On the existence of traveling waves for monomer chains with pre-compression Atanas Stefanov Department of Mathematics The University of Kansas 2nd LENCOS conference July 11th 2012, Seville
The Fermi-Pasta-Ulam (FPU) problem - I Infinite dimensional Hamiltonian system with ∞ q 2 ˙ j � H = 2 + V ( q j + 1 − q j ) j = −∞ Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V ( r ) = ar 2 + br 3 cubic FPU V ( r ) = c (( 1 + r ) − 12 − 2 ( 1 + r ) − 6 + 1 ) Lennard-Jones V ( r ) = α ( e β r − β r − 1 ) Toda
The Fermi-Pasta-Ulam (FPU) problem - I Infinite dimensional Hamiltonian system with ∞ q 2 ˙ j � H = 2 + V ( q j + 1 − q j ) j = −∞ Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V ( r ) = ar 2 + br 3 cubic FPU V ( r ) = c (( 1 + r ) − 12 − 2 ( 1 + r ) − 6 + 1 ) Lennard-Jones V ( r ) = α ( e β r − β r − 1 ) Toda
The Fermi-Pasta-Ulam (FPU) problem - I Infinite dimensional Hamiltonian system with ∞ q 2 ˙ j � H = 2 + V ( q j + 1 − q j ) j = −∞ Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V ( r ) = ar 2 + br 3 cubic FPU V ( r ) = c (( 1 + r ) − 12 − 2 ( 1 + r ) − 6 + 1 ) Lennard-Jones V ( r ) = α ( e β r − β r − 1 ) Toda
The Fermi-Pasta-Ulam (FPU) problem - I Infinite dimensional Hamiltonian system with ∞ q 2 ˙ j � H = 2 + V ( q j + 1 − q j ) j = −∞ Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V ( r ) = ar 2 + br 3 cubic FPU V ( r ) = c (( 1 + r ) − 12 − 2 ( 1 + r ) − 6 + 1 ) Lennard-Jones V ( r ) = α ( e β r − β r − 1 ) Toda
The Fermi-Pasta-Ulam (FPU) problem - II The equation of motion is q j ( t ) = V ′ ( q j + 1 − q j ) − V ′ ( q j − q j − 1 ) ¨ (1) In terms of the displacement functions r j = q j + 1 − q j , r j ( t ) = V ′ ( r j + 1 ) − 2 V ′ ( r j ) + V ′ ( r j − 1 ) ¨ (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.
The Fermi-Pasta-Ulam (FPU) problem - II The equation of motion is q j ( t ) = V ′ ( q j + 1 − q j ) − V ′ ( q j − q j − 1 ) ¨ (1) In terms of the displacement functions r j = q j + 1 − q j , r j ( t ) = V ′ ( r j + 1 ) − 2 V ′ ( r j ) + V ′ ( r j − 1 ) ¨ (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.
The Fermi-Pasta-Ulam (FPU) problem - II The equation of motion is q j ( t ) = V ′ ( q j + 1 − q j ) − V ′ ( q j − q j − 1 ) ¨ (1) In terms of the displacement functions r j = q j + 1 − q j , r j ( t ) = V ′ ( r j + 1 ) − 2 V ′ ( r j ) + V ′ ( r j − 1 ) ¨ (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.
The Fermi-Pasta-Ulam (FPU) problem - II The equation of motion is q j ( t ) = V ′ ( q j + 1 − q j ) − V ′ ( q j − q j − 1 ) ¨ (1) In terms of the displacement functions r j = q j + 1 − q j , r j ( t ) = V ′ ( r j + 1 ) − 2 V ′ ( r j ) + V ′ ( r j − 1 ) ¨ (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.
The Fermi-Pasta-Ulam (FPU) problem - II The equation of motion is q j ( t ) = V ′ ( q j + 1 − q j ) − V ′ ( q j − q j − 1 ) ¨ (1) In terms of the displacement functions r j = q j + 1 − q j , r j ( t ) = V ′ ( r j + 1 ) − 2 V ′ ( r j ) + V ′ ( r j − 1 ) ¨ (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.
Traveling waves for FPU systems Substituting r ( t , j ) = r c ( j − ct ) yields c 2 r ′′ V ′ ( r c ( x + 1 )) − 2 V ′ ( r c ( x )) + V ′ ( r c ( x − 1 )) c ( x ) = ∆ disc . V ′ ( r c ( · ))( x ) = Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞ ? Are the solutions r bell-shaped, i.e. positive, even and decaying in ( 0 , ∞ ) ?
