The stability of traveling wave solutions of parabolic equations

A method for determining by inspection the stability or instability of any solution u(t,x) = ɸ(x-ct) of any smooth equation of the form u_t = f(u_(xx),u_x,u where ∂/∂a f(a,b,c) > 0 for all arguments a,b,c, is developed. The connection between the mean wavespeed of solutions u(t,x) and their initial conditions u(0,x) is also explored. The mean wavespeed results and some of the stability results are then extended to include equations which contain integrals and also to include some special systems of equations. The results are applied to several physical examples.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Diffusion in glassy polymers

Fluid diffusion in glassy polymers proceeds in ways that are not explained by the standard diffusion model. Although the reasons for the anomalous effects are not known, much of the observed behavior is attributed to the long times that polymers below their glass transition temperature take to adjust to changes in their condition. The slow internal relaxations of the polymer chains ensure that the material properties are history-dependent, and also allow both local inhomogeneities and differential swelling to occur. Two models are developed in this thesis with the intent of accounting for these effects in the diffusion process.

In Part I, a model is developed to account for both the history dependence of the glassy polymer, and the dual sorption which occurs when gas molecules are immobilized by the local heterogeneities. A preliminary study of a special case of this model is conducted, showing the existence of travelling wave solutions and using perturbation techniques to investigate the effect of generalized diffusion mechanisms on their form. An integral averaging method is used to estimate the penetrant front position.

In Part II, a model is developed for particle diffusion along with displacements in isotropic viscoelastic materials. The nonlinear dependence of the materials on the fluid concentration is taken into account, while pure displacements are assumed to remain in the range of linear viscoelasticity. A fairly general model is obtained for three-dimensional irrotational movements, with the development of the model being based on the assumptions of irreversible thermodynamics. With the help of some dimensional analysis, this model is simplified to a version which is proposed to be studied for Case II behavior.

Two new integral transforms and their applications

This thesis is in two parts. In Part I the independent variable θ in the trigonometric form of Legendre's equation is extended to the range ( -∞, ∞). The associated spectral representation is an infinite integral transform whose kernel is the analytic continuation of the associated Legendre function of the second kind into the complex θ-plane. This new transform is applied to the problems of waves on a spherical shell, heat flow on a spherical shell, and the gravitational potential of a sphere. In each case the resulting alternative representation of the solution is more suited to direct physical interpretation than the standard forms.

In Part II separation of variables is applied to the initial-value problem of the propagation of acoustic waves in an underwater sound channel. The Epstein symmetric profile is taken to describe the variation of sound with depth. The spectral representation associated with the separated depth equation is found to contain an integral and a series. A point source is assumed to be located in the channel. The nature of the disturbance at a point in the vicinity of the channel far removed from the source is investigated.

Non-linear dispersive waves with a small dissipation

The general theory of Whitham for slowly-varying non-linear wavetrains is extended to the case where some of the defining partial differential equations cannot be put into conservation form. Typical examples are considered in plasma dynamics and water waves in which the lack of a conservation form is due to dissipation; an additional non-conservative element, the presence of an external force, is treated for the plasma dynamics example. Certain numerical solutions of the water waves problem (the Korteweg-de Vries equation with dissipation) are considered and compared with perturbation expansions about the linearized solution; it is found that the first correction term in the perturbation expansion is an excellent qualitative indicator of the deviation of the dissipative decay rate from linearity.

A method for deriving necessary and sufficient conditions for the existence of a general uniform wavetrain solution is presented and illustrated in the plasma dynamics problem. Peaking of the plasma wave is demonstrated, and it is shown that the necessary and sufficient existence conditions are essentially equivalent to the statement that no wave may have an amplitude larger than the peaked wave.

A new type of fully non-linear stability criterion is developed for the plasma uniform wavetrain. It is shown explicitly that this wavetrain is stable in the near-linear limit. The nature of this new type of stability is discussed.

Steady shock solutions are also considered. By a quite general method, it is demonstrated that the plasma equations studied here have no steady shock solutions whatsoever. A special type of steady shock is proposed, in which a uniform wavetrain joins across a jump discontinuity to a constant state. Such shocks may indeed exist for the Korteweg-de Vries equation, but are barred from the plasma problem because entropy would decrease across the shock front.

Finally, a way of including the Landau damping mechanism in the plasma equations is given. It involves putting in a dissipation term of convolution integral form, and parallels a similar approach of Whitham in water wave theory. An important application of this would be towards resolving long-standing difficulties about the "collisionless" shock.

