Simulating Solitons of the Sine-Gordon Equation using Variational Approximations and Hamiltonian Principles
|
|
- Cody Damon Tucker
- 5 years ago
- Views:
Transcription
1 Simulating Solitons of the Sine-Gordon Equation using Variational Approximations and Hamiltonian Principles By Evan Foley A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE STETSON UNIVERSITY 1
2 0.1 Acknowledgements I would like to thank Dr. Vogel for his patience, cooperation, and guidance with the research. I would like to thank the Stetson Undergraduate Research Experience (SURE) program for providing me the opportunity to work with Dr. Vogel in an one on one environment over the summer of
3 Contents 0.1 Acknowledgements Abstract Introduction 5 2 Approximation Method on Korteweg-de Vries Equations Original Modified Approximation Method on the Sine-Gordon Equation Gaussian Wave Trial Function Traveling Wave Solution Dynamic Solution Odd Oriented Gaussian Wave Trial Function Traveling Wave Solution Dynamic Solution Future Research 19 5 References 20 6 Appendices Mathematica Code List of Figures
4 0.2 Abstract Simulating Solitons of the Sine-Gordon Equation using Variational Approximations and Hamiltonian Principles By Evan Foley May 2013 Advisor: Dr. Thomas Vogel Department: Mathematics and Computer Science This project examines the underlying phenomena of solitons in partial differential equations using Hamilton s Principal and Variational Approximations. William Hamilton formulated the idea that physical models can be described in terms of energies. Using the concept of functionals, we can describe these energies in the form of the Lagrangian to find approximate solutions. Equations such as the Korteweg-de Vries (KdV), Modified KdV, and the Sine-Gordon will be examined to find localized structure (i.e. soliton) solutions. Interestingly enough, soltions behave differently than typical waves, due to their unique characteristics which will be addressed later. The solutions to these equations shall yield two results, prove the existence of solitons in the solution and provide an approximate solution for the PDEs that are being analyzed. These solutions will be compared to the exact solutions, to determine how accurate the approximation works on those particular equations. 4
5 1 Introduction In 1834, a Scottish naval engineer, named John Scott Russell, was conducting an experiment at the Union Canal in Scotland. He tried to establish a conversion factor between steampower and horsepower by tying two horses on opposite sides of the canal to a steamboat, while he observed on a horse on one side. While the horses were pulling the stationary steamboat, the ropes snapped and something unexpected happened. The boat was shaken around and a wave popped out in front of the boat, but the wave was not an ordinary wave. Russell described the wave as a rounded, smooth, and well-defined heap of water. He followed it for a couple of miles until the canal prevented him from moving onward. Russell tried to recreate the wave that he saw but had not been successful. Around that time, Newton and Bernoulli had formulated linear equations for explaining hydrodynamics, so Russells observation was rejected from physicists. By 1895, Diederik Korteweg and Gustav de Vries had derived a non-linear, partial differential equation, called the KdV equation, that would model the behavior of what will be known as a soliton wave observed by John Scott Russell [1]. A soliton can simply be described as a solitary wave with a localized structure that does not diminish over time. The soliton wave is also known as the Wave of Translation. A soliton wave is said to have 4 distinct properties: 1. Stable and travels over long distances. 2. Speed depends on height and width of wave on the depth of water. 3. Multiple waves never merge, rather, they pass through each other. 4. If the wave is too big for depth of water, it will split into two. William Hamilton developed the concept of energies describing physical systems and formalized Hamilton s Principle. Hamilton s Principle can be explained as follows: Of all possible paths along which a dynamical system may evolve within a specified time interval, the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energies which is called the action, S. S = t2 t 1 (T U)dt (1) 5
6 where L = T U (2) where L=f[x,y(x),y (x)] and is called the Lagrangian. In our case, we will integrate the action over R in the spatial domain. To minimize the action, we use the formula: δs δy = S y d dx S y = 0 (3) where y is a parameter of the action [1]. One thing to note is that the action is not considered a function, but a functional. In Variational Calculus, a functional is an operator that outputs numbers where the input is a function. For instance, a definite integral is considered a functional. Keeping the action in mind, there is a necessary condition that the lagrangian must satisfy called the Euler-Lagrange equation [2, p. 125]: L y d dx L y = 0 (4) where y=f(x). By applying the correct Lagrangian to this condition, the original partial differnetial equation should arise. 2 Approximation Method on Korteweg-de Vries Equations 2.1 Original The first equation that will be analyzed is the Korteweg-de Vries (KdV) Equation which is as follows [3]: u t + uu x + u xxx = 0 (5) This equation was originally designed to model the motion of a wave in shallow water. To start off, assume the soliton is in a traveling wave solution, meaning the only parameter 6
