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SUMMARY:Tumour growth model: Lie symmetries and exact solutions
DTSTART;VALUE=DATE-TIME:20181205T140000Z
DTEND;VALUE=DATE-TIME:20181205T142000Z
DTSTAMP;VALUE=DATE-TIME:20260818T125012Z
UID:indico-contribution-2-6@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Vasyl' Davydovych (Institute of Mathematics\,  NAS o
 f Ukraine)\nWe examine the tumour growth model proposed in [1]. In the 2D 
 case\, the governing equations after some simplifications take the form \n
  $\\begin{array} {l}\n  \\alpha_t + \\left(\\alpha u^1\\right)_{x}+\\left(
 \\alpha u^2\\right)_{y}=\n  S(\\alpha)\, \\\n  u^1_{x} + u^2_{y} = \\nabla
 \\cdot\\left(D(\\alpha)\\nabla p\\right)\,  \\\\\n  \\Big[(2+\\lambda)\\al
 pha u^1_{x} + \\lambda \\alpha u^2_{y}\\Big]_x + \\Big[\\alpha u^1_{y} + \
 \alpha u^2_{x}\\Big]_y =\n  p_x + (\\alpha\\Sigma(\\alpha))_x\, \\hskip1.2
 cm (1)\\\\\n  \\Big[\\alpha u^1_{y} +  \\alpha u^2_{x}\\Big]_x +\\Big[(2+\
 \lambda)\\alpha u^2_{y} +\n   \\lambda \\alpha u^1_{x}\\Big]_y = p_y + (\\
 alpha\\Sigma(\\alpha))_y\, \n  \\end{array}$ \nwhere $D\, \\ S$ and $\\Sig
 ma$ are some functions and their typical forms are listed in [1]. Assuming
  that the tumour boundary is prescribed by a curve $\\Gamma(t\,x\,y)=0$\, 
 where  $\\Gamma$ is an unknown function\, the boundary сonditions have th
 e form\n$\\begin{array} {l}\n   u^1\\Gamma_{x}+ u^2\\Gamma_{y}= -\\Gamma_t
 \, \\quad p=0\, \\\\\n  \\Big[(2+\\lambda) u^1_{x} + \\lambda  u^2_{y}\\Bi
 g]\\Gamma_x + \\Big[ u^1_{y} +  u^2_{x}\\Big]\\Gamma_y = 0\, \\hskip3.5cm 
 (2)\\\\\n  \\Big[ u^1_{y} +  u^2_{x}\\Big]\\Gamma_x +\\Big[(2+\\lambda) u^
 2_{y} + \\lambda  u^1_{x}\\Big]\\Gamma_y = 0.\n  \\end{array}$\nSo\, we ha
 ve the nonlinear boundary value problem (1)-(2) with the unknown moving bo
 undary $\\Gamma(t\,x\,y)=0$.\nUsing the definition proposed in [2] and ass
 uming $\\Gamma$ to be a closed curve for any $t\\geq 0$\, we examined the 
 Lie symmetry and constructed the exact solutions of the boundary value pro
 blem (1)-(2). For instance\, the following statement takes place:\n*The sy
 stem of nonlinear PDEs  (1) with arbitrary functions $D\, \\ S$ and  $\\Si
 gma$ is invariant with respect to the infinite-dimensional Lie algebra  ge
 nerated by the Lie symmetry operators\n $\\begin{array} {l}\n   \\partial_
 t\,  \\quad  F(t) \\partial_p\, \\quad\n    G_g = g(t)\\partial_x + \\dot 
 g\\partial_{u^1}\,  \\quad  G_h = h(t)\\partial_y + \\dot h\\partial_{u^2}
 \\\\\n   J_f = f(t) \\Big[y\\partial_x -   x\\partial_y +(u^2+\\frac{\\dot
  f}{f}y)\\partial_{u^1} -\n    (u^1+\\frac{\\dot f}{f}x)\\partial_{u^2}\\B
 ig].\n   \\end{array}$ \n   Here $ F\, f\, g\, $  and $h$ are arbitrary sm
 ooth functions and the upper dot means differentiation with respect to tim
 e.*\n\n [1]  H. Byrne\, J.R. King\, D.L.S. McElwain\, L. Preziosi. A two-p
 hase model of solid tumour growth. *Appl. Math. Letters.* **16** (2003)  5
 67-573.\n [2] R. Cherniha\, S. Kovalenko. Lie symmetries and reductions of
  multi-dimensional boundary value problems of the Stefan type. *J. Phys. A
 : Math. Theor.* **44** (2011) 485202 (25 pp.)\n\nhttps://indico.bitp.kiev.
 ua/event/2/contributions/6/
LOCATION:
URL:https://indico.bitp.kiev.ua/event/2/contributions/6/
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BEGIN:VEVENT
SUMMARY:Physics of singular self-adjoint extensions of one-dimensional Dir
 ac operator
DTSTART;VALUE=DATE-TIME:20181205T134000Z
DTEND;VALUE=DATE-TIME:20181205T140000Z
DTSTAMP;VALUE=DATE-TIME:20260818T125012Z
UID:indico-contribution-2-18@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Dmitry Panchenko (Odessa I.I.Mechnikov National Univ
 ersity)\nWe consider boundary conditions (self-adjoint extensions) corresp
 onding to point-like interactions for one-dimensional Dirac operator. Taki
 ng the non-relativistic limit we show how all possible point-like interact
 ions for one-dimensional Schrödinger operator of free spinless particle c
 an be obtained from the Dirac Hamiltonian. In case of spin-1/2 we show tha
 t there are boundary conditions with spin-flop mechanism. We suggest the p
 hysical interpretation these point-like extensions in terms of the Rashba 
 (spin-orbital) coupling.\n\nhttps://indico.bitp.kiev.ua/event/2/contributi
 ons/18/
LOCATION:
URL:https://indico.bitp.kiev.ua/event/2/contributions/18/
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