Slowly varying solitary waves. I. Korteweg-de Vries equation

1979 ◽  
Vol 368 (1734) ◽  
pp. 359-375 ◽  
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Asymptotic Solution ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Scale Method ◽  
De Vries

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient Korteweg-de Vries equation. A multiple scale method is used to determine the amplitude and phase of the wave to the second order in the perturbation parameter. The structure ahead and behind the solitary wave is also determined, and the results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

Slowly varying solitary waves in deep fluids

1981 ◽  
Vol 376 (1765) ◽  
pp. 319-332 ◽  
Keyword(s):  
Solitary Wave ◽  
Asymptotic Solution ◽  
Conservation Laws ◽  
Internal Waves ◽  
Solitary Waves ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Fluid Equation ◽  
Scale Method ◽  
Deep Fluid

The slowly varying solitary wave is constructed as an asymptotic solution of the deep fluid equation of Benjamin (1967), Davis & Acrivos (1967), and Ono (1975). A multiple scale method is used to determine the amplitude and phase of the wave to second order in the perturbation parameter. Behind the solitary wave a shelf develops. Outer expansions are intro­duced to remove certain non-uniformities in the expansion. The results are interpreted from conservation laws. Finally the effect of damping, either due to radiating internal waves or due to friction, is considered.

On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients

1973 ◽  
Vol 60 (4) ◽  
pp. 813-824 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Solitary Wave ◽  
Asymptotic Solution ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Variable Depth ◽  
Exponential Form ◽  
Varying Coefficients ◽  
Numerical Integrations ◽  
De Vries ◽  
Depth Water

The variable-coefficient Korteweg–de Vries equation \[ H_X + {\textstyle\frac{3}{2}}d^{-\frac{7}{4}}HH_{\xi} + {\textstyle\frac{1}{6}}\kappa d^{\frac{1}{2}}H_{\xi\xi\xi} = 0 \] with d = d(εX) is discussed for solitary-wave initial profiles. A straightforward asymptotic solution for ε → 0 is constructed and is shown to be non-uniform both ahead of and behind the solitary wave. The behaviour ahead is rectified by matching to the appropriate exponential form and, together with the use of conservation laws for the equation, the nature of the solution behind the solitary wave is discussed. This leads to the formulation of the solution in the oscillatory ‘tail’, which is again matched directly.The results are applied to the development of the solitary wave into variable-depth water, and the predictions are compared with those obtained, for example, by Grimshaw (1970, 1971). Finally, the asymptotic behaviour of both the solitary wave and the oscillatory tail are assessed in the light of some numerical integrations of the equation.

Slowly varying solitary waves. II. Nonlinear Schrödinger equation

1979 ◽  
Vol 368 (1734) ◽  
pp. 377-388 ◽  
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Scale Method

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient nonlinear Schrodinger equation. A multiple scale method is used to determine the amplitude and phases of the wave to the second order in the perturbation parameter. The method is similar to that used in (I) (R. Grimshaw 1979 Proc. R. Soc. Lond . A 368, 359). The results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

2004 ◽  
Vol 34 (12) ◽  
pp. 2774-2791 ◽  
Author(s):  
Roger Grimshaw ◽  
Efim Pelinovsky ◽  
Tatiana Talipova ◽  
Audrey Kurkin
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Asymptotic Theory ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Variable Depth ◽  
North West ◽  
Wave Parameters ◽  
De Vries

Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

scholarly journals Hamiltonian formulation for the description of interfacial solitary waves

1998 ◽  
Vol 5 (1) ◽  
pp. 3-12 ◽  
Author(s):  
R. Grimshaw ◽  
S. R. Pudjaprasetya
Keyword(s):  
Solitary Waves ◽  
Variable Coefficient ◽  
Hamiltonian Structure ◽  
Vries Equation ◽  
Hamiltonian Formulation ◽  
Constant Density ◽  
Two Fluids ◽  
De Vries ◽  
Lower Fluid ◽  
New Equation

