Existence of solitary-wave solutions to nonlocal equations

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	R. R. Coifman and Y. Meyer, Au-delà des Opérateurs Pseudo-différentiels, Astérisque, 1978.
[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.
[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.
[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.
[10]	R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Springer-Verlag, New York, 1995.
[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.

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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.
[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.
[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.
[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.
[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.
[6]	R. R. Coifman and Y. Meyer, Au-delà des Opérateurs Pseudo-différentiels, Astérisque, 1978.
[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.
[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.
[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.
[10]	R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Springer-Verlag, New York, 1995.
[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.

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