Almost Lyapunov functions
D. Liberzon, C. Ying and V. Zharnitsky, On Almost Lyapunov functions, ( PDF), in Proceedings of the 53rd IEEE
Conference on Decision and Control, Los Angeles, CA, Dec 2014, pp. 3083-3088.
D. Liberzon, S. Liu, V. Zharnitsky, On almost Lyapunov functions for non-vanishing vector fields, ( PDF),
in Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2016, pp. 5557-5562.
S. Liu, D. Liberzon, V. Zharnitsky, Almost Lyapunov functions for Nonlinear Systems, ( PDF), submitted in December 2018.
Linear search problem.
This problem was introduced by Beck and Bellman:
"An immobile hider is located on the real line according to a known probability distribution. A searcher, whose maximal velocity is one, starts from the origin and wishes to discover the hider in minimal expected time. It is assumed that the searcher can change the direction of his motion without any loss of time. It is also assumed that the searcher cannot see the hider until he actually reaches the point at which the hider is located and the time elapsed until this moment is the duration of the game." It is obvious that in order to find the hider the searcher has to go a distance x1 in one direction, return to the origin and go distance x2 in the other direction etc., (the length of the n-th step being denoted by xn), and to do it in an optimal way. (However, an optimal solution need not have a first step and could start with an infinite number of small 'oscillations'.) This problem is usually called the linear search problem and a search plan is called a trajectory. It has attracted much research, some of it quite recent ... (Wikipedia).
Y.M. Baryshnikov, V. Zharnitsky, Search on the brink of chaos,
( PDF), Nonlinearity 25 (2012) 3023-3047. In this work, we found a geometric characterization of the optimal search plan as an unstable separatrix in an associated Hamiltonian dynamical systems.
This shorter version ( PDF) has appeared in Proceedings of the Meeting on Analytic Algorithmics & Combinatorics, Kyoto, Japan, January 16, 2012.
High frequency dynamics of nonlinear dispersive waves.
M.B. Erdoǧan, V. Zharnitsky, Quasi-linear dynamics in nonlinear
Schrödinger equation with periodic boundary conditions, ( PDF), Commun. Math. Phys. 281, 655-673, 2008.
M.B. Erdoǧan, N. Tzirakis, V. Zharnitsky, Near-linear dynamics in KdV
with periodic boundary conditions, ( PDF), Nonlinearity 23, 1675-1694, 2010.
M.B. Erdoǧan, N. Tzirakis, V. Zharnitsky, Nearly linear dynamics of
nonlinear dispersive waves, ( PDF), Physica D 240, 1325-1333, 2011.
M.B. Erdoǧan, N. Tzirakis, V. Zharnitsky, High frequency perturbation of cnoidal waves in KdV, ( PDF), SIAM Jour. Math. Analysis, 44(6), 4147--4164, 2012.
E. VanderZee, A. N. Hirani, D. Guoy, V. Zharnitsky, E. Ramos, Geometric
and combinatorial properties of well-centered triangulations in three and higher dimensions, ( PDF), Computational Geometry: Theory and Applications 46, 700–724, 2013.
E. VanderZee, A. N. Hirani, V. Zharnitsky, D. Guoy, A dihedral acute triangulation of the cube, ( PDF), Computational Geometry: Theory and Applications 43, 445--452, 2010.
These papers deal with the classical billiard problem.
Y.M. Baryshnikov, V. Zharnitsky, Billiards and non-holonomic distributions,
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI) 300 (2003), Teor. Predst. Din. Sist. Spets. Vyp. 8, 56--64, 285.
Y.M. Baryshnikov, P. Heider, W. Parz, V. Zharnitsky, Whispering gallery modes inside the asymetric resonant cavities,( PDF) Phys. Rev. Lett. 93, 4839-42, 2004.
Y.M. Baryshnikov, V. Zharnitsky, Sub-Riemannian geometry and periodic orbits in classical billiards,( PDF) Mathematical Research Letters 13, no. 4, 587-598, 2006.
A. Tumanov, V. Zharnitsky, Periodic orbits in outer billiards,( PDF), International Mathematics Research Notices , vol. 2006, Article ID 67089, 2006
V. Blumen, K.Y. Kim, J. Nance, V. Zharnitsky, Three-period orbits in billiards on the surfaces of constant curvature (PDF), International Mathematics Research Notices, November, 2011.
S. Merenkov, V. Zharnitsky, Hausdorff dimension of three-period orbits in
Birkhoff billiards (PDF),
Nonlinearity 25 (2012) 1947-1954.
Y. Baryshnikov, V. Blumen, K.Y. Kim, V. Zharnitsky, Billiard dynamics of bouncing dumbbell, ( PDF), Physica D 269, 21-27, 2014.
Hamiltonian dynamics. Applications of KAM theory.
V. Zharnitsky, The Geometrical Description of the Nonlinear Dynamics of a Double Pendulum, (PDF) SIAM J. Appl. Math. 55, 1753-1763, 1995.
V. Zharnitsky, Quasi-periodic motions in the billiard problem with
a softened boundary,(PDF) Phys. Rev. Lett. 75, 4393-4396, 1995.
V. Zharnitsky, Breakdown of stability of motion in superquadratic potentials,(PDF) Commun. Math. Phys. 189, 165-204, 1997.
V. Zharnitsky, Resonances in a continuously forced anharmonic oscillator,(PDF) Phys. Lett. A 224, 264-270, 1997.
V. Zharnitsky, A note on adiabatic invariance in Hamiltonian systems depending singularly on the slow time, (PDF) Physica D 122, 62-72, 1998.