7 changing with respect with to time is it s position which can be described as ξ = x ct (6) To start the approximation, the trial function needs to be defined. Based on the geometry of a soliton wave, the gaussian wave will be used as the trial function: u(ξ) = Ae ξ 2 ρ 2 (7) Applying the necessary condition (3), the following Lagrangian can be found: L = 1 2 cu u3 1 2 (u ) 2 (8) The resulting action is: S = A2 π(2 3Aρ 2 9 2(cρ 2 + 1)) 36ρ (9) When minimizing the action, two Euler Lagrange equations show up since there are two parameters that we are interested in. S ρ = A2 π(2 3Aρ 2 9 2(cρ 2 1)) 36ρ 2 = 0 (10) S A = A π( 3Aρ 2 3 2(cρ 2 + 1)) = 0 (11) 6ρ Figure 1 shows the amplitude (red) and speed (blue) of the wave vary with respect to the wave width. Since the exact solution for the KdV equation is known, namely cξ u(ξ) = 3c sech 2 ( 2 ) (12) the approximations (blue) can be compared to the exact (red) solution in Figures 2 and 3. Looking at the graphs, the solutions obtained from the appoximation method very closely resemble the exact solutions, this shows how accurate this method is. 7
8 2.2 Modified Aside from the regular KdV equation, many modified versions have been created and tested, however, this research paper will only examine one [3]: u t + 6u 2 u x + u xxx = 0 (13) There will still be a the assumption that the soliton is in a stationary state from earlier, with the same trial function: u(ξ) = Ae ξ 2 ρ 2 (14) Applying the necessary condition (3), and the following Lagrangian is obtained L = 1 2 cu u4 1 2 (u ) 2 (15) The action becomes S = A2 π(a 2 ρ 2 2(cρ 2 + 1)) 4ρ (16) The two Euler Lagrange equations that spawn from minimizing the action are: S ρ = A2 π(a 2 ρ 2 + 2(1 cρ 2 )) 4ρ 2 = 0 (17) S A = A π(2a 2 ρ 2 2(cρ 2 + 1)) = 0 (18) 2ρ Figure 4 displays Amplitude (red) and speed (blue) vs. width of the soliton wave. Since the exact solution to this Modified KdV Equation is known, namely u(ξ) = c sech cξ (19) then a comparison is in order to determine how accurate the approximations (blue) are compared to exact (red) in Figures 5 and 6. Based on the graphs, the method used is producing very accurate results. 8
9 3 Approximation Method on the Sine-Gordon Equation Now that the method is known to produce reasonable results, it is time to finally test the method on the Sine-Gordon equation. Since there are multiple solutions for the Sine-Gordon Equation, we will examine a few of them. We will also use different trial functions since some of the solutions resemble the trial functions. The Sine Gordon equation is u tt u xx + sin u = 0 (20) 3.1 Gaussian Wave Trial Function The first trial funtion we will use is the same as the trial function used in the Korteweg-de Vries equations Traveling Wave Solution We first examine the Traveling Wave Solution (TWS), meaning the wave just moves along the x-axis and does not change shape or form. Recall the trial function being: u(ξ) = Ae ξ 2 ρ 2 (21) By applying the necessary condition (3) as well as the power series representation of sin(x), we get the following Lagrangian: L = 1 2 ( 1 c 2 ) (u ) 2 + u2 2! u4 4! + u6 6! u8 8! (22) This results in the following action: S = πa 2 (( 3 2A A A ) ρ c ) ρ (23) 9
10 The two following Euler-Lagrangian equations spawn when the action is minimized: (( S πa 3 2A 6 A = A A ) ρ c ) ρ ( π 18 2A A A ) A 2 ρ (24) S πa 2 ( ρ = 3 2A A A ) πa 2 (( 3 2A A A ) ρ c ) ρ 2 (25) Using numerics, the amplitude and the wave width can be approximated and solved for at varying speeds, which is shown in Figure Dynamic Solution Previously, the Sine-Gordon was examined and analyzed for a traveling wave solution, however, this assumes the wave s shape and speed do not change over time. It is more accurate to allow for breather soliton solutions to form. This is accomplished by allowing the wave s amplitude and width to vary with respect to time. Leaving behind the trial function: u(x, t) = A(t)e x 2 ρ(t) 2 (26) By taking this into effect, there is no transformation we can use that all of the previous examples used. Since the equation is not going to be transformed into an ODE, a new necessary condition for the lagrangian needs to be met. The new Euler-Lagrange equation becomes [2, p. 139] L u x L u x t L u t = 0 (27) Using the new necessary condition, the following Lagrangian can be obtained: L = 1 2 u2 t u2 x + u2 2! u4 4! + u6 6! u8 8! (28) 10
