Based on our recently developed theory of absolute and convective instabilities of spatially varying unbounded and semi–bounded flows and media with algebraically decaying tails, we develop in this paper a linear stability theory for continuous fronts whose tails have similar decay asymptotics at infinity. It is assumed that the base state of the front, W(x), tends to constant states, WR and WL ≠ WR, when x → ∞ and x→ −∞, respectively, and the tails, RR(x) and RL(x), in the governing equation linearized about W(x) decay as |x|−α, when x → ±∞, respectively, where α is sufficiently large. No restrictions on the rate of variability of the tails in the finite domain are imposed. The Laplace–transformed problem, Z x (x,ω)Z(x,ω)+G(x,ω),x∈, governing the perturbation dynamics of the front is treated by using the decompositions of the fundamental matrix of the system obtained by us previously, Φ(x,ω)= B R (x,ω) e R(ω)x[ B R (0,ω) ] , with the asymptotics B R (x,ω)=I+o( x -∈ ),∈>0, when x → ∞, and Φ(x,ω)= B L (x,ω) e A L (ω)x [ B L (0,ω) ] -1 , with the asymptotics BL(x,ω) = I + O(|x|−ε), when x → −∞ where I is the identity matrix. Here, Z(x,ω) denotes the Laplace–transformed perturbation, x ∈ R is the spatial coordinate, ω ∈ C is a frequency (and a Laplace transform parameter), G(x, ω) is the source function, AR(ω) = limx→∞ A(x,ω) and AL(ω) = limx→−∞ A(x,ω) ≠ AR(ω). The principal part of the analysis is the formulation of conditions equivalent to the boundary conditions of decay for Z(x,ω), when x → ±∞, derived by applying a result due to Kato (1980 Perturbation theory for linear operators). The boundary–value problem for Z(x,ω) is solved formally. Its solution has the form similar to that obtained in Brevdo (2003 Proc. R. Soc. Lond. A459, 1403–1425). The absolute and convective instabilities of, and signalling in, the front are studied by applying to the solution the treatments in the above papers, whereas new elements present due to the different limits in ±∞ of the matrix A(x,ω) are taken account of. We express the stability results in terms of the dispersion relation functions, Dn(ω), for the global normal modes, for the corresponding regions, Rn⊂ C, n ≥ 1, the dispersion relation functions of the associated uniform states, DR0(k, ω) = det[ikI −AR(ω)] and DL0(k, ω) = det[ikI − AL(ω)], and the singularities of the matrices BR(x,ω) and BL(x,ω), and of the projectors PR+(ω) and PL−(ω) related to the operators AR(ω) and AL(ω), respectively. Since all of the above objects controlling the instabilities are essentially global properties of the front, it is argued that the concept of local stability cannot be consistently defined for the fronts treated. A procedure for computing the instabilities is outlined.
基于我们最近发展的具有代数衰减尾的空间变化无界和半有界流动和介质的绝对不稳定性和对流不稳定性理论，我们发展了尾在无穷远处具有类似衰减渐近的连续锋面的线性稳定性理论。假设当x→∞和x→−∞时，前端的基态W(x)分别趋于恒定态WR和WL≠WR，当x→±∞时，当α足够大时，控制方程中关于W(x)线性化的尾部RR(x)和RL(x)衰减为|x|−α。在有限域内，对尾部的变异性率没有限制。Laplace-transformed问题,x (x,ω)Z Z (x,ω)+ G (x,ω),x∈,前面被扰动动态的管理通过使用系统的基本矩阵的分解得到我们以前,Φ(x,ω)= B (x,ω)e R(ω)x [B R(0,ω)],与渐近B R (x,ω)= I + o (x -∈)∈> 0,当x→∞,和Φ(x,ω)= B L (x,ω)e L(ω)x (B L(0,ω)]1、渐近的提单(x,ω)= I + o x(| |−ε),当x→−∞我是单位矩阵。其中，Z(x，ω)表示拉普拉斯变换微扰，x∈R为空间坐标，ω∈C为频率(和拉普拉斯变换参数)，G(x， ω)为源函数，AR(ω) = limx→∞a (x，ω)， AL(ω) = limx→−∞a (x，ω)≠AR(ω)。分析的主要部分是等效于x→±∞时Z(x，ω)衰减边界条件的条件的公式，通过应用Kato(1980年线性算子的摄动理论)的结果推导出来。对Z(x，ω)的边值问题进行了形式化求解。其解决方案与Brevdo (2003 Proc. R. Soc)中获得的形式相似。Lond。A459, 1403 - 1425)。通过将上述论文的处理方法应用于解，研究了锋面的绝对不稳定性和对流不稳定性以及信号的不稳定性，同时考虑了由于矩阵A(x，ω)在±∞上的不同极限而出现的新元素。我们用全局正模的色散关系函数Dn(ω)，对应区域Rn∧C, n≥1，相关均匀态的色散关系函数DR0(k， ω) = det[ikI−AR(ω)]和DL0(k， ω) = det[ikI−AL(ω)]，矩阵BR(x，ω)和BL(x，ω)的奇异性，以及分别与算子AR(ω)和AL(ω)相关的投影器PR+(ω)和PL−(ω)的奇异性来表示稳定性结果。由于上述控制不稳定性的所有对象本质上都是锋面的全局属性，因此认为局部稳定性的概念不能对所处理的锋面进行一致的定义。给出了计算不稳定性的程序。
