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Orbital stability

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Solution to a partial differential equation which remains close to the initial data

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form u ( x , t ) = e − i ω t ϕ ( x ) {\displaystyle u(x,t)=e^{-i\omega t}\phi (x)} is said to be orbitally stable if any solution with the initial data sufficiently close to ϕ ( x ) {\displaystyle \phi (x)} forever remains in a given small neighborhood of the trajectory of e − i ω t ϕ ( x ) . {\displaystyle e^{-i\omega t}\phi (x).}

Formal definition

Formal definition is as follows. Consider the dynamical system

i d u d t = A ( u ) , u ( t ) ∈ X , t ∈ R , {\displaystyle i{\frac {du}{dt}}=A(u),\qquad u(t)\in X,\quad t\in \mathbb {R} ,}

with X {\displaystyle X} a Banach space over C {\displaystyle \mathbb {C} } , and A : X → X {\displaystyle A:X\to X} . We assume that the system is U ( 1 ) {\displaystyle \mathrm {U} (1)} -invariant, so that A ( e i s u ) = e i s A ( u ) {\displaystyle A(e^{is}u)=e^{is}A(u)} for any u ∈ X {\displaystyle u\in X} and any s ∈ R {\displaystyle s\in \mathbb {R} } .

Assume that ω ϕ = A ( ϕ ) {\displaystyle \omega \phi =A(\phi )} , so that u ( t ) = e − i ω t ϕ {\displaystyle u(t)=e^{-i\omega t}\phi } is a solution to the dynamical system. We call such solution a solitary wave.

We say that the solitary wave e − i ω t ϕ {\displaystyle e^{-i\omega t}\phi } is orbitally stable if for any ϵ > 0 {\displaystyle \epsilon >0} there is δ > 0 {\displaystyle \delta >0} such that for any v 0 ∈ X {\displaystyle v_{0}\in X} with ‖ ϕ − v 0 ‖ X < δ {\displaystyle \Vert \phi -v_{0}\Vert _{X}<\delta } there is a solution v ( t ) {\displaystyle v(t)} defined for all t ≥ 0 {\displaystyle t\geq 0} such that v ( 0 ) = v 0 {\displaystyle v(0)=v_{0}} , and such that this solution satisfies

sup t ≥ 0 inf s ∈ R ‖ v ( t ) − e i s ϕ ‖ X < ϵ . {\displaystyle \sup _{t\geq 0}\inf _{s\in \mathbb {R} }\Vert v(t)-e^{is}\phi \Vert _{X}<\epsilon .}

Example

According to , the solitary wave solution e − i ω t ϕ ω ( x ) {\displaystyle e^{-i\omega t}\phi _{\omega }(x)} to the nonlinear Schrödinger equation

i ∂ ∂ t u = − ∂ 2 ∂ x 2 u + g ( | u | 2 ) u , u ( x , t ) ∈ C , x ∈ R , t ∈ R , {\displaystyle i{\frac {\partial }{\partial t}}u=-{\frac {\partial ^{2}}{\partial x^{2}}}u+g\!\left(|u|^{2}\right)u,\qquad u(x,t)\in \mathbb {C} ,\quad x\in \mathbb {R} ,\quad t\in \mathbb {R} ,}

where g {\displaystyle g} is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

d d ω Q ( ϕ ω ) < 0 , {\displaystyle {\frac {d}{d\omega }}Q(\phi _{\omega })<0,}

where

Q ( u ) = 1 2 ∫ R | u ( x , t ) | 2 d x {\displaystyle Q(u)={\frac {1}{2}}\int _{\mathbb {R} }|u(x,t)|^{2}\,dx}

is the charge of the solution u ( x , t ) {\displaystyle u(x,t)} , which is conserved in time (at least if the solution u ( x , t ) {\displaystyle u(x,t)} is sufficiently smooth).

It was also shown, that if d d ω Q ( ω ) < 0 {\textstyle {\frac {d}{d\omega }}Q(\omega )<0} at a particular value of ω {\displaystyle \omega } , then the solitary wave e − i ω t ϕ ω ( x ) {\displaystyle e^{-i\omega t}\phi _{\omega }(x)} is Lyapunov stable, with the Lyapunov function given by L ( u ) = E ( u ) − ω Q ( u ) + Γ ( Q ( u ) − Q ( ϕ ω ) ) 2 {\displaystyle L(u)=E(u)-\omega Q(u)+\Gamma (Q(u)-Q(\phi _{\omega }))^{2}} , where E ( u ) = 1 2 ∫ R ( | ∂ u ∂ x | 2 + G ( | u | 2 ) ) d x {\displaystyle E(u)={\frac {1}{2}}\int _{\mathbb {R} }\left(\left|{\frac {\partial u}{\partial x}}\right|^{2}+G\!\left(|u|^{2}\right)\right)dx} is the energy of a solution u ( x , t ) {\displaystyle u(x,t)} , with G ( y ) = ∫ 0 y g ( z ) d z {\textstyle G(y)=\int _{0}^{y}g(z)\,dz} the antiderivative of g {\displaystyle g} , as long as the constant Γ > 0 {\displaystyle \Gamma >0} is chosen sufficiently large.

See also

References

  1. Manoussos Grillakis; Jalal Shatah & Walter Strauss (1990). "Stability theory of solitary waves in the presence of symmetry". J. Funct. Anal. 94 (2): 308–348. doi:10.1016/0022-1236(90)90016-E.
  2. T. Cazenave & P.-L. Lions (1982). "Orbital stability of standing waves for some nonlinear Schrödinger equations". Comm. Math. Phys. 85 (4): 549–561. Bibcode:1982CMaPh..85..549C. doi:10.1007/BF01403504. S2CID 120472894.
  3. Jerry Bona; Panagiotis Souganidis & Walter Strauss (1987). "Stability and instability of solitary waves of Korteweg-de Vries type". Proceedings of the Royal Society A. 411 (1841): 395–412. Bibcode:1987RSPSA.411..395B. doi:10.1098/rspa.1987.0073. S2CID 120894859.
  4. Michael I. Weinstein (1986). "Lyapunov stability of ground states of nonlinear dispersive evolution equations". Comm. Pure Appl. Math. 39 (1): 51–67. doi:10.1002/cpa.3160390103.
  5. Richard Jordan & Bruce Turkington (2001). "Statistical equilibrium theories for the nonlinear Schrödinger equation". Advances in Wave Interaction and Turbulence. Contemp. Math. Vol. 283. South Hadley, MA. pp. 27–39. doi:10.1090/conm/283/04711. ISBN 9780821827147.{{cite book}}: CS1 maint: location missing publisher (link)
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