11 This results in the following action π S = ρ(t) ( A(t)ρ(t)A (t)ρ (t) ρ(t) 2 A (t) A(t) 2 ( 3ρ (t) 2 + 4ρ(t) 2 + 4) + 3 2A(t) 8 ρ(t) A(t) 6 ρ(t) A(t) 4 ρ(t) 2 ) (29) with the two Euler-Lagrangian equations S A(t) d πρ dt S (t) A(t) = ρ(t) 2 ( ρ(t)A (t)ρ (t) A(t)( 3ρ (t) 2 + 4ρ(t) 2 + 4) A(t) 7 ρ(t) A(t) 5 ρ(t) A(t) 3 ρ(t) 2 ) π ρ(t) ( ρ(t)A (t)ρ (t) A(t)( 3ρ (t) 2 + 4ρ(t) 2 + 4) A(t) 7 ρ(t) A(t) 5 ρ(t) A(t) 3 ρ(t) 2 ) π ρ(t) ( ρ(t)A (t)ρ (t) ρ(t)A (t)ρ (t) A (t)ρ (t) A (t)( 3ρ (t) 2 + 4ρ(t) 2 + 4) A(t) 6 ρ(t) 2 A (t) A(t) 4 ρ(t) 2 A (t) A(t) 2 ρ(t) 2 A (t) A(t) 7 ρ(t)ρ (t) A(t) 5 ρ(t)ρ (t) A(t) 3 ρ(t)ρ (t) A(t)(8ρ(t)ρ (t) 6ρ (t)ρ (t))) (30) 11
12 S ρ(t) d dt S 1 ρ(t) = πρ ρ(t) 2 (t)(6 2ρ(t)A(t) ρ(t)A(t) ρ(t)A(t) ρ(t)A(t) A (t)ρ (t)a(t) ρ(t)A (t) 2 ) π ρ(t) (6 2ρ(t)A(t) ρ(t)A(t) ρ(t)A(t) ρ(t)A(t) A (t)ρ (t)a(t) ρ(t)A (t) 2 ) πρ (t) ρ(t) 3 (3 2ρ(t) 2 A(t) ρ(t) 2 A(t) ρ(t) 2 A(t) (4ρ(t) 2 3ρ (t) 2 + 4)A(t) ρ(t)A (t)ρ (t)a(t) π 2ρ(t) 2 A (t) 2 ) ρ(t) 2 (3 2ρ(t) 2 A(t) ρ(t) 2 A(t) ρ(t) 2 A(t) (4ρ(t) 2 3ρ (t) 2 + 4)A(t) ρ(t)A (t)ρ (t)a(t) ρ(t) 2 A (t) 2 ) π ρ(t) (6 2ρ (t)a(t) ρ(t)A (t)a(t) ρ (t)a(t) ρ(t)A (t)a(t) ρ (t)a(t) ρ(t)A (t)a(t) ρ (t)a(t) ρ(t)A (t)a(t) ρ (t)a (t)a(t) A (t)ρ (t)a(t) A (t) 2 ρ (t) ρ(t)A (t)a (t)) π ρ(t) 2 (6 2ρ(t)ρ (t)a(t) ρ(t) 2 A (t)a(t) ρ(t)ρ (t)a(t) ρ(t) 2 A (t)a(t) ρ(t)ρ (t)a(t) ρ(t) 2 A (t)a(t) (8ρ(t)ρ (t) 6ρ (t)ρ (t))a(t) A (t)ρ (t) 2 A(t) A (t)(4ρ(t) 2 3ρ (t) 2 + 4)A(t) ρ(t)ρ (t)a (t)a(t) ρ(t)A (t)ρ (t)a(t) ρ(t)A (t) 2 ρ (t) ρ(t) 2 A (t)a (t)) (31) In order to numerically solve for the amplitude and width of the wave using the two ODEs that were just calculated, four initial conditions must be known. It is somewhat displeasing to just plug in random numbers and see where the solution is heading towards. It is more beneficial to use initial conditions that had some sort of relevance to one each other. This can be accomplished by taking an ordered triplet of values from the TWS section. By taking a particular speed, amplitude, and width trio, we can use these values as initial conditions. However, we still need two more conditions. In general, it can be observed that a wave s amplitude tends to change more often and rapidly than it s width. With this idea, we can assume the initial change in the wave width is roughly zero. With that in mind, now we need one more condition. We would like the last initial condition to be 12
13 how fast the wave s amplitude is changing initially. This can be acquired by observation and some simple algebra. Let us examine the derivatives of u. u t (x, t) = A (t)e x 2 ρ(t) 2 + 2x2 e x 2 ρ 2 A(t)ρ (t) ρ(t) 3 (32) u x (x, t) = x 2 2xe ρ(t) 2 A(t) ρ(t) 2 (33) Thus, u t (x, t) = A (t)e x 2 ρ(t) 2 xρ (t) ρ(t) u x(x, t) (34) Observe that x = 0 (35) u x (x, t) = 0 (36) u t (x, 0) c (37) (37) holds true since u t represents how fast the wave is moving with respect to time, which is the same as the wave speed. Plugging in x = t = 0 into (34) yields A (0) = c (38) The final four intital conditions to the problem are A(0) = A (39) ρ(0) = ρ (40) A (0) = c (41) ρ (0) = 0 (42) where A, ρ, c are chosen values from the TWS trial earlier. With these values, A(t) and ρ(t) can be numerically calculated. Unfortunately, according to the calculations, the wave immediately dies 13
14 off after a fraction of a second. Possible solutions to this problem will be addressed later. 3.2 Odd Oriented Gaussian Wave Trial Function There is a set of solutions that do not resemble the trial that was previously being used. This other trial function can be thought of as a gaussian wave oriented as an odd function, hence the title Traveling Wave Solution With a slight modification, and the Lorentzian transformation, the new trial function that will be used is u(ξ) = Aξe ξ 2 ρ 2 (43) Once again we have the Lagrangian L = 1 2 (1 c2 )(u ) 2 + u2 2! u4 4! + u6 6! u8 8! (44) Hence the action is S = A2 πρ( (1 c 2 ) ρ A 2 ρ A 4 ρ A 6 ρ 8 ) (45) The two Euler Lagrange equations become S (46) A = A2 πρ( Aρ A 3 ρ A 5 ρ 8 ) A πρ( (1 c 2 ) ρ A 2 ρ A 4 ρ A 6 ρ 8 ) S ρ = A2 πρ( ρ A 2 ρ A 4 ρ A 6 ρ 7 ) A2 π( (1 c 2 ) ρ A 2 ρ A 4 ρ A 6 ρ 8 ) (47) Once again, letting the wave speed vary, we can solve for the amplitude and width, represented in Figure 8. 14
15 3.2.2 Dynamic Solution Earlier, the Sine-Gordon was examined and analyzed for a traveling wave solution, however, this assumes that the wave s shape and speed do not change over time, only position would change. It is better to include the extra possibilities such as a change in amplitude, width, and speed. I.e.: u(x, t) = A(t)xe (x ζ(t)) 2 ρ(t) 2 (48) The Lagrangian for the dynamically changing Sine-Gordon Wave is L = 1 2 u2 t u2 x + u2 2! u4 4! + u6 6! u8 8! (49) The following action can be calculated π S = ρ(t) ( A(t)ρ(t)A (t)(8ζ(t)ρ(t)ζ (t) + 4ζ(t) 2 ρ (t) + 3ρ(t) 2 ρ (t)) ρ(t) 2 A (t) 2 (4ζ(t) 2 + ρ(t) 2 ) A(t) 2 (4ζ(t) 2 ( 4ζ (t) 2 3ρ (t) 2 + 4ρ(t) 2 + 4) 48ζ(t)ρ(t)ζ (t)ρ (t) + ρ(t) 2 ( 12ζ (t) 2 15ρ (t) 2 + 4ρ(t) )) +27 2A(t) 8 ρ(t) 2 (114688ζ(t) 6 ρ(t) ζ(t) 4 ρ(t) ζ(t) 2 ρ(t) ζ(t) ρ(t) 8 ) A(t) 6 ρ(t) 2 (720ζ(t) 4 ρ(t) ζ(t) 2 ρ(t) ζ(t) 6 + 5ρ(t) 6 ) A(t) 4 (48ζ(t) 2 ρ(t) ζ(t) 4 ρ(t) 2 + 3ρ(t) 6 )) (50) 15
