Wave propagation in an elastic medium:Generalized Davey-Stewartson equation

Elastik bir ortamda dalga yayılımı: Genelleştirilmiş Davey-Stewartson denklemleri

Abstract

It is well-known that the envelope of a (1+1) (one spatial and one temporal) dimensional quasi-monochromatic wave train is governed by the single nonlinear Schrödinger equation $i A_t+pA_{xx}+q backslash A backslash^2 A=0$ where t is time, x is the spatial coordinate and Adenotes the complex amplitude. The nonlinear Schrödinger equation appears to be a generic equation describing unidirectional wave modulation. If modulations transverse to the wave propagation direction are also allowed, second spatial coordinate effect should be taken into account and new (2+1) evolution equations should be derived. A natural way to obtain two-dimensional modulations of nonlinear waves is simply to replace the one dimensional dispersive term with a two dimensional dispersive term $i A_t+pA_{xx}+sA_{xy}+rA_{yy}+q backslash A backslash^2 A=0$ However, in many two dimensional systems both short waves and long waves may co-exist and the modulation of such a system can be characterized by Davey-Stewartson equations $i A_t+pA_{xx}+rA_{yy}+q backslash A backslash^2 A=bAphi_x$, $phi_{xx}+m phi_{yy}=(backslash Abackslash^2)_x$, where A is the complex amplitude of the short wave and $theta$ is the long wave amplitude. The Davey- Stewartson system is a model for the evolution of weakly nonlinear packets of water waves that travel in one direction but in which the amplitudes of waves are modulated in two spatial directions. The main purpose of the present study is to extent the analysis of Davey and Stewartson to describe (2+1) dimensional wave motion in a bulk medium composed of an elastic material with couple stresses. To this aim, a multi-scale expansion of quasi-monochromatic wave solutions is used to derive (2+1) non-linear model equations for the description of elastic waves in the far field. By using the reductive perturbation method, the contribution of the second spatial coordinate effect on the propagation of a quasi-monochromatic wave and zero harmonic modes is determined. It is shown that the modulation of waves in the above mentioned medium is governed by the following system of three non-linear evolution equations which may be called the “generalized Davey-Stewartson equations”: $A_t+delta A_{xx}+A_{yy}=chi backslash A backslash^2A+b(phi_{1,x}+phi_{2,y})A $ $phi_{1,xx}+m_2phi_{1,yy}+nphi_{2,xy}=(backslash A backslash^2)_x$, $lambdaphi_{2,xx}+m_1phi_{2,yy}+nphi_{1,xy}=(backslash A backslash^2)_y$ where A is the complex amplitude of the free short transverse wave mode whereas $phi_1$ and $phi_2$are the free long longitudinal and free long transverse wave modes, respectively. These coupled equations govern (2+1) dimensional weakly nonlinear waves in a generalized elastic solid, that travel mostly in the x direction and whose amplitudes are slowly modulating in both x and y directions. Since the nonlinear interaction of the quasi-monochromatic transverse wave and the zero harmonic transverse and longitudinal waves is considered, i.e., a free short transverse, a free long longitudinal and a free long transverse wave modes are included, these evolution equations present a generalized form of the Davey-Stewartson equations. It is also shown that under some restrictions on the parameter values, the generalized Davey-Stewartson equations is reduced to the nonlinear Schrödinger equation and to the Davey-Stewartson equations. Finally, some special solutions of the generalized Davey-Stewartson equations are obtained. By using the traveling wave transformations, the partial differential equations are reduced to ordinary differential equations and the special solutions are given in terms of Jacobian elliptic functions. It is also shown that, these solutions involve secant hyperbolic and tangent hyperbolic type solitary wave solutions for some values of the parameters. These solutions can be given briefly as follows:$A(zeta)=mp a sech (b_1zeta + D)exp(itheta)$, $phi_1(zeta)=-a_1 tanh (b_1zeta + D)$, $phi_2(zeta)=-a_2 tanh (b_1zeta + D)$,and$A(zeta)=mp overline{a} sech (b_2zeta + D)exp(itheta)$, $phi_1(zeta)=-overline{a_1} tanh (b_2zeta + D)$, $phi_2(zeta)=-overline{a_2} tanh (b_2zeta + D)$.

Özet

Bu çalışmada, iki uzay ve bir zaman boyutlu nonlineer dalga yayılımı problemi ele alınmıştır. Sonsuz, homojen, zayıf nonlineer ve zayıf dispersif elastik bir ortamda (2+1) (iki uzay ve bir zaman) boyutlu dalgaların modülasyonu incelenmiştir. Modülasyon problemi için dalgaların asimptotik davranışını tanımlayan (2+1) boyutlu nonlineer evolüsyon denklemleri türetilmiştir. Denklemler türetilirken indirgeyici pertürbasyon yöntemi olarak adlandırılan bir asimptotik yöntem kullanılmış ve dalgaların modülasyonu probleminin üçlü bir nonlineer kısmi diferansiyel denklem sistemi ile karakterize edildiği gösterilmiştir. Bu denklemler, bir kısa enine dalga, bir uzun enine dalga ve bir uzun boyuna dalga olmak üzere üç dalganın etkileşimlerini içermiş ve “genelleştirilmiş Davey- Stewartson denklemleri” olarak adlandırılmıştır. Türetilen denklemlerde beliren parametre değerleri üzerinde alınan bazı kısıtlar altında, genelleştirilmiş Davey-Stewartson denklemlerinin literatürde sıkça karşılaşılan nonlineer Schrödinger denklemine veya Davey-Stewartson denklemlerine indirgendiği gösterilmiştir. Ayrıca, türetilen genelleştirilmiş Davey-Stewartson denklemlerinin bazı özel çözümleri elde edilmiştir. Özel çözümler hesaplanırken, gezen dalga dönüşümlerini esas alan bir yöntem yardımı ile kısmi diferansiyel denklemler adi diferansiyel denklemlere indirgenmiş ve çözümler Jacobi eliptik fonksiyonları cinsinden verilmiştir. Son olarak, elde edilen özel çözümlerin bazı durumlarda hiperbolik fonksiyonlara indirgendiği gösterilmiş ve parametre değerlerinde alınan kimi kısıtlar altında sech-tanh-tanh ve tanh-tanh-tanh yapılarındaki yalnız dalga (solitary wave) çözümlerini içerdiği ifade edilmiştir.