Some approximate solutions of dynamic problems in the linear theory of thin elastic shells

Some aspects of wave propagation in thin elastic shells are considered. The governing equations are derived by a method which makes their relationship to the exact equations of linear elasticity quite clear. Finite wave propagation speeds are ensured by the inclusion of the appropriate physical effects.

The problem of a constant pressure front moving with constant velocity along a semi-infinite circular cylindrical shell is studied. The behavior of the solution immediately under the leading wave is found, as well as the short time solution behind the characteristic wavefronts. The main long time disturbance is found to travel with the velocity of very long longitudinal waves in a bar and an expression for this part of the solution is given.

When a constant moment is applied to the lip of an open spherical shell, there is an interesting effect due to the focusing of the waves. This phenomenon is studied and an expression is derived for the wavefront behavior for the first passage of the leading wave and its first reflection.

For the two problems mentioned, the method used involves reducing the governing partial differential equations to ordinary differential equations by means of a Laplace transform in time. The information sought is then extracted by doing the appropriate asymptotic expansion with the Laplace variable as parameter.

Promoting innovative stormwater solutions for coastal plain communities

In 2008, the Center for Watershed Protection (CWP) surveyed seventy-three coastal plain communities to determine their current practices and need for watershed planning and low impact development (LID). The survey found that communities had varying watershed planning effectiveness and need better stormwater management, land use planning, and watershed management communication. While technical capacity is improving, stormwater programs are under staffed and innovative site designs may be prohibited under current regulations. In addition, the unique site constraints (e.g., sandy soils, low relief, tidal influence, vulnerability to coastal hazards, etc.) and lack of local examples are common LID obstacles along the coast (Vandiver and Hernandez, 2009). LID stormwater practices are an innovative approach to stormwater management that provide an alternative to structural stormwater practices, reduce runoff, and maintain or restores hydrology. The term LID is typically used to refer to the systematic application of small, distributed practices that replicate pre-development hydrologic functions. Examples of LID practices include: downspout disconnection, rain gardens, bioretention areas, dry wells, and vegetated filter strips. In coastal communities, LID practices have not yet become widely accepted or applied. The geographic focus for the project is the Atlantic and Gulf coastal plain province which includes nearly 250,000 square miles in portions of fifteen states from New Jersey to Texas (Figure 1). This project builds on CWP’s “Coastal Plain Watershed Network: Adapting, Testing, and Transferring Effective Tools to Protect Coastal Plain Watersheds” that developed a coastal land cover model, conducted a coastal plain community needs survey (results are online here: http://www.cwp.org/#survey), created a coastal watershed Network, and adapted the 8 Tools for Watershed Protection Framework for coastal areas. (PDF contains 4 pages)

Local solutions to local problems: A stormwater education collaborative

Currently completing its fifth year, the Coastal Waccamaw Stormwater Education Consortium (CWSEC) helps northeastern South Carolina communities meet National Pollutant Discharge Elimination System (NPDES) Phase II permit requirements for Minimum Control Measure 1 - Public Education and Outreach - and Minimum Control Measure 2 - Public Involvement. Coordinated by Coastal Carolina University, six regional organizations serve as core education providers to eight coastal localities including six towns and cities and two large counties. CWSEC recently finished a needs assessment to begin the process of strategizing for the second NPDES Phase II 5-year permit cycle in order to continue to develop and implement effective, results-oriented stormwater education and outreach programs to meet federal requirements and satisfy local environmental and economic needs. From its conception in May 2004, CWSEC set out to fulfill new federal Clean Water Act requirements associated with the NPDES Phase II Stormwater Program. Six small municipal separate storm sewer systems (MS4s) located within the Myrtle Beach Urbanized Area endorsed a coordinated approach to regional stormwater education, and participated in a needs assessment resulting in a Regional Stormwater Education Strategy and a Phased Education Work Plan. In 2005, CWSEC was formally established and the CWSEC’s Coordinator was hired. The Coordinator, who is also the Environmental Educator at Coastal Carolina University’s Waccamaw Watershed Academy, organizes six regional agencies who serve as core education providers for eight coastal communities. The six regional agencies working as core education providers to the member MS4s include Clemson Public Service and Carolina Clear Program, Coastal Carolina University’s Waccamaw Watershed Academy, Murrells Inlet 2020, North Inlet-Winyah Bay National Estuarine Research Reserve’s Coastal Training and Public Education Programs, South Carolina Sea Grant Consortium, and Winyah Rivers Foundation’s Waccamaw Riverkeeper®. CWSEC’s organizational structure results in a synergy among the education providers, achieving greater productivity than if each provider worked separately. The member small MS4s include City of Conway, City of North Myrtle Beach, City of Myrtle Beach, Georgetown County, Horry County, Town of Atlantic Beach, Town of Briarcliffe Acres, and Town of Surfside Beach. Each MS4 contributes a modest annual fee toward the salary of the Coordinator and operational costs. (PDF contains 3 pages)

