restriction on the Lax pair eigenfunction or, alternatively, as a Darboux transformation of the known algebro-geometric solution for the Dirichlet boundary. Finally, the collision of sine-Gordon solitons with a Robin type boundary is examined. This boundary is typically non-integrable but becomes an integrable Neumann or Dirichlet boundary for certain values of a boundary parameter. Depending on the boundary param- eter and initial velocity an antikink may be reflected into various combinations of kinks, antikinks and breathers. The soliton content of the field after the collision is numerically determined by computing the discrete scattering data associated with the inverse scatter- ing method. A highlight of this investigation is the discovery of an intricate structure of resonance windows caused by the production of a breather which can collide multiple times with the boundary before escaping as a lighter breather or antikink. Contents Abstract . .2 Contents . .3 List of Figures . .6 Introduction . 10 Acknowledgements . 18 Declaration . 19 1 Integrable defects 20 1.1 Sine-Gordon equation . 20 1.1.1 Phase-shifted soliton solutions . 22 1.1.2 Soliton emission . 24 1.1.3 One-to-two soliton solution . 25 1.1.4 Defects, momentum and Bäcklund transformations . 25 1.2 Korteweg-de Vries equation . 27 1.2.1 Phase-shifted soliton solutions . 28 1.2.2 One-to-two soliton solution . 30 2 Algebro-geometric solutions of integrable systems 31 2.1 Lax pairs and algebraic curves . 31 2.2 Algebraic curves and Riemann surfaces . 32 2.2.1 Hyperelliptic curves . 34 2.2.2 Basis of cycles . 36 2.3 Abelian differentials and integrals . 38 2.3.1 Abelian integrals . 39 2.3.2 Riemann’s bilinear identity . 40 2.3.3 Basis of Abelian differentials . 41 2.3.4 Normalised basis of holomorphic differentials . 42 2.3.5 Normalised differentials of the second and third kind . 43 2.4 Theta functions . 44 2.4.1 Abel map, divisors, and the Jacobi inversion problem . 44 2.5 Baker-Akhiezer functions . 45 2.6 Algebro-geometric solutions for KdV . 46 2.6.1 Explicit formulas for integrals of the second kind . 50 2.6.2 Reality and regularity conditions . 51 2.7 Algebro-geometric solutions for sine-Gordon . 53 2.7.1 Explicit formulas for integrals of the second kind . 55 3 Contents 2.7.2 Reality conditions . 56 2.8 Numerical evaluation . 57 3 Phase-shifted elliptic solutions for sine-Gordon with a defect 59 3.1 The ‘reality gap’ . 62 3.2 Soliton limit . 65 3.3 Effectiveness of the phase shift ansatz . 65 4 Defects and Darboux transformations 66 4.1 KdV . 67 4.1.1 Multi-soliton solutions . 68 4.1.2 Algebro-geometric Darboux transformation . 69 4.1.3 Reality and smoothness conditions . 70 4.1.4 Interpretation . 71 4.1.5 Soliton limit . 71 4.2 Sine-Gordon . 73 4.2.1 Multi-soliton solutions . 74 4.2.2 Algebro-geometric Darboux transformation . 75 4.2.3 Reality conditions . 76 4.2.4 Phase-shifted and soliton limits . 77 5 Defects in 2+1 dimensions 80 5.1 Conserved quantities . 81 5.2 Defect along y = 0 ................................ 82 5.3 Comments on defects along x = 0 ........................ 83 6 Integrable boundaries and algebro-geometric solutions 85 6.1 The Dirichlet boundary problem . 87 6.1.1 An adjustment . 91 6.2 The Ghoshal-Zamolodchikov boundary problem . 91 6.2.1 An alternate approach . 92 7 Sine-Gordon with Robin boundary conditions 96 7.1 Linearised model . 96 7.2 Conservation laws . 97 7.3 Vacuua . 100 7.4 Numerical method . 103 7.5 Integrable boundary . 107 7.6 Forces . 108 7.7 Robin boundary overview for k > 0 ....................... 110 7.8 Resonance structure . 118 7.9 Comments on Robin boundary with k < 0 ................... 123 8 Discussion 125 4 Contents A Full soliton limit 130 A.1 KdV . 131 A.2 Sine-Gordon . 133 B Partial soliton limit 134 B.1 Sine-Gordon . 136 B.2 KdV . 138 C Finding the roots of complex analytic functions: cxroots 140 C.1 Theory . 140 C.2 Using cxroots .................................. 144 5 List of Figures 1 Some solutions to the KdV equation. 