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July  2020, 40(7): 4113-4130. doi: 10.3934/dcds.2020174

Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

Ola I. H. Maehlen

Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received  February 2019 Revised  January 2020 Published  April 2020

Fund Project: The author acknowledges the support from grant no. 250070 from the Research Council of Norway

We prove existence of solitary-wave solutions to the equation
$ \begin{equation*} u_t+ (Lu - n(u))_x = 0\,, \end{equation*} $
for weak assumptions on the dispersion
$ L $
 and the nonlinearity
$ n $
. The symbol
$ m $
 of the Fourier multiplier
$ L $
 is allowed to be of low positive order (
$ s > 0 $
), while
$ n $
 need only be locally Lipschitz and asymptotically homogeneous at zero. We shall discover such solutions in Sobolev spaces contained in
$ H^{1+s} $
.
Keywords: Solitary waves, weak dispersion, capillary Whitham equation, water waves, concentration compactness.
Mathematics Subject Classification: Primary: 35A01; Secondary: 35A15, 35Q53, 76B03, 76B15.
Citation: Ola I. H. Maehlen. Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4113-4130. doi: 10.3934/dcds.2020174
References:
[1]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, in Applied Analysis, Contemp. Math., 221, Amer. Math. Soc., Providence, RI, 1999, 1–29. doi: 10.1090/conm/221/03116.

[2]

J. P. Albert, J. L. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Roy. Soc. London Ser. A, 453 (1997), 1233-1260.  doi: 10.1098/rspa.1997.0068.

[3]

M. N. Arnesen, Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510.  doi: 10.3934/dcds.2016.36.3483.

[4]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.

[5]

G. Bourdaud, M. Moussai and W. Sickel, Composition operators acting on Besov spaces on the real line, Ann. Mat. Pura Appl. (4), 193 (2014), 1519-1554.  doi: 10.1007/s10231-013-0342-x.

[6]

H. Chen and J. L. Bona, Periodic traveling-wave solutions of nonlinear dispersive evolution equations, Discrete Contin. Dyn. Syst., 33 (2013), 4841-4873.  doi: 10.3934/dcds.2013.33.4841.

[7]

M. Ehrnström, M. D. Groves and E. Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 2903-2936.  doi: 10.1088/0951-7715/25/10/2903.

[8]

M. Ehrnström, M. A. Johnson, O. I. H. Maehlen and F. Remonato, On the bifurcation diagram of the capillary–gravity Whitham equation, Water Waves, 1 (2019), 275-313.  doi: 10.1007/s42286-019-00019-4.

[9]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.

[10]

M. D. Groves and E. Wahlén, On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water, J. Math. Fluid Mech., 13 (2011), 593-627.  doi: 10.1007/s00021-010-0034-x.

[11]

M. A. Johnson and J. D. Wright, Generalized solitary waves in the gravity-capillary Whitham equation, Studies in Applied Mathematics, 144 (2020), 102-130.  doi: 10.1111/sapm.12288.

[12]

D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.

[13]

F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations. I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.

[14]

F. Linares, D. Pilod and J.-C. Saut, Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Differential Equations, 20 (2015), 835-858. 

[15]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[16]

M. Moussai, Composition operators on Besov spaces in the limiting case $s=1+1/p$, Studia Math., 241 (2018), 1-15.  doi: 10.4064/sm8136-4-2017.

[17]

S. M. Sun, Non-existence of truly solitary waves in water with small surface tension, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2191-2228.  doi: 10.1098/rspa.1999.0399.

[18]

M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173.  doi: 10.1080/03605308708820522.

[19]

L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Differential Equations, 188 (2003), 1-32.  doi: 10.1016/S0022-0396(02)00061-X.

show all references

References:
[1]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, in Applied Analysis, Contemp. Math., 221, Amer. Math. Soc., Providence, RI, 1999, 1–29. doi: 10.1090/conm/221/03116.

[2]

J. P. Albert, J. L. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Roy. Soc. London Ser. A, 453 (1997), 1233-1260.  doi: 10.1098/rspa.1997.0068.

[3]

M. N. Arnesen, Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510.  doi: 10.3934/dcds.2016.36.3483.

[4]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.

[5]

G. Bourdaud, M. Moussai and W. Sickel, Composition operators acting on Besov spaces on the real line, Ann. Mat. Pura Appl. (4), 193 (2014), 1519-1554.  doi: 10.1007/s10231-013-0342-x.

[6]

H. Chen and J. L. Bona, Periodic traveling-wave solutions of nonlinear dispersive evolution equations, Discrete Contin. Dyn. Syst., 33 (2013), 4841-4873.  doi: 10.3934/dcds.2013.33.4841.

[7]

M. Ehrnström, M. D. Groves and E. Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 2903-2936.  doi: 10.1088/0951-7715/25/10/2903.

[8]

M. Ehrnström, M. A. Johnson, O. I. H. Maehlen and F. Remonato, On the bifurcation diagram of the capillary–gravity Whitham equation, Water Waves, 1 (2019), 275-313.  doi: 10.1007/s42286-019-00019-4.

[9]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.

[10]

M. D. Groves and E. Wahlén, On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water, J. Math. Fluid Mech., 13 (2011), 593-627.  doi: 10.1007/s00021-010-0034-x.

[11]

M. A. Johnson and J. D. Wright, Generalized solitary waves in the gravity-capillary Whitham equation, Studies in Applied Mathematics, 144 (2020), 102-130.  doi: 10.1111/sapm.12288.

[12]

D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.

[13]

F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations. I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.

[14]

F. Linares, D. Pilod and J.-C. Saut, Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Differential Equations, 20 (2015), 835-858. 

[15]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[16]

M. Moussai, Composition operators on Besov spaces in the limiting case $s=1+1/p$, Studia Math., 241 (2018), 1-15.  doi: 10.4064/sm8136-4-2017.

[17]

S. M. Sun, Non-existence of truly solitary waves in water with small surface tension, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2191-2228.  doi: 10.1098/rspa.1999.0399.

[18]

M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173.  doi: 10.1080/03605308708820522.

[19]

L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Differential Equations, 188 (2003), 1-32.  doi: 10.1016/S0022-0396(02)00061-X.

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