Abstract. We consider solitary waves propagating on the interface between two fluids, each of constant density, for the case when the upper fluid is bounded above by a rigid horizontal plane, but the lower fluid has a variable depth. It is well-known that in this situation, the solitary waves can be described by a variable-coefficient Korteweg-de Vries equation. Here we reconsider the derivation of this equation and present a formulation which preserves the Hamiltonian structure of the underlying system. The result is a new variable-coefficient Korteweg-de Vries equation, which conserves energy to a higher order than the more conventional well-known equation. The new equation is used to describe the transformation of an interfacial solitary wave which propagates into a region of decreasing depth.

scholarly journals The stability of solitary waves

1972 ◽  
Vol 328 (1573) ◽  
pp. 153-183 ◽  
Keyword(s):  
Solitary Wave ◽  
Spectral Theory ◽  
Solitary Waves ◽  
Vries Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Long Time ◽  
Stability Of Solitary Waves ◽  
The Stability

The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona & Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.

scholarly journals The modified Korteweg - de Vries equation in the theory of large - amplitude internal waves

1997 ◽  
Vol 4 (4) ◽  
pp. 237-250 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
T. Talipova
Keyword(s):  
Solitary Wave ◽  
Internal Waves ◽  
Solitary Waves ◽  
Large Amplitude ◽  
Nonlinear Term ◽  
Vries Equation ◽  
Nonlinear Effects ◽  
Wave Generation ◽  
Density Stratification ◽  
De Vries

Abstract. The propagation of large- amplitude internal waves in the ocean is studied here for the case when the nonlinear effects are of cubic order, leading to the modified Korteweg - de Vries equation. The coefficients of this equation are calculated analytically for several models of the density stratification. It is shown that the coefficient of the cubic nonlinear term may have either sign (previously only cases of a negative cubic nonlinearity were known). Cubic nonlinear effects are more important for the high modes of the internal waves. The nonlinear evolution of long periodic (sine) waves is simulated for a three-layer model of the density stratification. The sign of the cubic nonlinear term influences the character of the solitary wave generation. It is shown that the solitary waves of both polarities can appear for either sign of the cubic nonlinear term; if it is positive the solitary waves have a zero pedestal, and if it is negative the solitary waves are generated on the crest and the trough of the long wave. The case of a localised impulsive initial disturbance is also simulated. Here, if the cubic nonlinear term is negative, there is no solitary wave generation at large times, but if it is positive solitary waves appear as the asymptotic solution of the nonlinear wave evolution.

Weakly nonlinear internal wave fronts trapped in contractions

2000 ◽  
Vol 415 ◽  
pp. 323-345 ◽  
Author(s):  
S. R. CLARKE ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Internal Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Wave Fronts ◽  
Nonlinear Internal Wave ◽  
Weakly Nonlinear ◽  
Virtual Control ◽  
De Vries

The propagation of weakly nonlinear, long internal wave fronts in a contraction is considered in the transcritical limit as a model for the establishment of virtual controls. It is argued that the appropriate equation to describe this process is a variable coefficient Korteweg–de Vries equation. The solutions of this equation are then considered for compressive and rarefaction fronts. Rarefaction fronts exhibit both normal and virtual control solutions. However, the interaction of compressive fronts with contractions is intrinsically unsteady. Here the dynamics take two forms, interactions with the bulk of the front and interactions with individual solitary waves separating off from a front trapped downstream of the contraction.

scholarly journals Painleve analysis for a forced Korteveg-de Vries equation arisen in fluid dynamics of internal solitary waves

2015 ◽  
Vol 19 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
Sheng Zhang ◽  
Mei-Tong Chen ◽  
Wei-Yi Qian
Keyword(s):  
Fluid Dynamics ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Painlevé Analysis ◽  
New Exact Solutions ◽  
De Vries ◽  
Painleve Analysis ◽  
Truncated Painlevé Expansion

In this paper, Painleve analysis is used to test the Painleve integrability of a forced variable-coefficient extended Korteveg-de Vries equation which can describe the weakly-non-linear long internal solitary waves in the fluid with continuous stratification on density. The obtained results show that the equation is integrable under certain conditions. By virtue of the truncated Painleve expansion, a pair of new exact solutions to the equation is obtained.

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