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July  2019, 39(7): 3671-3716. doi: 10.3934/dcds.2019150

Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations

Shengfu Deng 1,2,

1. 

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China
2. 

Department of Mathematics, lingnan Normal University, Zhanjiang, Guangdong 524048, China

Received  October 2017 Published  April 2019

The KdV-KdV system of Boussinesq equations belongs to the class of Boussinesq equations modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid. It has been numerically shown that this system possesses solutions with two humps which tend to a periodic solution with much smaller amplitude at infinity (called generalized two-hump wave solutions). This paper presents the first rigorous proof. The traveling form of this system can be formulated into a dynamical system with dimension 4. The classical dynamical system approach provides the existence of a solution with an exponentially decaying part and an oscillatory part (small-amplitude periodic solution) at positive infinity, which has a single hump at the origin and is reversible near negative infinity if some free constants, such as the amplitude and the phase shit of the periodic solution, are activated. This eventually yields a generalized two-hump wave solution. The method here can be applied to obtain generalized $2^k$-hump wave solutions for any positive integer $k$.

Keywords: KdV-KdV system, Boussinesq equations, multi-hump, solitary wave solutions, homoclinic solutions, periodic solutions.
Mathematics Subject Classification: Primary: 34C37, 76B25; Secondary: 35B32, 37C29.
Citation: Shengfu Deng. Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3671-3716. doi: 10.3934/dcds.2019150
References:
[1]

A. Ali and H. Kalisch, Mechanical balance laws for Boussinesq models of surface water waves, J. Nonlinear Sci., 22 (2012), 371-398.  doi: 10.1007/s00332-011-9121-2.  Google Scholar

[2]

D. C. Antonopoulos, V. A. Dougalis and D. E. Mitsotakis, Numerical solution of Boussinesq systems of the Bona-Smith family, Appl. Numer. Math., 60 (2010), 633-669.  doi: 10.1016/j.apnum.2009.03.002.  Google Scholar

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J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

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J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

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J. L. Bona, V. A. Dougalis and D. Mitsotakis, Numerical solution of the KdV-KdV systems of Boussinesq equations Ⅰ. Numerical schemes and generalized solitary waves, Math. Comput. Simulation, 74 (2007), 214-228.  doi: 10.1016/j.matcom.2006.10.004.  Google Scholar

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J. L. Bona, V. A. Dougalis and D. Mitsotakis, Numerical solution of Boussinesq systems of KdV-KdV type. Ⅱ. Evolution of radiating solitary waves, Nonlinearity, 21 (2008), 2825-2848.  doi: 10.1088/0951-7715/21/12/006.  Google Scholar

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J. L. Bona, Z. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discrete Contin. Dyn. Syst., 26 (2010), 1121-1139.  doi: 10.3934/dcds.2010.26.1121.  Google Scholar

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M. Chen, Exact traveling-wave solutions to bi-directional wave equations, Int. J. Theor. Phys., 37 (1998), 1547-1567.  doi: 10.1023/A:1026667903256.  Google Scholar

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J. W. Choi, D. S. Lee, S. H. Oh, S. M. Sun and S. I. Whang, Multi-hump solutions of some singularly perturbed equations of KdV type, Discrete Contin. Dyn. Syst., 34 (2014), 5181-5209.  doi: 10.3934/dcds.2014.34.5181.  Google Scholar

[10]

S. Deng and S. M. Sun, Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation, Nonlinearity, 30 (2017), 765-809.  doi: 10.1088/1361-6544/aa525a.  Google Scholar

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V. A. Dougalis and D. E. Mitsotakis, Theory and numerical analysis of Boussinesq systems: A review, in: N. A. Kampanis, V. A. Dougalis and J. A. Ekaterinaris (eds.), Effective Computational Methods in Wave Propagation, CRC Press, 5 (2008), 63–110. doi: 10.1201/9781420010879.ch3.  Google Scholar

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B. Karasozen and G. Simsek, Energy preserving integration of KdV-KdV systems, TWMS J. App. Eng. Math., 2 (2012), 219-227.   Google Scholar

[13]