12 1.1 Placement of a defect for a 1+1 dimensional field theory. 20 1.2 Examples of an initial soliton (a) incident on an initially continuous type I integrable defect being transmitted (b), transmitted and flipped (c) and cap- tured (d) for different values of the defect parameter σ = exp( η) according − to (1.9). 23 1.3 Examples of an initial soliton (a) incident on an initially excited type I integrable defect being transmitted (b), flipped (c) and captured (d) for different values of the defect parameter σ = exp( η) according to (1.10). 24 − 1.4 Examples of outcomes for a single KdV soliton (1.21) passing through the type I integrable defect (1.19) placed at x = 0 where the field to the right of the defect is the phase-shifted one soliton solution (1.23). 29 2.1 Examples of hyperelliptic Riemann surfaces whose branch points are λ N f igi=1 (and for (b)) together with a basis of cycles a ; b .............. 36 1 i i 2.2 Intersection numbers at a point of intersection, P ............... 37 2.3 This cycle bounds a region of the double torus (half of it in this case) so it is homologous to zero even though it is not homotopic to a point. 37 2.4 Cycles can be divided into closed portions whose sum is homologous to the original cycle: γ γ + γ ............................. 37 ∼ 1 2 2.5 A canonical basis of cycles ai, bi for a hyperelliptic Riemann surface plotted in the complex plane of λ ............................ 38 2.6 Gluing a torus. 40 2.7 Illustration for the proof of the Riemann bilinear identity. 41 2.8 Basis of cycles, as in Fig. (2.5), but pictured for KdV specifically. 51 2.9 A basis of cycles, as in Fig. (2.5), but pictured for sine-Gordon specifically. 56 3.1 The two values of the phase shift ∆, satisfying (3.10), corresponding to a range of values for the defect parameter σ for a pair of (a) real and (b) conjugate branch points. 62 3.2 Examples of elliptic, purely phase-shifted solutions to the type I integrable defect equations (1.7) for sine-Gordon plotted at t = 0 with the defect placed at x = 0...................................... 63 3.3 The imaginary part of the phase shift ∆, given by (3.20), plotted for exam- ples of real (a) and conjugate (b) branch points. 64 6 List of Figures 4.1 Examples of genus 1 (a), 2 (b), 3 (c) and 4 (d) algebro-geometric solutions to the KdV equation with a type I integrable defect at x = 0. In each of these examples the field to the right contains a soliton which is the trough at x 10..................................... 72 ≈ 4.2 Examples of genus 1 (a), 2 (b), 3 (c), 5 (d) algebro-geometric solutions to the sine-Gordon equation with a type I integrable defect at x = 0. In each of these examples the field to the right of the defect contains a soliton at x 10........................................ 77 ≈ 6.1 Some examples of algebro-geometric solutions satisfying the boundary con- dition u(0; t) = 2.................................. 90 6.2 Some examples of algebro-geometric solutions satisfying the Ghoshal-Zamolodchikov type boundary condition u (0; t) = 0:8 sin((u(0; t) 1:2)=2).......... 93 x − 7.1 The graphical solution of (7.27) with sin(u =2) plotted in red and ku in j 0 j 0 blue for k = 0:25 (solid line), and for the first three critical values of k (dashed lines). 101 7.2 The lowest energy static fields corresponding to the solutions of (7.26) for the critical value of k = k 0:064187...................... 101 3 ≈ 7.3 The energy, E, of a static antikink u(x) with u(0) = u0 as given by (7.29) with k = k 0:064187, the third critical value of k shown in Fig. (7.1). 3 ≈ The vertical dotted lines indicate the solutions of (7.27), which are also the stationary points of E(u0)............................. 102 7.4 Antikinks satisfying u(0) = u0 for u0 close to 2π. This illustrates the dif- ference in approaching u 2π from above and below. For u < 2π, 0 ! 0 u(x) = 4 arctan(exp(x x)) with a suitably chosen x while for u > 2π, 0 − 0 0 u(x) = 4 arctan(exp(x x)) + 2π......................
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