{"title":"Linear stability theory for fronts with algebraically decaying tails","authors":"L. Brevdo","doi":"10.1098/rspa.2004.1332","DOIUrl":"https://doi.org/10.1098/rspa.2004.1332","url":null,"abstract":"Based on our recently developed theory of absolute and convective instabilities of spatially varying unbounded and semi–bounded flows and media with algebraically decaying tails, we develop in this paper a linear stability theory for continuous fronts whose tails have similar decay asymptotics at infinity. It is assumed that the base state of the front, W(x), tends to constant states, WR and WL ≠ WR, when x → ∞ and x→ −∞, respectively, and the tails, RR(x) and RL(x), in the governing equation linearized about W(x) decay as |x|−α, when x → ±∞, respectively, where α is sufficiently large. No restrictions on the rate of variability of the tails in the finite domain are imposed. The Laplace–transformed problem, Z x (x,ω)Z(x,ω)+G(x,ω),x∈, governing the perturbation dynamics of the front is treated by using the decompositions of the fundamental matrix of the system obtained by us previously, Φ(x,ω)= B R (x,ω) e R(ω)x[ B R (0,ω) ] , with the asymptotics B R (x,ω)=I+o( x -∈ ),∈>0, when x → ∞, and Φ(x,ω)= B L (x,ω) e A L (ω)x [ B L (0,ω) ] -1 , with the asymptotics BL(x,ω) = I + O(|x|−ε), when x → −∞ where I is the identity matrix. Here, Z(x,ω) denotes the Laplace–transformed perturbation, x ∈ R is the spatial coordinate, ω ∈ C is a frequency (and a Laplace transform parameter), G(x, ω) is the source function, AR(ω) = limx→∞ A(x,ω) and AL(ω) = limx→−∞ A(x,ω) ≠ AR(ω). The principal part of the analysis is the formulation of conditions equivalent to the boundary conditions of decay for Z(x,ω), when x → ±∞, derived by applying a result due to Kato (1980 Perturbation theory for linear operators). The boundary–value problem for Z(x,ω) is solved formally. Its solution has the form similar to that obtained in Brevdo (2003 Proc. R. Soc. Lond. A459, 1403–1425). The absolute and convective instabilities of, and signalling in, the front are studied by applying to the solution the treatments in the above papers, whereas new elements present due to the different limits in ±∞ of the matrix A(x,ω) are taken account of. We express the stability results in terms of the dispersion relation functions, Dn(ω), for the global normal modes, for the corresponding regions, Rn⊂ C, n ≥ 1, the dispersion relation functions of the associated uniform states, DR0(k, ω) = det[ikI −AR(ω)] and DL0(k, ω) = det[ikI − AL(ω)], and the singularities of the matrices BR(x,ω) and BL(x,ω), and of the projectors PR+(ω) and PL−(ω) related to the operators AR(ω) and AL(ω), respectively. Since all of the above objects controlling the instabilities are essentially global properties of the front, it is argued that the concept of local stability cannot be consistently defined for the fronts treated. A procedure for computing the instabilities is outlined.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78466200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the title problem, the temperature satisfies Laplac's equation within the solid and liquid, and Newton's law of cooling at the liquid–vapour phase interface. That boundary condition defines a length L ∼ 10nm on which the interface changes from being, in effect, adiabatic at the contact line to isothermal at infinity. We give an exact solution showing how this change affects the temperature field when the solid occupies a half–space, and the liquid a quarter–space (so the liquid–solid contact angle θ=π/2); the liquid–solid conductivity ratio k is arbitrary. We use this solution to verify the predictions of existing analysis of the limit ? → 0 with θ fixed but arbitrary. In the limit r/L → 0 of vanishing distance from the contact line, the new solution reduces to an existing local solution.