A. Khare and A. Saxena, Novel PT-invariant solutions for a large number of real nonlinear equations, Phys. Lett. A, 380 (2015), 856-862.  doi: 10.1016/j.physleta.2015.12.007.  Google Scholar

[14]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer-Verlag, Berlin, 2003. Google Scholar

[15]

M. O. Korpusov, Blowup of solutions of nonlinear equations and systems of nonlinear equations in wave theory, Theoret. and Math. Phys., 174 (2013), 307-314.  doi: 10.1007/s11232-013-0028-y.  Google Scholar

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F. Linares, D. Pilod and J.-C. Saut, Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4195-4221.  doi: 10.1137/110828277.  Google Scholar

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E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, with Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Mathematics, Vol. 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar

[18]

A. Mielke, P. Holmes and O. O'Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center, J. Dynam. Differ. Equ., 4 (1992), 95-126.  doi: 10.1007/BF01048157.  Google Scholar

[19]

M. Ming, J.-C. Saut and P. Zhang, Long-time existence of solutions to Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4078-4100.  doi: 10.1137/110834214.  Google Scholar

[20]

T. E. Mogorosi, B. Muatjetjeja and C. M. Khalique, Conservation laws for a generalized coupled Boussinesq system of KdV-KdV type, Springer International Publishing, 117 (2015), 315-321.   Google Scholar

[21]

H. Y. Nguyen and F. Dias, A Boussinesq system for two-way propagation of interfacial waves, Phys. D, 237 (2008), 2365-2389.  doi: 10.1016/j.physd.2008.02.020.  Google Scholar

[22]

J.-C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface waves type Boussinesq systems Ⅱ, SIAM J. Math. Anal., 49 (2017), 2321-2386.  doi: 10.1137/15M1050203.  Google Scholar

[23]

J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl., 97 (2012), 635-662.  doi: 10.1016/j.matpur.2011.09.012.  Google Scholar

[24]

E. V. Yushkov, Blowup in Korteweg-de Vries-type systems, Theoret. Math. Phys., 173 (2012), 197-206.  doi: 10.1007/s11232-012-0129-z.  Google Scholar

[25]

V. C. Zelati and M. Macrì, Multibump solutions homoclinic to periodic orbits of large energy in a centre manifold, Nonlinearity, 18 (2005), 2409-2445.  doi: 10.1088/0951-7715/18/6/001.  Google Scholar

[26]

Y. Zhou, M. Wang and Y. Wang, Periodic wave solutions to coupled KdV equations with variable coefficients, Phys. Lett. A, 308 (2003), 31-36.  doi: 10.1016/S0375-9601(02)01775-9.  Google Scholar

show all references

References:
[1]

A. Ali and H. Kalisch, Mechanical balance laws for Boussinesq models of surface water waves, J. Nonlinear Sci., 22 (2012), 371-398.  doi: 10.1007/s00332-011-9121-2.  Google Scholar

[2]

D. C. Antonopoulos, V. A. Dougalis and D. E. Mitsotakis, Numerical solution of Boussinesq systems of the Bona-Smith family, Appl. Numer. Math., 60 (2010), 633-669.  doi: 10.1016/j.apnum.2009.03.002.  Google Scholar

[3]

J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[4]

J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[5]

J. L. Bona, V. A. Dougalis and D. Mitsotakis, Numerical solution of the KdV-KdV systems of Boussinesq equations Ⅰ. Numerical schemes and generalized solitary waves, Math. Comput. Simulation, 74 (2007), 214-228.  doi: 10.1016/j.matcom.2006.10.004.  Google Scholar

[6]

J. L. Bona, V. A. Dougalis and D. Mitsotakis, Numerical solution of Boussinesq systems of KdV-KdV type. Ⅱ. Evolution of radiating solitary waves, Nonlinearity, 21 (2008), 2825-2848.  doi: 10.1088/0951-7715/21/12/006.  Google Scholar

[7]

J. L. Bona, Z. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discrete Contin. Dyn. Syst., 26 (2010), 1121-1139.  doi: 10.3934/dcds.2010.26.1121.  Google Scholar

[8]