16 which leaves the three Euler-Lagrangian equations: S A(t) d dt S A(t) πρ (t) = ρ(t) 2 (216 2ρ(t) 2 (65536ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t) ρ(t) 8 )A(t) ρ(t) 2 (576ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) 2 + 5ρ(t) 6 )A(t) (3ρ(t) ζ(t) 2 ρ(t) ζ(t) 4 ρ(t) 2 )A(t) (4(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4)ζ(t) 2 48ρ(t)ζ (t)ρ (t)ζ(t) + ρ(t) 2 (4ρ(t) 2 12ζ (t) 2 15ρ (t) ))A(t) ρ(t)A (t)(4ρ (t)ζ(t) 2 + 8ρ(t)ζ (t)ζ(t) + 3ρ(t) 2 ρ (t))) π ρ(t) (216 2ρ(t) 2 (65536ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t) ρ(t) 8 )A(t) ρ(t) 2 (576ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) 2 + 5ρ(t) 6 )A(t) (3ρ(t) ζ(t) 2 ρ(t) ζ(t) 4 ρ(t) 2 )A(t) (4(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4)ζ(t) 2 48ρ(t)ζ (t)ρ (t)ζ(t) + ρ(t) 2 (4ρ(t) 2 12ζ (t) 2 15ρ (t) ))A(t) ρ(t)A (t)(4ρ (t)ζ(t) 2 + 8ρ(t)ζ (t)ζ(t) + 3ρ(t) 2 ρ (t))) π ρ(t) (432 2ρ(t)(65536ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t) ρ(t) 8 )ρ (t)a(t) ρ(t) 2 (524288ζ (t)ζ(t) ρ(t)ρ (t)ζ(t) ρ(t) 2 ζ (t)ζ(t) ρ(t) 3 ρ (t)ζ(t) ρ(t) 4 ζ (t)ζ(t) ρ(t) 5 ρ (t)ζ(t) ρ(t) 6 ζ (t)ζ(t) + 840ρ(t) 7 ρ (t))a(t) ρ(t) 2 (65536ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t) ρ(t) 8 )A (t)a(t) ρ(t)(576ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) 2 + 5ρ(t) 6 )ρ (t)a(t) ρ(t) 2 (3456ζ (t)ζ(t) ρ(t)ρ (t)ζ(t) ρ(t) 2 ζ (t)ζ(t) ρ(t) 3 ρ (t)ζ(t) ρ(t) 4 ζ (t)ζ(t) + 30ρ(t) 5 ρ (t))a(t) ρ(t) 2 (576ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) 2 + 5ρ(t) 6 )A (t)a(t) (18ρ (t)ρ(t) ζ(t)ζ (t)ρ(t) ζ(t) 2 ρ (t)ρ(t) ζ(t) 3 ζ (t)ρ(t) ζ(t) 4 ρ (t)ρ(t))a(t) (3ρ(t) ζ(t) 2 ρ(t) ζ(t) 4 ρ(t) 2 )A (t)a(t) (4(8ρ(t)ρ (t) 6ρ (t)ρ (t) 8ζ (t)ζ (t))ζ(t) 2 48ζ (t)ρ (t) 2 ζ(t) + 8ζ (t)(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4)ζ(t) 48ρ(t)ρ (t)ζ (t)ζ(t) 48ρ(t)ζ (t)ρ (t)ζ(t) 48ρ(t)ζ (t) 2 ρ (t) + 2ρ(t)ρ (t)(4ρ(t) 2 12ζ (t) 2 15ρ (t) ) + ρ(t) 2 (8ρ(t)ρ (t) 30ρ (t)ρ (t) 24ζ (t)ζ (t)))a(t) A (t)ρ (t)(4ρ (t)ζ(t) 2 + 8ρ(t)ζ (t)ζ(t) + 3ρ(t) 2 ρ (t)) A (t)(4(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4)ζ(t) 2 48ρ(t)ζ (t)ρ (t)ζ(t) + ρ(t) 2 (4ρ(t) 2 12ζ (t) 2 15ρ (t) )) ρ(t)(4ρ (t)ζ(t) 2 + 8ρ(t)ζ (t)ζ(t) + 3ρ(t) 2 ρ (t))a (t) ρ(t)A (t)(4ρ (t)ζ(t) ζ (t)ρ (t)ζ(t) + 8ρ(t)ζ (t)ζ(t) + 8ρ(t)ζ (t) 2 + 6ρ(t)ρ (t) 2 + 3ρ(t) 2 ρ (t))) (51) 16
17 S ρ(t) d dt S ρ(t) = Refer to Appendix 1 for Mathematica Code (52) S ζ(t) d dt S ζ(t) πρ (t) = ρ(t) 2 (27 2ρ(t) 2 (524288ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t))a(t) ρ(t) 2 (3456ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t))a(t) (96ζ(t)ρ(t) ζ(t) 3 ρ(t) 2 )A(t) (8ζ(t)(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4) 48ρ(t)ζ (t)ρ (t))a(t) ρ(t)A (t)(8ρ(t)ζ (t) + 8ζ(t)ρ (t))a(t) ζ(t)ρ(t) 2 A (t) 2 ) π ρ(t) (27 2ρ(t) 2 (524288ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t))a(t) ρ(t) 2 (3456ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t))a(t) (96ζ(t)ρ(t) ζ(t) 3 ρ(t) 2 )A(t) (8ζ(t)(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4) 48ρ(t)ζ (t)ρ (t))a(t) ρ(t)A (t)(8ρ(t)ζ (t) + 8ζ(t)ρ (t))a(t) ζ(t)ρ(t) 2 A (t) 2 ) π ρ(t) (54 2ρ(t)(524288ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t))ρ (t)a(t) ρ(t) 2 ( ζ (t)ζ(t) ρ(t)ρ (t)ζ(t) ρ(t) 2 ζ (t)ζ(t) ρ(t) 3 ρ (t)ζ(t) ρ(t) 4 ζ (t)ζ(t) ρ(t) 5 ρ (t)ζ(t) ρ(t) 6 ζ (t))a(t) ρ(t) 2 (524288ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t) ρ(t) 6 ζ(t))a (t)a(t) ρ(t)(3456ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t))ρ (t)a(t) ρ(t) 2 (17280ζ (t)ζ(t) ρ(t)ρ (t)ζ(t) ρ(t) 2 ζ (t)ζ(t) ρ(t) 3 ρ (t)ζ(t) + 360ρ(t) 4 ζ (t))a(t) ρ(t) 2 (3456ζ(t) ρ(t) 2 ζ(t) ρ(t) 4 ζ(t))a (t)a(t) (96ζ (t)ρ(t) ζ(t)ρ (t)ρ(t) ζ(t) 2 ζ (t)ρ(t) ζ(t) 3 ρ (t)ρ(t))a(t) (96ζ(t)ρ(t) ζ(t) 3 ρ(t) 2 )A (t)a(t) ( 48ζ (t)ρ (t) 2 48ρ(t)ζ (t)ρ (t) + 8ζ (t)(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4) 48ρ(t)ζ (t)ρ (t) + 8ζ(t)(8ρ(t)ρ (t) 6ρ (t)ρ (t) 8ζ (t)ζ (t)))a(t) A (t)ρ (t)(8ρ(t)ζ (t) + 8ζ(t)ρ (t))a(t) A (t)(8ζ(t)(4ρ(t) 2 4ζ (t) 2 3ρ (t) 2 + 4) 48ρ(t)ζ (t)ρ (t))a(t) ρ(t)(8ρ(t)ζ (t) + 8ζ(t)ρ (t))a (t)a(t) ρ(t)A (t)(16ζ (t)ρ (t) + 8ρ(t)ζ (t) + 8ζ(t)ρ (t))a(t) ρ(t) 2 A (t) 2 ζ (t) ζ(t)ρ(t)A (t) 2 ρ (t) ρ(t)A (t) 2 (8ρ(t)ζ (t) + 8ζ(t)ρ (t)) ζ(t)ρ(t) 2 A (t)a (t)) (53) In order to solve this system of ODEs, there needs to be initial conditions, which can be obtained by examining the Traveling Wave Solution, similar to the Gaussian Wave Trial case. We need to 17