San Francisco Bay regional sediment management strategy development

The San Francisco Bay Conservation and Development Commission (BCDC), in continued partnership with the San Francisco Bay Long Term Management Strategies (LTMS) Agencies, is undertaking the development of a Regional Sediment Management Plan for the San Francisco Bay estuary and its watershed (estuary). Regional sediment management (RSM) is the integrated management of littoral, estuarine, and riverine sediments to achieve balanced and sustainable solutions to sediment related needs. Regional sediment management recognizes sediment as a resource. Sediment processes are important components of coastal and riverine systems that are integral to environmental and economic vitality. It relies on the context of the sediment system and forecasting the long-range effects of management actions when making local project decisions. In the San Francisco Bay estuary, the sediment system includes the Sacramento and San Joaquin delta, the bay, its local tributaries and the near shore coastal littoral cell. Sediment flows from the top of the watershed, much like water, to the coast, passing through rivers, marshes, and embayments on its way to the ocean. Like water, sediment is vital to these habitats and their inhabitants, providing nutrients and the building material for the habitat itself. When sediment erodes excessively or is impounded behind structures, the sediment system becomes imbalanced, and rivers become clogged or conversely, shorelines, wetlands and subtidal habitats erode. The sediment system continues to change in response both to natural processes and human activities such as climate change and shoreline development. Human activities that influence the sediment system include flood protection programs, watershed management, navigational dredging, aggregate mining, shoreline development, terrestrial, riverine, wetland, and subtidal habitat restoration, and beach nourishment. As observed by recent scientific analysis, the San Francisco Bay estuary system is changing from one that was sediment rich to one that is erosional. Such changes, in conjunction with increasing sea level rise due to climate change, require that the estuary sediment and sediment transport system be managed as a single unit. To better manage the system, its components, and human uses of the system, additional research and knowledge of the system is needed. Fortunately, new sediment science and modeling tools provide opportunities for a vastly improved understanding of the sediment system, predictive capabilities and analysis of potential individual and cumulative impacts of projects. As science informs management decisions, human activities and management strategies may need to be modified to protect and provide for existing and future infrastructure and ecosystem needs. (PDF contains 3 pages)

Diffraction integral formulas of the pulsed wave field in the temporal domain

To resolve the diffraction problems of the pulsed wave field directly in the temporal domain, we extend the Rayleigh diffraction integrals to the temporal domain and then discuss the approximation condition of this diffraction formula. (C) 1997 Optical Society of America.

Oscillatory integral operators related to pointwise convergence of Schrödinger operators

In this thesis we consider smooth analogues of operators studied in connection with the pointwise convergence of the solution, u(x,t), (x,t) ∈ ℝ^n x ℝ, of the free Schrodinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we develop strategies to capture the oscillations and obtain sharp L^2 → L^2 bounds. We then consider, for fixed smooth t(x), the restriction of u to the surface (x,t(x)). We find that u(x,t(x)) ∈ L^2(D^n) when the initial data is in a suitable L^2-Sobolev space H^8 (ℝ^n), where s depends on conditions on t.

Sustainable energy solutions for irrigation and harvesting in developing countries

One of the critical problems currently being faced by agriculture industry in developing nations is the alarming rate of groundwater depletion. Irrigation accounts for over 70% of the total groundwater withdrawn everyday. Compounding this issue is the use of polluting diesel generators to pump groundwater for irrigation. This has made irrigation not only the biggest consumer of groundwater but also one of the major contributors to green house gases. The aim of this thesis is to present a solution to the energy-water nexus. To make agriculture less dependent on fossil fuels, the use of a solar-powered Stirling engine as the power generator for on-farm energy needs is discussed. The Stirling cycle is revisited and practical and ideal Stirling cycles are compared. Based on agricultural needs and financial constraints faced by farmers in developing countries, the use of a Fresnel lens as a solar-concentrator and a Beta-type Stirling engine unit is suggested for sustainable power generation on the farms. To reduce the groundwater consumption and to make irrigation more sustainable, the conceptual idea of using a Stirling engine in drip irrigation is presented. To tackle the shortage of over 37 million tonnes of cold-storage in India, the idea of cost-effective solar-powered on-farm cold storage unit is discussed.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.