V. Zharnitsky, Instability in Fermi-Ulam ``ping-pong'' problem,(PDF) Nonlinearity 11, 1481-87, 1998.
V. Zharnitsky, Quasiperiodic motions in the Hamiltonian systems of billiard type,(PDF) Phys. Rev. Lett. 81, 4839-42, 1998.
V. Zharnitsky, Invariant tori in the Hamiltonian systems with impacts,(PDF) Commun. Math. Phys. 211, 289-302, 2000.
V. Zharnitsky, Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem,(PDF) Nonlinearity 13, 1123-1136, 2000.
M. Arnold, V. Zharnitsky, Pinball dynamics: unlimited energy growth in switching Hamiltonian systems ,(PDF), Communications in Mathematical Physics 338 (2), 501-521, 2015.
sine-Gordon equation and physics applications.
V. Zharnitsky, I. Mitkov, M. Levi, Parametrically Forced Sine-Gordon Equation and Domain Walls Dynamics in Ferromagnets, (PDF) Phys. Rev. B 57, 5033-35, 1998.
I. Mitkov, V. Zharnitsky, $\pi$-Kinks in the parametrically driven sine-Gordon equation and applications,(PDF) Physica D 123, 301-307, 1998.
V. Zharnitsky, I. Mitkov, N. Gronbech-Jensen, $\pi$-kinks in strongly ac driven sine-Gordon systems, (PDF) Phys. Rev. E 58, R52-R55, 1998.
Nonlinear optics. Ground state solutions.
S. Turitsyn, A. Aceves, C.K.R.T. Jones, V. Zharnitsky, Average
dynamics of the optical soliton in communication lines with dispersion
management: Analytical results, (PDF) Phys. Rev. E 58, R48-R51, 1998.
Kim O. Rasmussen, V. Zharnitsky, I. Mitkov, N. Gronbech-Jensen, Higher
order Shapiro steps in ac-driven Josephson junctions,(PDF) Phys. Rev. B 59, 58-61, 1999.
S. K. Turitsyn, A. B. Aceves, C.K.R.T. Jones, V. Zharnitsky, and
V. K. Mezentsev, Hamiltonian averaging in soliton-bearing systems with a
periodically varying dispersion, (PDF) Phys. Rev. E 59, R3843-R3846, 1999.
V. Zharnitsky, E. Grenier, S. K. Turitsyn, C.K.R.T. Jones, Jan
S. Hesthaven, Ground states of the dispersion managed nonlinear Schroedinger
equation, (PDF) Phys. Rev. E 62, November 2000.
V. Zharnitsky, E. Grenier, C.K.R.T. Jones, S. K. Turitsyn, Stabilizing effects of dispersion management, (PDF) Physica D 152-3, 794-817, 2001 (volume dedicated to Vladimir Zakharov on the occasion of his 60-th birthday)
V. Zharnitsky, Averaging for split-step scheme, (PDF) Nonlinearity 16, 1359-1366, 2003.
D.E. Pelinovsky, V. Zharnitsky, Averaging of dispersion managed solitons: existence, stability and propagation, (PDF) SIAM Jour. Appl. Math. 63, 745-776, 2003.
R. Jackson, C.K.R.T. Jones, V. Zharnitsky, Critical points in the
averaged variational principle for Dispersion Managed Soliton, (PDF) Physica D 190, Issues 1-2, March 15, 63-77, 2004.
J.T. Moeser, C.K.R.T. Jones, V. Zharnitsky, Stable pulse
solutions for the nonlinear Schroedinger equation with higher order dispersion management, (PDF), SIAM Jour. Math. Analysis. 35, Number 6, 1486-1511. 2004
D.E. Pelinovsky, P.G. Kevrekidis, D.J. Frantzeskakis and V. Zharnitsky, Averaging for solitons with nonlinearity management revisited,( PDF) Phys. Rev. E 70, 2004.
M.I. Weinstein, V. Zharnitsky, A reaction dispersion system and Raman interactions, (PDF), SIAM Jour. Math. Analysis., 36, Number 6, 1742-1771. 2005.
M. Kunze, J. Moeser, V. Zharnitsky, Ground states for the higher order dispersion managed NLS equation in the absence of averaged dispersion,
(PDF), Jour. Diff. Eq. 209, 77-100,
V. Zharnitsky, D.E. Pelinovsky, Averaging of nonlinearity managed pulses,( PDF) Chaos 15 (3): Art. No. 037105, 2005.
D. Hundertmark, V. Zharnitsky, On sharp Strichartz inequalities in low dimensions,( PDF) International Mathematics Research Notices, Vol. 2006, Art. ID 34080, 2006.
D. Hundertmark, Young-Ran Lee, Tobias Ried, Vadim Zharnitsky, Dispersion managed solitons in the presence of saturated nonlinearity,
( PDF), Physica D , 356--357, 65-69 2017.
D. Hundertmark, Young-Ran Lee, Tobias Ried, Vadim Zharnitsky, Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearity,
( PDF), Journal of Differential Equations , 2017.
C.E. Wayne, V. Zharnitsky, Critical points in Strichartz functional,
( PDF), Experimental Mathematics , 1-23, 2018.
M. Arnold, V. Zharnitsky, Cyclic evasion in the three bug problem, ( PDF ),
The American Mathematical Monthly, Vol. 122, No. 04 (April 2015), pp. 377-380. 2015.
M. Arnold, M. Golich, A. Grim, L. Vargas, V. Zharnistky, Square and bow-tie configurations in the cyclic evasion problem,
(PDF), Nonlinearity Vol 30, Number 5. , 2017.