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

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

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 $k$.

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.

[2]

D. C. AntonopoulosV. 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.

[3]

J. L. BonaM. 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.

[4]

J. L. BonaM. 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.

[5]

J. L. BonaV. 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.

[6]

J. L. BonaV. 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.

[7]

J. L. BonaZ. 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.

[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.

[9]

J. W. ChoiD. S. LeeS. H. OhS. 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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

F. LinaresD. 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.

[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.

[18]

A. MielkeP. 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.

[19]

M. MingJ.-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.

[20]

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

[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.

[22]

J.-C. SautC. 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.

[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.

[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.

[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.

[26]

Y. ZhouM. 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.

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.

[2]

D. C. AntonopoulosV. 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.

[3]

J. L. BonaM. 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.

[4]

J. L. BonaM. 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.

[5]

J. L. BonaV. 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.

[6]

J. L. BonaV. 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.

[7]

J. L. BonaZ. 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.

[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.

[9]

J. W. ChoiD. S. LeeS. H. OhS. 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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

F. LinaresD. 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.

[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.

[18]

A. MielkeP. 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.

[19]

M. MingJ.-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.

[20]

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

[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.

[22]

J.-C. SautC. 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.

[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.

[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.

[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.

[26]

Y. ZhouM. 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.

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