M. Chen, Exact traveling-wave solutions to bi-directional wave equations, Int. J. Theor. Phys., 37 (1998), 1547-1567.  doi: 10.1023/A:1026667903256.  Google Scholar

[9]

J. W. Choi, D. S. Lee, S. H. Oh, S. M. Sun and S. I. Whang, Multi-hump solutions of some singularly perturbed equations of KdV type, Discrete Contin. Dyn. Syst., 34 (2014), 5181-5209.  doi: 10.3934/dcds.2014.34.5181.  Google Scholar

[10]

S. Deng and S. M. Sun, Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation, Nonlinearity, 30 (2017), 765-809.  doi: 10.1088/1361-6544/aa525a.  Google Scholar

[11]

V. A. Dougalis and D. E. Mitsotakis, Theory and numerical analysis of Boussinesq systems: A review, in: N. A. Kampanis, V. A. Dougalis and J. A. Ekaterinaris (eds.), Effective Computational Methods in Wave Propagation, CRC Press, 5 (2008), 63–110. doi: 10.1201/9781420010879.ch3.  Google Scholar

[12]

B. Karasozen and G. Simsek, Energy preserving integration of KdV-KdV systems, TWMS J. App. Eng. Math., 2 (2012), 219-227.   Google Scholar

[13]

A. Khare and A. Saxena, Novel PT-invariant solutions for a large number of real nonlinear equations, Phys. Lett. A, 380 (2015), 856-862.  doi: 10.1016/j.physleta.2015.12.007.  Google Scholar

[14]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer-Verlag, Berlin, 2003. Google Scholar

[15]

M. O. Korpusov, Blowup of solutions of nonlinear equations and systems of nonlinear equations in wave theory, Theoret. and Math. Phys., 174 (2013), 307-314.  doi: 10.1007/s11232-013-0028-y.  Google Scholar

[16]

F. Linares, D. Pilod and J.-C. Saut, Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4195-4221.  doi: 10.1137/110828277.  Google Scholar

[17]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, with Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Mathematics, Vol. 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar

[18]

A. Mielke, P. Holmes and O. O'Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center, J. Dynam. Differ. Equ., 4 (1992), 95-126.  doi: 10.1007/BF01048157.  Google Scholar

[19]

M. Ming, J.-C. Saut and P. Zhang, Long-time existence of solutions to Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4078-4100.  doi: 10.1137/110834214.  Google Scholar

[20]

T. E. Mogorosi, B. Muatjetjeja and C. M. Khalique, Conservation laws for a generalized coupled Boussinesq system of KdV-KdV type, Springer International Publishing, 117 (2015), 315-321.   Google Scholar

[21]

H. Y. Nguyen and F. Dias, A Boussinesq system for two-way propagation of interfacial waves, Phys. D, 237 (2008), 2365-2389.  doi: 10.1016/j.physd.2008.02.020.  Google Scholar

[22]

J.-C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface waves type Boussinesq systems Ⅱ, SIAM J. Math. Anal., 49 (2017), 2321-2386.  doi: 10.1137/15M1050203.  Google Scholar

[23]

J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl., 97 (2012), 635-662.  doi: 10.1016/j.matpur.2011.09.012.  Google Scholar

[24]

E. V. Yushkov, Blowup in Korteweg-de Vries-type systems, Theoret. Math. Phys., 173 (2012), 197-206.  doi: 10.1007/s11232-012-0129-z.  Google Scholar

[25]

V. C. Zelati and M. Macrì, Multibump solutions homoclinic to periodic orbits of large energy in a centre manifold, Nonlinearity, 18 (2005), 2409-2445.  doi: 10.1088/0951-7715/18/6/001.  Google Scholar

[26]

Y. Zhou, M. Wang and Y. Wang, Periodic wave solutions to coupled KdV equations with variable coefficients, Phys. Lett. A, 308 (2003), 31-36.  doi: 10.1016/S0375-9601(02)01775-9.  Google Scholar

Figure 1.  (1) Homoclinic solutions. (2) Generalized homoclinic solutions
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Figure 2.  Generalized two-hump homoclinic solutions
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Figure 3.  Eigenvalues of $ L $
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