Singular solitons, shock waves, and other solutions to potential KdV equation

Wang, Gangwei; Xu, Tianzhou; Ebadi, Ghodrat; Johnson, Stephen L.; Strong, Andre; Biswas, Anjan

doi:10.1007/s11071-013-1189-9

Cited by 53 publications

(23 citation statements)

References 20 publications

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“…(13a) and (13b) are derived using (4) and (5) and considering that in our case h = 1 [see (1)]. Moreover, the four models described before are also valid in the present work, but since h is assumed to be a constant, the terms associated with the coefficients B 3 and B 4 disappear [see (5i) and (5j)].…”

Section: Nonlinear Differential Boussinesq Equation Of Sixth Ordermentioning

confidence: 92%

“…Other nonlinear evolution equations have also been used to model these physical phenomena. Some of them are the Korteweg-de Vries equation [1][2][3]5,6], Kawahara equation [7], Bretherton equation [8], Benjamin-Bona-Mahony equation [9,10], CamassaHolm equation [11] and other types of Boussinesq equations [4,[12][13][14][15][16][17]. These nonlinear evolution equations support solitons and other types of travelling wave solutions sparking worldwide interest in the mathematical physics community.…”

Section: Introductionmentioning

confidence: 99%

“…[5,9,11,12,[18][19][20][21]). Several mathematical techniques have been used in the literature to integrate analytically nonlinear evolution equations such as the mapping method [12], variational iteration method [22], Hirota's bilinear method [23], Adomian decomposition method [24][25][26], Fan's F-expansion method [27], G /G-method [28], tanh-method [28], exp-method [29], inverse scattering method [30], Kudryashov's method [28] and the Weierstrass elliptic function method [12] as well as the Lie symmetry approach [12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

Pereira

Lopes

Trabucho³

2015

Nonlinear Dyn

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An improved class of nonlinear bidirectional Boussinesq equations of sixth order using a wave surface elevation formulation is derived. Exact travelling wave solutions for the proposed class of nonlinear evolution equations are deduced. A new exact travelling wave solution is found which is the uniform limit of a geometric series. The ratio of this series is proportional to a classical soliton-type solution of the form of the square of a hyperbolic secant function. This happens for some values of the wave propagation velocity. However, there are other values of this velocity which display this new type of soliton, but the classical soliton structure vanishes in some regions of the domain. Exact solutions of the form of the square of the classical soliton are also deduced. In some cases, we find that the ratio between the amplitude of this wave and the amplitude of the classical soliton is equal to 35/36. It is shown that different families of travelling wave solutions are associated with different values of the parameters introduced in the improved equations.

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Section: Nonlinear Differential Boussinesq Equation Of Sixth Ordermentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

Pereira

Lopes

Trabucho³

2015

Nonlinear Dyn

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“…In general, Lie symmetries can be used to reduce the order as well number of independent variables of original equation (system of equations). For further details, readers are referred to [1][2][3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

2015

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“…Here we just mention some of the recent work. Biswas [5] studied the solitary wave solution for KdV equation with power law nonlinearity and time-dependent coefficients, [6] investigated the solitons, shock waves for the potential KdV equation, while [7] studied the solitary wave solutions for the generalized KdV equation. In addition to the theoretical studies, readers can refer to [8,9] for the numerical simulations of the KdV equation and the generalized KdV equation.…”

Section: Introductionmentioning

confidence: 99%

New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

2015

Nonlinear Dyn

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Recently, Biswas et al. (Phys Wave Phenom 19:24-29, 2011) derived exact bright and dark solitary solutions for the Rosenau-Kawahara equation with power law nonlinearity, and Hu et al. (Adv Math Phys 11, Article ID 217393, 2014) proposed two conservative finite difference schemes for the usual RosenauKawahara equation. In this paper, we obtain another set of exact solitary solutions for the Rosenau-Kawahara equation with power law nonlinearity. More importantly, a conservative finite difference method is presented. The fundamental energy conservation law is preserved by the current numerical scheme. And the existence and uniqueness of the numerical solution are proved. The convergence and stability of the numerical solution are also shown. The method is second-order convergent both in time and in space. Finally, numerical results confirm well with the theoretical results and show that the current method can be well used to study the solitary wave at long time.

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Singular solitons, shock waves, and other solutions to potential KdV equation

Cited by 53 publications

References 20 publications

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

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