18 choose a trio of values for the amplitude and width relative to speed to use as the initial conditions. However, this only satisifies three out of the six initial conditions necessary. The third condition that is met is the ζ (0) term which can be considered as the initial speed of the wave, in other words ζ (0) u t (x, 0) c (54) Just like in the Gaussian trial case, we will assume ρ (0) = 0 since the wave s width does not change much. For the ζ(0) case, we need to be careful about what value to start it at. It may not be obvious, but choosing ζ(0) = 0 is dangerous because undefined answers appear during the solving process, so we need to choose another value. It s value must be something relatable to the other conditions in some way, so we will choose a value to allow for simplicity in some sort of calculation. Suppose ζ(0) = ρ(0) 2 (55) the reason behind this assumption may become apparent in the upcoming calculations. Using the same argument and methods as in the Gaussian Trial case, meaning we solve for A (0) by writing u t in terms of u x, this leads to A (0) = 2c (1 + A(0)) (56) ρ(0) This gives six initial conditions to use A(0) = A (57) A (0) = 2c (1 + A) (58) ρ ρ(0) = ρ (59) ρ (0) = 0 (60) ζ(0) = ρ 2 (61) ζ (0) = c (62) By numerically solving the 3 Ordinary Differential Equations with the initial conditions above, we come to the conclusion that the soliton once again dies quickly. A solution to the problem will be addressed later. 18
19 4 Future Research Negative amplitudes will be accounted for due to loss of generality. More values of speed will be examined in the Traveling Wave Solution case to see if there are other solutions besides what is in the interval between 0 and 1. There still needs to be a comparison to an exact solution to see how accurate the results really are. There is the idea of examining multi-soliton solutions, once this single soliton case is completed, however that may put off for future research in graduate school and such. To address the issues during the Dynamic solution cases, it seems the initial values are not forming a practical system, it appears the system is stiff. More values of A, ρ, c, and the initial conditions need to be tested in the system until a viable solution is obtained. 19
20 5 References References [1] D.J. Kaup, T.K. Vogel, Quantitative Measurement of Variational Approximations, Physics Letters A PLA 362 (2007) [2] J. David Logan Applied Mathematics. 2nd ed. A Wiley-Interscience Publication [3] P.G. Drazin, R.S. Johnson Solitons: an introduction Cambridge University Press Appendices 6.1 Mathematica Code U[x,t]:=-A[t]*x*Eˆ((-(x-ζ[t])ˆ2)((ρ[t])ˆ2)) Ux=D[U[x,t],x] Ut=D[U[x,t],t] Lp=- 1 2 *(Ut)ˆ *(Ux)ˆ *(U[x,t])ˆ2-1 4! ((U[x,t])ˆ4)+ 1 6! ((U[x,t])ˆ6)- 1 8! ((U[x,t])ˆ8) S=Assuming[ρ[t]>0,Integrate[Lp,x,-Infinity,Infinity]] el1=d[s,a[t]]-d[d[s,a[t]],t] el2=d[s,[t]]-d[d[s,ρ[t]],t] el3=d[s,[t]]-d[d[s,ζ[t]],t] NDSolve[(el1==0,el2==0,el3==0,A[0]== ,ρ[0]== , ζ[0]== (sqrt[2]),ζ [0]==.5,ρ [0]==0,A [0]== ), (A[t],ρ[t],ζ[t]),(t,0,10)] 6.2 List of Figures 20
21 Figure 1: A plot of the wave s amplitude and width vs. it s speed Figure 2: A plot of the wave s approximate and exact amplitude vs. it s speed. 21
22 Figure 3: A plot of the wave s approximate and exact width vs. it s speed Figure 4: A plot of the wave s amplitude and speed vs. it s width. 22
23 Figure 5: A plot of the wave s approximate and exact amplitude vs. it s speed Figure 6: A plot of the wave s approximate and exact width vs. it s speed. 23
24 Figure 7: The wave s amplitude and width paired with varying values of speed Figure 8: The wave s amplitude and width paired with varying values of speed. 24
Solitons : An Introduction
Solitons : An Introduction p. 1/2 Solitons : An Introduction Amit Goyal Department of Physics Panjab University Chandigarh Solitons : An Introduction p. 2/2 Contents Introduction History Applications Solitons
More informationWhat we do understand by the notion of soliton or rather solitary wave is a wave prole which is very stable in the following sense
Introduction to Waves and Solitons What we do understand by the notion of soliton or rather solitary wave is a wave prole which is very stable in the following sense i) It is localized, which means it
More informationNotes on the Inverse Scattering Transform and Solitons. November 28, 2005 (check for updates/corrections!)
Notes on the Inverse Scattering Transform and Solitons Math 418 November 28, 2005 (check for updates/corrections!) Among the nonlinear wave equations are very special ones called integrable equations.
More informationUniversity of Central Florida. Quantifying Variational Solutions
University of Central Florida Institute for Simulation & Training and epartment of Mathematics and CREOL.J. Kaup and Thomas K. Vogel Quantifying Variational Solutions (Preprint available at http://gauss.math.ucf.edu/~kaup/)
More informationConvergence of operator splitting for the KdV equation
Convergence of operator splitting for the KdV equation H. Holden Norwegian University of Science and Technology Trondheim, Norway Joint work with K.H. Karlsen, N. H. Risebro, and T. Tao Institute for Mathematics
More informationShock Waves & Solitons
1 / 1 Shock Waves & Solitons PDE Waves; Oft-Left-Out; CFD to Follow Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from
More informationAre Solitary Waves Color Blind to Noise?