Traveling waves for FPU systems Substituting r ( t , j ) = r c ( j − ct ) yields c 2 r ′′ V ′ ( r c ( x + 1 )) − 2 V ′ ( r c ( x )) + V ′ ( r c ( x − 1 )) c ( x ) = ∆ disc . V ′ ( r c ( · ))( x ) = Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞ ? Are the solutions r bell-shaped, i.e. positive, even and decaying in ( 0 , ∞ ) ?
Traveling waves for FPU systems Substituting r ( t , j ) = r c ( j − ct ) yields c 2 r ′′ V ′ ( r c ( x + 1 )) − 2 V ′ ( r c ( x )) + V ′ ( r c ( x − 1 )) c ( x ) = ∆ disc . V ′ ( r c ( · ))( x ) = Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞ ? Are the solutions r bell-shaped, i.e. positive, even and decaying in ( 0 , ∞ ) ?
Traveling waves for FPU systems Substituting r ( t , j ) = r c ( j − ct ) yields c 2 r ′′ V ′ ( r c ( x + 1 )) − 2 V ′ ( r c ( x )) + V ′ ( r c ( x − 1 )) c ( x ) = ∆ disc . V ′ ( r c ( · ))( x ) = Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞ ? Are the solutions r bell-shaped, i.e. positive, even and decaying in ( 0 , ∞ ) ?
Traveling waves for FPU systems Substituting r ( t , j ) = r c ( j − ct ) yields c 2 r ′′ V ′ ( r c ( x + 1 )) − 2 V ′ ( r c ( x )) + V ′ ( r c ( x − 1 )) c ( x ) = ∆ disc . V ′ ( r c ( · ))( x ) = Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞ ? Are the solutions r bell-shaped, i.e. positive, even and decaying in ( 0 , ∞ ) ?
Previous results Toda - explicit formulas in the “Toda” case, due to the integrability. Friesecke-Wattis [Comm. Math. Phys.’94] - existence of waves with prescribed average potential energy � ∞ K = V ( u ( x − 1 ) − u ( x )) dx −∞ Also work of Smets-Willem, [J. Funct. Anal.’97] clarifying FW about the set of speeds. Friesecke-Pego [Nonlinearity 99, 02, 04,04], Existence and stability for “slightly supersonic speeds”, i.e. � V ′′ ( 0 ) . c s + ǫ > | c | ≥ c s :=
Previous results Toda - explicit formulas in the “Toda” case, due to the integrability. Friesecke-Wattis [Comm. Math. Phys.’94] - existence of waves with prescribed average potential energy � ∞ K = V ( u ( x − 1 ) − u ( x )) dx −∞ Also work of Smets-Willem, [J. Funct. Anal.’97] clarifying FW about the set of speeds. Friesecke-Pego [Nonlinearity 99, 02, 04,04], Existence and stability for “slightly supersonic speeds”, i.e. � V ′′ ( 0 ) . c s + ǫ > | c | ≥ c s :=
Previous results Toda - explicit formulas in the “Toda” case, due to the integrability. Friesecke-Wattis [Comm. Math. Phys.’94] - existence of waves with prescribed average potential energy � ∞ K = V ( u ( x − 1 ) − u ( x )) dx −∞ Also work of Smets-Willem, [J. Funct. Anal.’97] clarifying FW about the set of speeds. Friesecke-Pego [Nonlinearity 99, 02, 04,04], Existence and stability for “slightly supersonic speeds”, i.e. � V ′′ ( 0 ) . c s + ǫ > | c | ≥ c s :=
Starosvetsky-Vakakis - PRE’10, existence of periodic waves. More results for dimers and numerics. G. James - J. Nonl. Sci.’12, existence of periodic traveling waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in oscillatory chains with Hertzian interactions.
Starosvetsky-Vakakis - PRE’10, existence of periodic waves. More results for dimers and numerics. G. James - J. Nonl. Sci.’12, existence of periodic traveling waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in oscillatory chains with Hertzian interactions.
Starosvetsky-Vakakis - PRE’10, existence of periodic waves. More results for dimers and numerics. G. James - J. Nonl. Sci.’12, existence of periodic traveling waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in oscillatory chains with Hertzian interactions.
Starosvetsky-Vakakis - PRE’10, existence of periodic waves. More results for dimers and numerics. G. James - J. Nonl. Sci.’12, existence of periodic traveling waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in oscillatory chains with Hertzian interactions.
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