{"title":"Heat flow near the triple junction of an evaporating meniscus and a substrate","authors":"S. Morris","doi":"10.1098/rspa.2004.1308","DOIUrl":"https://doi.org/10.1098/rspa.2004.1308","url":null,"abstract":"In the title problem, the temperature satisfies Laplac's equation within the solid and liquid, and Newton's law of cooling at the liquid–vapour phase interface. That boundary condition defines a length L ∼ 10nm on which the interface changes from being, in effect, adiabatic at the contact line to isothermal at infinity. We give an exact solution showing how this change affects the temperature field when the solid occupies a half–space, and the liquid a quarter–space (so the liquid–solid contact angle θ=π/2); the liquid–solid conductivity ratio k is arbitrary. We use this solution to verify the predictions of existing analysis of the limit ? → 0 with θ fixed but arbitrary. In the limit r/L → 0 of vanishing distance from the contact line, the new solution reduces to an existing local solution.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78027009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some universal identities for plane elastic waves in free and clamped homogeneous plates of arbitrary anisotropy are obtained and analysed. Insight is gained by linking the dispersion of guided–wave phase velocity (or, more precisely, its derivative in wavenumber or frequency) to the Stroh matrix, i.e. to the coefficients of the governing system of wave motion equations in the sextic form, on the one hand, and to the energetic parameters, on the other. The derivation also involves the residues of the plate admittance (Gree's function in the transform domain) along a dispersion branch. Combining these complementary perspectives enables a general criterion for increasing or decreasing trends in the dispersion branches and provides useful interpretations of the difference between the phase velocity and the in–plane group velocity. Explicit examples at low, high and cut–off frequencies are presented. Limitations for the case of transversely inhomogeneous plates are discussed.
{"title":"General relationships for guided acoustic waves in anisotropic plates","authors":"A. Shuvalov","doi":"10.1098/rspa.2004.1319","DOIUrl":"https://doi.org/10.1098/rspa.2004.1319","url":null,"abstract":"Some universal identities for plane elastic waves in free and clamped homogeneous plates of arbitrary anisotropy are obtained and analysed. Insight is gained by linking the dispersion of guided–wave phase velocity (or, more precisely, its derivative in wavenumber or frequency) to the Stroh matrix, i.e. to the coefficients of the governing system of wave motion equations in the sextic form, on the one hand, and to the energetic parameters, on the other. The derivation also involves the residues of the plate admittance (Gree's function in the transform domain) along a dispersion branch. Combining these complementary perspectives enables a general criterion for increasing or decreasing trends in the dispersion branches and provides useful interpretations of the difference between the phase velocity and the in–plane group velocity. Explicit examples at low, high and cut–off frequencies are presented. Limitations for the case of transversely inhomogeneous plates are discussed.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87442092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The dynamics of a circular liquid lamella resulting from the collision of a water drop with a small disc–like target was studied experimentally and theoretically. Such a type of collision also acts as a model of drop impacts on plane surfaces in the absence of liquid friction, and therefore for more widespread collisions of drops of inviscid liquid with solid surfaces. We propose a simple model to describe the dynamics of the lamella resulting from the drop impact and also predict the structure of the liquid flow in the lamella. It is based on the observations that during the drop collision with the target, the liquid is ejected at an approximately constant flow rate with a velocity that significantly decreases in time. The resulting distributions of velocities, local flow rates and film thickness in the lamella are calculated. Besides, we have measured the distribution of the local Weber numbers by generating Mach–like rupture waves (we have called them Mach–Taylor waves) in the lamella, which follows the Taylor theory of disintegration of fluid sheets. Unknown parameters of the model are obtained from the comparison between the theoretical expression for local Weber number and the experimental data. The time evolution of the lamella diameter was obtained by numerical integration of the model. It was found that during the lamella life, zones of metastability could be formed in the lamella. In these zones a propagating rupture hole cannot be transported away by the flow and it yields to destabilization. One metastability zone expands from the target towards the external rim, and it is the opposite for the other one.