Are Solitary Waves Color Blind to Noise? Dr. Russell Herman Department of Mathematics & Statistics, UNCW March 29, 2008 Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise? 3
More informationFurther Ordinary Differential Equations
Advanced Higher Notes (Unit ) Prerequisites: Standard integrals; integration by substitution; integration by parts; finding a constant of integration; solving quadratic equations (with possibly complex
More informationMACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
. MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Willy Hereman Mathematics Department and Center for the Mathematical Sciences University of Wisconsin at
More informationDispersion relations, stability and linearization
Dispersion relations, stability and linearization 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient partial differential
More informationMaking Waves in Vector Calculus
Making Waves in Vector Calculus J. B. Thoo Yuba College 2014 MAA MathFest, Portland, OR This presentation was produced
More informationGroup analysis, nonlinear self-adjointness, conservation laws, and soliton solutions for the mkdv systems
ISSN 139-5113 Nonlinear Analysis: Modelling Control, 017, Vol., No. 3, 334 346 https://doi.org/10.15388/na.017.3.4 Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions for the
More informationPropagation of Solitons Under Colored Noise
Propagation of Solitons Under Colored Noise Dr. Russell Herman Departments of Mathematics & Statistics, Physics & Physical Oceanography UNC Wilmington, Wilmington, NC January 6, 2009 Outline of Talk 1
More informationSolutions of differential equations using transforms
Solutions of differential equations using transforms Process: Take transform of equation and boundary/initial conditions in one variable. Derivatives are turned into multiplication operators. Solve (hopefully
More informationAlexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011
Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan,
More informationChapter 7. Homogeneous equations with constant coefficients
Chapter 7. Homogeneous equations with constant coefficients It has already been remarked that we can write down a formula for the general solution of any linear second differential equation y + a(t)y +
More informationThe Solitary Wave Solutions of Zoomeron Equation
Applied Mathematical Sciences, Vol. 5, 011, no. 59, 943-949 The Solitary Wave Solutions of Zoomeron Equation Reza Abazari Deparment of Mathematics, Ardabil Branch Islamic Azad University, Ardabil, Iran
More informationA Guided Tour of the Wave Equation
A Guided Tour of the Wave Equation Background: In order to solve this problem we need to review some facts about ordinary differential equations: Some Common ODEs and their solutions: f (x) = 0 f(x) =
More informationNorth Carolina State University
North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
More informationCompacton-like solutions in some nonlocal hydrodynamic-type models
Compacton-like solutions in some nonlocal hydrodynamic-type models Vsevolod Vladimirov AGH University of Science and technology, Faculty of Applied Mathematics Protaras, October 26, 2008 WMS AGH Compactons
More informationNonlinear Optics (WiSe 2015/16) Lecture 7: November 27, 2015
Review Nonlinear Optics (WiSe 2015/16) Lecture 7: November 27, 2015 Chapter 7: Third-order nonlinear effects (continued) 7.6 Raman and Brillouin scattering 7.6.1 Focusing 7.6.2 Strong conversion 7.6.3
More informationTowards Soliton Computer Based on Solitary Wave Solution of Maxwell-Dirac equation: A Plausible Alternative to Manakov System
Towards Soliton Computer Based on Solitary Wave Solution of Maxwell-Dirac equation: A Plausible Alternative to Manakov System Victor Christianto* 1, Florentin Smarandache 1 Malang Institute of Agriculture
More informationMultitime sine-gordon solitons via geometric characteristics
Multitime sine-gordon solitons via geometric characteristics Laura Gabriela Matei, Constantin Udrişte Abstract. Our paper introduces and studies the idea of multitime evolution in the context of solitons.
More informationSoliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric and Hyperbolic Function Methods.
ISSN 1749-889 (print), 1749-897 (online) International Journal of Nonlinear Science Vol.14(01) No.,pp.150-159 Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric
More informationPartial Differential Equations Summary
Partial Differential Equations Summary 1. The heat equation Many physical processes are governed by partial differential equations. temperature of a rod. In this chapter, we will examine exactly that.
More informationMath 2930 Worksheet Final Exam Review
Math 293 Worksheet Final Exam Review Week 14 November 3th, 217 Question 1. (* Solve the initial value problem y y = 2xe x, y( = 1 Question 2. (* Consider the differential equation: y = y y 3. (a Find the
More informationMATH141: Calculus II Exam #4 review solutions 7/20/2017 Page 1
MATH4: Calculus II Exam #4 review solutions 7/0/07 Page. The limaçon r = + sin θ came up on Quiz. Find the area inside the loop of it. Solution. The loop is the section of the graph in between its two
More informationKdV equation obtained by Lie groups and Sturm-Liouville problems
KdV equation obtained by Lie groups and Sturm-Liouville problems M. Bektas Abstract In this study, we solve the Sturm-Liouville differential equation, which is obtained by using solutions of the KdV equation.
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationPart 1. The simple harmonic oscillator and the wave equation
Part 1 The simple harmonic oscillator and the wave equation In the first part of the course we revisit the simple harmonic oscillator, previously discussed in di erential equations class. We use the discussion
More informationThe Stochastic KdV - A Review
The Stochastic KdV - A Review Russell L. Herman Wadati s Work We begin with a review of the results from Wadati s paper 983 paper [4] but with a sign change in the nonlinear term. The key is an analysis
More informationHere we consider soliton solutions of the Korteweg-de Vries (KdV) equation. This equation is given by
17 Solitons Lab Objective: We study traveling wave solutions of the Korteweg-de Vries (KdV) equation, using a pseudospectral discretization in space and a Runge-Kutta integration scheme in time. Here we
More informationConservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations.
Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Alexey Shevyakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon,
More informationA Very General Hamilton Jacobi Theorem
University of Salerno 9th AIMS ICDSDEA Orlando (Florida) USA, July 1 5, 2012 Introduction: Generalized Hamilton-Jacobi Theorem Hamilton-Jacobi (HJ) Theorem: Consider a Hamiltonian system with Hamiltonian
More informationThe General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics 015; 4(5): 335-341 Published online August 16 015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.0150405.11 ISSN: 38-5605 (Print); ISSN: 38-5613
More informationSimulation for Different Order Solitons in Optical Fibers and the Behaviors of Kink and Antikink Solitons
Simulation for Different Order Solitons in Optical Fibers and the Behaviors of Kink and Antikink Solitons MOHAMMAD MEHDI KARKHANEHCHI and MOHSEN OLIAEE Department of Electronics, Faculty of Engineering
More informationSketchy Notes on Lagrangian and Hamiltonian Mechanics
Sketchy Notes on Lagrangian and Hamiltonian Mechanics Robert Jones Generalized Coordinates Suppose we have some physical system, like a free particle, a pendulum suspended from another pendulum, or a field
More informationSpectral stability of periodic waves in dispersive models
in dispersive models Collaborators: Th. Gallay, E. Lombardi T. Kapitula, A. Scheel in dispersive models One-dimensional nonlinear waves Standing and travelling waves u(x ct) with c = 0 or c 0 periodic
More informationSolving Nonlinear Wave Equations and Lattices with Mathematica. Willy Hereman
Solving Nonlinear Wave Equations and Lattices with Mathematica Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs home/whereman/
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
More informationComputer Problems for Fourier Series and Transforms
Computer Problems for Fourier Series and Transforms 1. Square waves are frequently used in electronics and signal processing. An example is shown below. 1 π < x < 0 1 0 < x < π y(x) = 1 π < x < 2π... and
More informationLecture Notes on the Fermi-Pasta-Ulam-Tsingou Problem
Lecture Notes on the Fermi-Pasta-Ulam-Tsingou Problem Andrew Larkoski November 9, 016 Because this is a short week, we re going to just discuss some fun aspects of computational physics. Also, because
More informationMath 575-Lecture 26. KdV equation. Derivation of KdV
Math 575-Lecture 26 KdV equation We look at the KdV equations and the so-called integrable systems. The KdV equation can be written as u t + 3 2 uu x + 1 6 u xxx = 0. The constants 3/2 and 1/6 are not
More informationHow to Use Calculus Like a Physicist
How to Use Calculus Like a Physicist Physics A300 Fall 2004 The purpose of these notes is to make contact between the abstract descriptions you may have seen in your calculus classes and the applications
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More informationNew Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equations
ISSN 1749-3889 print), 1749-3897 online) International Journal of Nonlinear Science Vol.008) No.1,pp.4-5 New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Euations
More informationNew approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations
Volume 28, N. 1, pp. 1 14, 2009 Copyright 2009 SBMAC ISSN 0101-8205 www.scielo.br/cam New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations HASSAN A. ZEDAN Mathematics
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationNonlinear Optics (WiSe 2017/18) Lecture 12: November 28, 2017
7.6 Raman and Brillouin scattering 7.6.1 Focusing Nonlinear Optics (WiSe 2017/18) Lecture 12: November 28, 2017 7.6.2 Strong conversion 7.6.3 Stimulated Brillouin scattering (SBS) 8 Optical solitons 8.1
More informationPartial Differential Equations
Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives with respect to those variables. Most (but
More informationMaking Waves in Multivariable Calculus
Making Waves in Multivariable Calculus J. B. Thoo Yuba College 2014 CMC3 Fall Conference, Monterey, Ca This presentation
More informationSoliton and Numerical Solutions of the Burgers Equation and Comparing them
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 52, 2547-2564 Soliton and Numerical Solutions of the Burgers Equation and Comparing them Esmaeel Hesameddini and Razieh Gholampour Shiraz University of
More informationDark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate
Dark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate Indu Satija, George Mason Univ & National Institute of Standard and Tech ( NIST) collaborator:radha Balakrishnan, Institute of Mathematical
More informationHere we consider soliton solutions of the Korteweg-de Vries (KdV) equation. This equation is given by u t + u u x + 3 u
Lab 3 Solitons Lab Objective: We study traveling wave solutions of the Korteweg-de Vries (KdV) equation, using a pseudospectral discretization in space and a Runge-Kutta integration scheme in time. Here
More informationSecond-Order Homogeneous Linear Equations with Constant Coefficients
15 Second-Order Homogeneous Linear Equations with Constant Coefficients A very important class of second-order homogeneous linear equations consists of those with constant coefficients; that is, those
More informationNonlinear Optics (WiSe 2018/19) Lecture 7: November 30, 2018
Nonlinear Optics (WiSe 2018/19) Lecture 7: November 30, 2018 7 Third-order nonlinear effects (continued) 7.6 Raman and Brillouin scattering 7.6.1 Focusing 7.6.2 Strong conversion 7.6.3 Stimulated Brillouin
More informationCalculus Favorite: Stirling s Approximation, Approximately
Calculus Favorite: Stirling s Approximation, Approximately Robert Sachs Department of Mathematical Sciences George Mason University Fairfax, Virginia 22030 rsachs@gmu.edu August 6, 2011 Introduction Stirling
More informationA Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation
A Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation arxiv:math/6768v1 [math.ap] 6 Jul 6 Claire David, Rasika Fernando, and Zhaosheng Feng Université Pierre et Marie Curie-Paris
More informationAlgebra Year 10. Language
Algebra Year 10 Introduction In Algebra we do Maths with numbers, but some of those numbers are not known. They are represented with letters, and called unknowns, variables or, most formally, literals.
