هذا البحث يناقش الانتشار للموجات المنفردة والموجات الدورية غير الخطية في المسافات الضحلة على تدرج المنحدر مع الحركة الحادثة في إطار معادلة كورتيفيج-دي فريس المعامل المتغير. نحن نستخدم طريقة التوحيد الويتهام، باستخدام تطور حديث من هذه النظرية للمعادلات المتغيرة المتقطعة. يمكن لهذا النهج العام أيضًا تحسين النتائج المعروفة حول تطور آدياباتي للموجات المنفردة والموجات الدورية المنفصلة في وجود التخطيط المتغير والحركة الحادثة التي يتم تحديدها بواسطة قانون شيزي، ولكن أهمية أكبر، لدراسة تأثير هذه العوامل على انتشار الأوندولار بور، الذي يعتبر غير مستقر في النظام المرجعي. وبالإضافة إلى ذلك، يظهر أن العمل المشترك للتخطيط المتغير والحركة الحادثة بشكل عام يقيد القيود العالمية على انتشار الأوندولار بور، بحيث يمكن أن يكون تطور الموجة المنفردة الرائدة مختلفًا بشكل كبير عن تطور الموجة المنفردة المنفصلة التي لها نفس الامتداد الأولي. وهذا التأثير غير المحلي يرجع إلى التفاعلات الموجية الغير خطية داخل الأوندولار بور ويمكن أن يؤدي إلى نمو إضافي لامتداد الموجة المنفردة، والذي لا يمكن التنبؤ به في إطار النهج الآدياباتي التقليدي لانتشار الموجات المنفردة في الوسائط التي تتغير ببطء.
This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modeled by the Chezy law, but also importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.
We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su-Gardner (or one-dimensional Green-Naghdi) system. Our analysis is based on the properties of the characteristics of the associated Whitham modulation system which describes an intermediate undular bore stage of the evolution. The resulting formula represents a non-integrable analogue of the well-known semi-classical distribution for the Korteweg-de Vries equation, which is usually obtained through the inverse scattering transform. Our analytical results are shown to agree with the results of direct numerical simulations of the Su-Gardner system. Our analysis can be generalised to other weakly dispersive, fully nonlinear systems which are not necessarily completely integrable.
We study asymptotic stability of solitary wave solutions in the one-dimensional Benney-Luke equation, a formally valid approximation for describing two-way water wave propagation. For this equation, as for the full water wave problem, the classic variational method for proving orbital stability of solitary waves fails dramatically due to the fact that the second variation of the energy-momentum functional is infinitely indefinite. We establish nonlinear stability in energy norm under the spectral stability hypothesis that the linearization admits no non-zero eigenvalues of non-negative real part. We also verify this hypothesis for waves of small energy.
When a $(1+1)$-dimensional nonlinear PDE in real function $eta(x,t)$ admits localized traveling solutions we can consider $L$ to be the average width of the envelope, $A$ the average value of the amplitude of the envelope, and $V$ the group velocity of such a solution. The replacement rule (RR or nonlinear dispersion relation) procedure is able to provide a simple qualitative relation between these three parameters, without actually solve the equation. Examples are provided from KdV, C-H and BBM equations, but the procedure appears to be almost universally valid for such $(1+1)$-dimensional nonlinear PDE and their localized traveling solutions cite{3}.
We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrodinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing, and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approximation and validated by NLSE simulations. Our result will contribute to understand dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media.