zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global existence for a new periodic integrable equation. (English) Zbl 1033.35121
Summary: We establish the local well-posedness for a new periodic integrable equation $$\alignat2 &u_t- u_{txx}+ 4uu_x= 3u_x u_{xx}+ uu_{xxx},\quad &&t> 0,\ x\in\Bbb R,\\ & u(0,x)= u_0(x),\quad &&x\in\Bbb R,\\ & u(t, x+1)= u(t,x),\quad &&t\ge 0,\ x\in\bbfR.\endalignat$$ We show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time.

35Q58Other completely integrable PDE (MSC2000)
35B40Asymptotic behavior of solutions of PDE
35B10Periodic solutions of PDE
35B30Dependence of solutions of PDE on initial and boundary data, parameters
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
Full Text: DOI
[1] Bona, J. L.; Smith, R.: The initial-value problem for the Korteweg--de Vries equation. Philos. trans. Roy. soc. London 278, 555-601 (1975) · Zbl 0306.35027
[2] Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solitons. Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521
[3] Constantin, A.: The Cauchy problem for the periodic Camassa--Holm equation. J. differential equations 141, 218-235 (1997) · Zbl 0889.35022
[4] Constantin, A.; Escher, J.: Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. pure appl. Math. 51, 475-504 (1998) · Zbl 0934.35153
[5] Constantin, A.; Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta math. 181, 229-243 (1998) · Zbl 0923.76025
[6] Constantin, A.; Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 79-91 (2000) · Zbl 0954.35136
[7] Constantin, A.; Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. phys. A 35, R51-R79 (2002) · Zbl 1039.37068
[8] Constantin, A.; Mckean, H. P.: A shallow water equation on the circle. Comm. pure appl. Math. 52, 949-982 (1999) · Zbl 0940.35177
[9] A. Degasperis, D.D. Holm, A.N.W. Hone, A new integral equation with Peakon solutions, in: Theoretical and Mathematical Physics, NEEDS 2001 Proceedings, to appear
[10] Degasperis, A.; Procesi, M.: Asymptotic integrability. Symmetry and perturbation theory, 23-37 (1999) · Zbl 0963.35167
[11] F. Gesztesy, H. Holden, Algebro-geometric solutions of the Camassa--Holm hierarchy, Rev. Mat. lberoamericana, to appear · Zbl 1153.37427
[12] Gesztesy, F.; Holden, H.: Soliton equations and their algebro-geometric solutions. (2003) · Zbl 1061.37056
[13] Fokas, A.; Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47-66 (1981) · Zbl 1194.37114
[14] Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Lecture notes in math. 448, 25-70 (1975)
[15] Kato, T.: On the Korteweg--de Vries equation. Manuscripta math. 28, 89-99 (1979) · Zbl 0415.35070
[16] Kenig, C.; Ponce, G.; Vega, L.: Well-posedness and scattering results for the generalized Korteweg--de Vries equation via the contraction principle. Comm. pure appl. Math. 46, 527-620 (1993) · Zbl 0808.35128
[17] Mckean, H. P.: Integrable systems and algebraic curves. Lecture notes in math. 755, 83-200 (1979)
[18] Misiolek, G.: A shallow water equation as a geoderic flows on the Bott--Virasoro group. J. geom. Phys. 24, 203-208 (1998)
[19] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., to appear · Zbl 1061.35142