{"title":"Dynamics of a liquid lamella resulting from the impact of a water drop on a small target","authors":"A. Rozhkov, B. Prunet-Foch, M. Vignes-Adler","doi":"10.1098/rspa.2004.1293","DOIUrl":"https://doi.org/10.1098/rspa.2004.1293","url":null,"abstract":"The dynamics of a circular liquid lamella resulting from the collision of a water drop with a small disc–like target was studied experimentally and theoretically. Such a type of collision also acts as a model of drop impacts on plane surfaces in the absence of liquid friction, and therefore for more widespread collisions of drops of inviscid liquid with solid surfaces. We propose a simple model to describe the dynamics of the lamella resulting from the drop impact and also predict the structure of the liquid flow in the lamella. It is based on the observations that during the drop collision with the target, the liquid is ejected at an approximately constant flow rate with a velocity that significantly decreases in time. The resulting distributions of velocities, local flow rates and film thickness in the lamella are calculated. Besides, we have measured the distribution of the local Weber numbers by generating Mach–like rupture waves (we have called them Mach–Taylor waves) in the lamella, which follows the Taylor theory of disintegration of fluid sheets. Unknown parameters of the model are obtained from the comparison between the theoretical expression for local Weber number and the experimental data. The time evolution of the lamella diameter was obtained by numerical integration of the model. It was found that during the lamella life, zones of metastability could be formed in the lamella. In these zones a propagating rupture hole cannot be transported away by the flow and it yields to destabilization. One metastability zone expands from the target towards the external rim, and it is the opposite for the other one.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82242936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two–dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross–section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function.
{"title":"On the two-dimensional sloshing problem","authors":"V. Kozlov, N. Kuznetsov, O. Motygin","doi":"10.1098/rspa.2004.1303","DOIUrl":"https://doi.org/10.1098/rspa.2004.1303","url":null,"abstract":"We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two–dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross–section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78007888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A prevalent though unexpected asymptotic phenomenon occurs near anti–Stokes lines, on which two exponentials contributing to a function have the same absolute value: the subdominant exponential contribution can be larger than that from the dominant exponential. The phenomenon arises because the factors multiplying the two exponentials have different asymptotic forms. The boundary of the region of dominance by the subdominant exponential (DSE) is a line, for which an explicit general form is given; this shows that the region of DSE is asymptotically infinitely wide. The DSE line contains the zeros of the function, resulting from complete destructive interference between the two exponential contributions. Several examples are given; two have a physical origin in diffraction physics, and illustrate the fact that DSE can explain observed optical phenomena.
{"title":"Asymptotic dominance by subdominant exponentials","authors":"M. Berry","doi":"10.1098/rspa.2004.1343","DOIUrl":"https://doi.org/10.1098/rspa.2004.1343","url":null,"abstract":"A prevalent though unexpected asymptotic phenomenon occurs near anti–Stokes lines, on which two exponentials contributing to a function have the same absolute value: the subdominant exponential contribution can be larger than that from the dominant exponential. The phenomenon arises because the factors multiplying the two exponentials have different asymptotic forms. The boundary of the region of dominance by the subdominant exponential (DSE) is a line, for which an explicit general form is given; this shows that the region of DSE is asymptotically infinitely wide. The DSE line contains the zeros of the function, resulting from complete destructive interference between the two exponential contributions. Several examples are given; two have a physical origin in diffraction physics, and illustrate the fact that DSE can explain observed optical phenomena.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73213471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A set of model equations for water–wave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, the resulting set of equations has three free parameters, allowing for an optimization with known analytical properties of water waves. The optimized model equations show good linear wave characteristics up to kh ≈ 6, while the second–order nonlinear behaviour is captured to kh ≈ 6 as well. A numerical algorithm for solving the model equations is developed and tested against one– and two–horizontal–dimension cases. Agreement with laboratory data is excellent.
