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July  2016, 36(7): 3483-3510. doi: 10.3934/dcds.2016.36.3483

Existence of solitary-wave solutions to nonlocal equations

Mathias Nikolai Arnesen 1,

1. 

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

Received  March 2015 Revised  November 2015 Published  March 2016

We prove existence and conditional energetic stability of solitary-wave solutions for the two classes of pseudodifferential equations \begin{equation*} u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 \end{equation*} and \begin{equation*} u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, \end{equation*} where $f$ is a nonlinear term, typically of the form $c|u|^p$ or $cu|u|^{p-1}$, and $L$ is a Fourier multiplier operator of positive order. The former class includes for instance the Whitham equation with capillary effects and the generalized Korteweg-de Vries equation, and the latter the Benjamin-Bona-Mahony equation. Existence and conditional energetic stability results have earlier been established using the method of concentration-compactness for a class of operators with symbol of order $s\geq 1$. We extend these results to symbols of order $0 < s < 1$, thereby improving upon the results for general operators with symbol of order $s\geq 1$ by enlarging both the class of linear operators and nonlinearities admitting existence of solitary waves. Instead of using abstract operator theory, the new results are obtained by direct calculations involving the nonlocal operator $L$, something that gives us the bounds and estimates needed for the method of concentration-compactness.
Keywords: concentration-compactness., Solitary waves, capillary Whitham.
Mathematics Subject Classification: Primary: 35Q53, 35A15; Secondary: 76B1.
Citation: Mathias Nikolai Arnesen. Existence of solitary-wave solutions to nonlocal equations. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3483-3510. doi: 10.3934/dcds.2016.36.3483
References:
[1]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Contemp. Math., 221 (1999), 1-29. doi: 10.1090/conm/221/03116.  Google Scholar

[2]

J. P. Albert, J. L. Bona and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Phys. D, 24 (1987), 343-366. doi: 10.1016/0167-2789(87)90084-4.  Google Scholar

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

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J. Angulo Pava, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/surv/156.  Google Scholar

[5]

H. Chen and J. L. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Differential Equations, 3 (1998), 51-84.  Google Scholar

[6]

R. R. Coifman and Y. Meyer, Au-delà des Opérateurs Pseudo-différentiels, Astérisque, 1978.  Google Scholar

[7]

M. Ehrnström, J. Escher and L. Pei, A note on the local well-posedness for the Whitham equation, Elliptic and Parabolic Equations, Springer Proceedings in Mathematics & Statistics, 119 (2015), 63-75. doi: 10.1007/978-3-319-12547-3_3.  Google Scholar

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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.  Google Scholar

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M. Ehrnström and H. Kalisch, Global bifurcation for the Whitham equation, Math. Model. Nat. Phenom., 8 (2013), 13-30. doi: 10.1051/mmnp/20138502.  Google Scholar

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

[11]

B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53 (2012), 083702, 15pp. doi: 10.1063/1.4746806.  Google Scholar

[12]

V. M. Hur and M. A. Johnson, Modulational instability in the Whitham equation for water waves, Studies in Applied Mathematics, 134 (2015), 120-143. doi: 10.1111/sapm.12061.  Google Scholar

[13]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Physica D, 295/296 (2015), 46-65. doi: 10.1016/j.physd.2014.12.004.  Google Scholar

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D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equations, Kinet. Relat. Models, 6 (2013), 989-1009. doi: 10.3934/krm.2013.6.989.  Google Scholar

[15]

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.  Google Scholar

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincaré Anal. Non Linéare, 1 (1984), 109-145.  Google Scholar

[17]

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

[18]

E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Springer-Verlag, New York, 1995.  Google Scholar

[19]

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

show all references

References:
[1]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Contemp. Math., 221 (1999), 1-29. doi: 10.1090/conm/221/03116.  Google Scholar

[2]

J. P. Albert, J. L. Bona and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Phys. D, 24 (1987), 343-366. doi: 10.1016/0167-2789(87)90084-4.  Google Scholar

[3]

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

[4]

J. Angulo Pava, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/surv/156.  Google Scholar

[5]

H. Chen and J. L. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Differential Equations, 3 (1998), 51-84.  Google Scholar

[6]

R. R. Coifman and Y. Meyer, Au-delà des Opérateurs Pseudo-différentiels, Astérisque, 1978.  Google Scholar

[7]

M. Ehrnström, J. Escher and L. Pei, A note on the local well-posedness for the Whitham equation, Elliptic and Parabolic Equations, Springer Proceedings in Mathematics & Statistics, 119 (2015), 63-75. doi: 10.1007/978-3-319-12547-3_3.  Google Scholar

[8]

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.  Google Scholar

[9]

M. Ehrnström and H. Kalisch, Global bifurcation for the Whitham equation, Math. Model. Nat. Phenom., 8 (2013), 13-30. doi: 10.1051/mmnp/20138502.  Google Scholar

[10]

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

[11]

B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53 (2012), 083702, 15pp. doi: 10.1063/1.4746806.  Google Scholar

[12]

V. M. Hur and M. A. Johnson, Modulational instability in the Whitham equation for water waves, Studies in Applied Mathematics, 134 (2015), 120-143. doi: 10.1111/sapm.12061.  Google Scholar

[13]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Physica D, 295/296 (2015), 46-65. doi: 10.1016/j.physd.2014.12.004.  Google Scholar

[14]

D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equations, Kinet. Relat. Models, 6 (2013), 989-1009. doi: 10.3934/krm.2013.6.989.  Google Scholar

[15]

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.  Google Scholar

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincaré Anal. Non Linéare, 1 (1984), 109-145.  Google Scholar

[17]

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

[18]

E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Springer-Verlag, New York, 1995.  Google Scholar

[19]

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

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