7 research outputs found
On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation
The Conte-Musette method has been modified for the search of only elliptic
solutions to systems of differential equations. A key idea of this a priory
restriction is to simplify calculations by means of the use of a few Laurent
series solutions instead of one and the use of the residue theorem. The
application of our approach to the quintic complex one-dimensional
Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave
form. We also find restrictions on coefficients, which are necessary conditions
for the existence of elliptic solutions for the CGLE5. Using the investigation
of the CGLE5 as an example, we demonstrate that to find elliptic solutions the
analysis of a system of differential equations is more preferable than the
analysis of the equivalent single differential equation.Comment: LaTeX, 21 page
ΠΠ²ΠΎΡΠΊΠΎ-ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΌΠ΅ΡΠΎΠΌΠΎΡΡΠ½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ Π°Π²ΡΠΎΠ½ΠΎΠΌΠ½ΡΡ Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΡΡ Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΡΡ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ
The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ Π·Π°Π΄Π°ΡΠ° ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΠΈ ΠΊΠ»Π°ΡΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ ΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΡΡ
Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ. ΠΠΏΠΈΡΡΠ²Π°Π΅ΡΡΡ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΠΉ ΠΌΠ΅ΡΠΎΠ΄, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΠΉ Π½Π°Ρ
ΠΎΠ΄ΠΈΡΡ Π»ΡΠ±ΠΎΠ΅ ΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ Π°Π²ΡΠΎΠ½ΠΎΠΌΠ½ΠΎΠ³ΠΎ Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ³ΠΎ ΠΎΠ±ΡΠΊΠ½ΠΎΠ²Π΅Π½Π½ΠΎΠ³ΠΎ Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ. ΠΠ΅ΡΠΎΠ΄ Π½Π΅ ΡΡΠ΅Π±ΡΠ΅Ρ ΠΈΠ½ΡΠ΅Π³ΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ Π΄ΠΎΠΏΠΎΠ»Π½ΠΈΡΠ΅Π»ΡΠ½ΡΡ
Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ. ΠΠΎΠ»ΡΡΠΎΠ΅ Π²Π½ΠΈΠΌΠ°Π½ΠΈΠ΅ ΡΠ΄Π΅Π»ΡΠ΅ΡΡΡ ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΠ΅ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Ρ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΈΠΌΠΈ ΠΏΠΎΠ»ΡΡΠ°ΠΌΠΈ Π² ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΠΎΠ³ΡΠ°ΠΌΠΌΠ΅ ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΎΠ². Π‘ ΠΏΠΎΠΌΠΎΡΡΡ Π΄Π°Π½Π½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Π° Π½Π°ΠΉΠ΄Π΅Π½ ΡΠ²Π½ΡΠΉ Π²ΠΈΠ΄ Π²ΡΠ΅Ρ
ΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π΄ΠΎ ΡΠ΅ΡΠ²Π΅ΡΡΠΎΠ³ΠΎ ΠΏΠΎΡΡΠ΄ΠΊΠ° Π²ΠΊΠ»ΡΡΠΈΡΠ΅Π»ΡΠ½ΠΎ Π΄Π»Ρ ΠΎΠ±ΡΠΊΠ½ΠΎΠ²Π΅Π½Π½ΠΎΠ³ΠΎ Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ, ΠΈΠΌΠ΅ΡΡΠ΅Π³ΠΎ ΡΡΠ΄ ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΏΡΠΈΠ»ΠΎΠΆΠ΅Π½ΠΈΠΉ. Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΡΠΉ ΠΌΠ΅ΡΠΎΠ΄ Π΄ΠΎΠΏΡΡΠΊΠ°Π΅Ρ Π΅ΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠ΅ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ Π½Π° ΡΠ»ΡΡΠ°ΠΉ ΡΠΈΡΡΠ΅ΠΌ ΠΎΠ±ΡΠΊΠ½ΠΎΠ²Π΅Π½Π½ΡΡ
Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ
ΠΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΡΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΡΠ°Π²Π½ΠΎΠ²Π΅ΡΠ½ΡΡ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΉ ΡΠΎΡΠ΅ΡΠ½ΡΡ Π²ΠΈΡ ΡΠ΅ΠΉ Π½Π° ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ
The problem of constructing and classifying stationary and translating configurations of point vortices with an arbitrary choice of circulations is studied. The polynomial method enabling one to find any such configuration is described in detail. Stationary configurations for vortex systems with circulations Ξ, β¡Πare classified in the case of integer Β΅. New configurations are obtained.Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ Π²ΠΎΠΏΡΠΎΡ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΠΈ ΠΊΠ»Π°ΡΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ ΡΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΈ ΡΠ°Π²Π½ΠΎΠΌΠ΅ΡΠ½ΠΎ Π΄Π²ΠΈΠΆΡΡΠΈΡ
ΡΡ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΉ ΡΠΎΡΠ΅ΡΠ½ΡΡ
Π²ΠΈΡ
ΡΠ΅ΠΉ Π½Π° ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ ΠΏΡΠΈ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ»ΡΠ½ΠΎΠΌ Π²ΡΠ±ΠΎΡΠ΅ ΠΈΠ½ΡΠ΅Π½ΡΠΈΠ²Π½ΠΎΡΡΠ΅ΠΉ Π²ΠΈΡ
ΡΠ΅ΠΉ. ΠΠ°Π΅ΡΡΡ Π΄Π΅ΡΠ°Π»ΡΠ½ΠΎΠ΅ ΠΎΠΏΠΈΡΠ°Π½ΠΈΠ΅ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Π°, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠ΅Π³ΠΎ Π½Π°Ρ
ΠΎΠ΄ΠΈΡΡ Π»ΡΠ±ΡΡ ΡΠ°ΠΊΡΡ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΡ. ΠΡΠΎΠ²ΠΎΠ΄ΠΈΡΡΡ ΠΊΠ»Π°ΡΡΠΈΡΠΈΠΊΠ°ΡΠΈΡ ΡΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΉ Π΄Π»Ρ Π²ΠΈΡ
ΡΠ΅ΠΉ Ρ ΠΈΠ½ΡΠ΅Π½ΡΠΈΠ²Π½ΠΎΡΡΡΠΌΠΈ Ξ, β¡ΠпΡΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΈ, ΡΡΠΎ Β΅ β ΡΠ΅Π»ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ, Π° ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²ΠΎ Π²ΠΈΡ
ΡΠ΅ΠΉ Π½Π΅ ΠΏΡΠ΅Π²ΡΡΠ°Π΅Ρ Π΄Π΅ΡΡΡΠΈ. ΠΠΎΠ»ΡΡΠ΅Π½Ρ Π½ΠΎΠ²ΡΠ΅ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΈ
Polynomial Method for Constructing Equilibrium Configurations of Point Vortices in the Plane
The problem of constructing and classifying stationary and translating configurations of point vortices with an arbitrary choice of circulations is studied. The polynomial method enabling one to find any such configuration is described in detail. Stationary configurations for vortex systems with circulations Ξ, β¡Πare classified in the case of integer Β΅. New configurations are obtained.</p
Doubly Periodic Meromorphic Solutions of Autonomous Nonlinear Differential Equations
The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations
Polynomial Method for Constructing Equilibrium Configurations of Point Vortices in the Plane
The problem of constructing and classifying stationary and translating configurations of point vortices with an arbitrary choice of circulations is studied. The polynomial method enabling one to find any such configuration is described in detail. Stationary configurations for vortex systems with circulations Ξ, β¡Πare classified in the case of integer Β΅. New configurations are obtained