{"title":"A two-layer approach to wave modelling","authors":"P. Lynett, P. Liu","doi":"10.1098/rspa.2004.1305","DOIUrl":"https://doi.org/10.1098/rspa.2004.1305","url":null,"abstract":"A set of model equations for water–wave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, the resulting set of equations has three free parameters, allowing for an optimization with known analytical properties of water waves. The optimized model equations show good linear wave characteristics up to kh ≈ 6, while the second–order nonlinear behaviour is captured to kh ≈ 6 as well. A numerical algorithm for solving the model equations is developed and tested against one– and two–horizontal–dimension cases. Agreement with laboratory data is excellent.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76510305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Hadamard expansion procedure applied to Laplace–type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents. This mode of expansion is illustrated by the evaluation of the Bessel functions Jv(? x) and Yv(v x) of large order and argument when x is bounded away from unity. The limit x → 1, corresponding to the coalescence of the active saddles in the integral representations of the Bessel functions, translates into a progressive loss of exponential separation between the different levels of the Hadamard expansion, which renders computation in this limit more difficult. It is shown how a simple modification to this procedure can be employed to deal with the coalescence of the active saddles when x → 1.
{"title":"Exactification of the method of steepest descents: the Bessel functions of large order and argument","authors":"R. Paris","doi":"10.1098/rspa.2004.1307","DOIUrl":"https://doi.org/10.1098/rspa.2004.1307","url":null,"abstract":"The Hadamard expansion procedure applied to Laplace–type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents. This mode of expansion is illustrated by the evaluation of the Bessel functions Jv(? x) and Yv(v x) of large order and argument when x is bounded away from unity. The limit x → 1, corresponding to the coalescence of the active saddles in the integral representations of the Bessel functions, translates into a progressive loss of exponential separation between the different levels of the Hadamard expansion, which renders computation in this limit more difficult. It is shown how a simple modification to this procedure can be employed to deal with the coalescence of the active saddles when x → 1.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86158728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper refines Johnso's implementation of Constantin's method for solving the Camassa–Holm equation for a multiple–soliton solution. An analytical formula for the q(y) and an explicit relation between x and y are found. An algorithm of solving for u(y) is presented. How to introduce time variable t into the solution is also clearly explained.
{"title":"The multiple-soliton solution of the Camassa-Holm equation","authors":"Yi-shen Li, Jin E. Zhang","doi":"10.1098/rspa.2004.1331","DOIUrl":"https://doi.org/10.1098/rspa.2004.1331","url":null,"abstract":"This paper refines Johnso's implementation of Constantin's method for solving the Camassa–Holm equation for a multiple–soliton solution. An analytical formula for the q(y) and an explicit relation between x and y are found. An algorithm of solving for u(y) is presented. How to introduce time variable t into the solution is also clearly explained.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75805420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new theory for transverse instability of bright solitons of equations of nonlinear Schrödinger (NLS) type is presented, based on a natural deformation of the solitons into a four–parameter family. This deformation induces a set of four diagnostic functionals which encode information about transverse instability. These functionals include the deformed power, the deformed momentum and two new functionals. The main result is that a sufficient condition for long–wave transverse instability is completely determined by these functionals. Whereas longitudinal instability is determined by a single partial derivative (the Vakhitov–Kolokolov criterion), the condition for transverse instability requires 10 partial derivatives. The theory is illustrated by application to scalar NLS equations with general potential, and vector NLS equations for optical media with ξ(2) nonlinearity.
{"title":"On the susceptibility of bright nonlinear Schrödinger solitons to long–wave transverse instability","authors":"T. Bridges","doi":"10.1098/rspa.2004.1330","DOIUrl":"https://doi.org/10.1098/rspa.2004.1330","url":null,"abstract":"A new theory for transverse instability of bright solitons of equations of nonlinear Schrödinger (NLS) type is presented, based on a natural deformation of the solitons into a four–parameter family. This deformation induces a set of four diagnostic functionals which encode information about transverse instability. These functionals include the deformed power, the deformed momentum and two new functionals. The main result is that a sufficient condition for long–wave transverse instability is completely determined by these functionals. Whereas longitudinal instability is determined by a single partial derivative (the Vakhitov–Kolokolov criterion), the condition for transverse instability requires 10 partial derivatives. The theory is illustrated by application to scalar NLS equations with general potential, and vector NLS equations for optical media with ξ(2) nonlinearity.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79558293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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