More informationKdV soliton solutions to a model of hepatitis C virus evolution
KdV soliton solutions to a model of hepatitis C virus evolution T. Telksnys, Z. Navickas, M. Ragulskis Kaunas University of Technology Differential Equations and Applications, Brno 2017 September 6th,
More informationTraffic Flow I. Kurt Bryan
Introduction Traffic Flow I Kurt Bryan Consider a stretch of freeway with no entrance or eit ramps. On such a stretch of freeway we clearly have conservation of cars, barring any wrecks. Thus in our conservation
More informationA Novel Nonlinear Evolution Equation Integrable by the Inverse Scattering Method
Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part, 384 39 A Novel Nonlinear Evolution Equation Integrable by the Inverse Scattering Method Vyacheslav VAKHNENKO and John PARKES
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 6
Strauss PDEs 2e: Section 3 - Exercise Page of 6 Exercise Carefully derive the equation of a string in a medium in which the resistance is proportional to the velocity Solution There are two ways (among
More informationTHE SECANT METHOD. q(x) = a 0 + a 1 x. with
THE SECANT METHOD Newton s method was based on using the line tangent to the curve of y = f (x), with the point of tangency (x 0, f (x 0 )). When x 0 α, the graph of the tangent line is approximately the
More informationMITOCW MITRES18_005S10_DiffEqnsMotion_300k_512kb-mp4
MITOCW MITRES18_005S10_DiffEqnsMotion_300k_512kb-mp4 PROFESSOR: OK, this lecture, this day, is differential equations day. I just feel even though these are not on the BC exams, that we've got everything
More informationWhat Is a Soliton? by Peter S. Lomdahl. Solitons in Biology
What Is a Soliton? by Peter S. Lomdahl A bout thirty years ago a remarkable discovery was made here in Los Alamos. Enrico Fermi, John Pasta, and Stan Ulam were calculating the flow of energy in a onedimensional
More informationBoundary value problems for partial differential equations
Boundary value problems for partial differential equations Henrik Schlichtkrull March 11, 213 1 Boundary value problem 2 1 Introduction This note contains a brief introduction to linear partial differential
More informationChapter Thirteen: Traveling Waves and the Wave Equation in Air, Water and the Ether
Chapter Thirteen: Traveling Waves and the Wave Equation in Air, Water and the Ether We have been discussing the waves that are found in the string of a guitar or violin and how they are the physical reality
More informationMath53: Ordinary Differential Equations Autumn 2004
Math53: Ordinary Differential Equations Autumn 2004 Unit 2 Summary Second- and Higher-Order Ordinary Differential Equations Extremely Important: Euler s formula Very Important: finding solutions to linear
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationMath 124A October 11, 2011
Math 14A October 11, 11 Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This corresponds to a string of infinite length. Although
More informationThe (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics
Vol.3, Issue., Jan-Feb. 3 pp-369-376 ISSN: 49-6645 The ('/) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics J.F.Alzaidy Mathematics Department, Faculty
More informationClass Meeting # 2: The Diffusion (aka Heat) Equation
MATH 8.52 COURSE NOTES - CLASS MEETING # 2 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 2: The Diffusion (aka Heat) Equation The heat equation for a function u(, x (.0.). Introduction
More informationContinuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China
Continuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China the 3th GCOE International Symposium, Tohoku University, 17-19
More informationAn Invitation to Mathematics Prof. Sankaran Vishwanath Institute of Mathematical Sciences, Chennai. Unit I Polynomials Lecture 1A Introduction
An Invitation to Mathematics Prof. Sankaran Vishwanath Institute of Mathematical Sciences, Chennai Unit I Polynomials Lecture 1A Introduction Hello and welcome to this course titled An Invitation to Mathematics.
More informationTWO SOLITON SOLUTION OF TZITZEICA EQUATION. Bulgarian Academy of Sciences, Sofia, Bulgaria
TWO SOLITON SOLUTION OF TZITZEICA EQUATION Corina N. Babalic 1,2, Radu Constantinescu 1 and Vladimir S. Gerdjikov 3 1 Dept. of Physics, University of Craiova, Romania 2 Dept. of Theoretical Physics, NIPNE,
More informationSystems of Linear ODEs
P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here
More informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
More informationMATHEMATICAL STRUCTURES IN CONTINUOUS DYNAMICAL SYSTEMS
MATHEMATICAL STRUCTURES IN CONTINUOUS DYNAMICAL SYSTEMS Poisson Systems and complete integrability with applications from Fluid Dynamics E. van Groesen Dept. of Applied Mathematics University oftwente
More informationFinite Mathematics : A Business Approach
Finite Mathematics : A Business Approach Dr. Brian Travers and Prof. James Lampes Second Edition Cover Art by Stephanie Oxenford Additional Editing by John Gambino Contents What You Should Already Know
More informationApplication of linear combination between cubic B-spline collocation methods with different basis for solving the KdV equation
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 4, No. 3, 016, pp. 191-04 Application of linear combination between cubic B-spline collocation methods with different basis
More informationz x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.
Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These
More informationTHE WAVE EQUATION. F = T (x, t) j + T (x + x, t) j = T (sin(θ(x, t)) + sin(θ(x + x, t)))
THE WAVE EQUATION The aim is to derive a mathematical model that describes small vibrations of a tightly stretched flexible string for the one-dimensional case, or of a tightly stretched membrane for the
More informationConservation laws for the geodesic equations of the canonical connection on Lie groups in dimensions two and three
Appl Math Inf Sci 7 No 1 311-318 (013) 311 Applied Mathematics & Information Sciences An International Journal Conservation laws for the geodesic equations of the canonical connection on Lie groups in
More informationLagrangian for Central Potentials
Physics 411 Lecture 2 Lagrangian for Central Potentials Lecture 2 Physics 411 Classical Mechanics II August 29th 2007 Here we will review the Lagrange formulation in preparation for the study of the central
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define
More informationChapter 6. Second order differential equations
Chapter 6. Second order differential equations A second order differential equation is of the form y = f(t, y, y ) where y = y(t). We shall often think of t as parametrizing time, y position. In this case
More informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Equations. David Levermore Department of Mathematics University of Maryland
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Equations David Levermore Department of Mathematics University of Maryland 14 March 2012 Because the presentation of this material
More informationBACKGROUND IN SYMPLECTIC GEOMETRY
BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations
More informationLS.5 Theory of Linear Systems
LS.5 Theory of Linear Systems 1. General linear ODE systems and independent solutions. We have studied the homogeneous system of ODE s with constant coefficients, (1) x = Ax, where A is an n n matrix of
More informationPeriodic, hyperbolic and rational function solutions of nonlinear wave equations
Appl Math Inf Sci Lett 1, No 3, 97-101 (013 97 Applied Mathematics & Information Sciences Letters An International Journal http://dxdoiorg/101785/amisl/010307 Periodic, hyperbolic and rational function
More informationFirst Order Linear Ordinary Differential Equations
First Order Linear Ordinary Differential Equations The most general first order linear ODE is an equation of the form p t dy dt q t y t f t. 1 Herepqarecalledcoefficients f is referred to as the forcing
More information1D Wave Equation General Solution / Gaussian Function
D Wave Euation General Solution / Gaussian Function Phys 375 Overview and Motivation: Last time we derived the partial differential euation known as the (one dimensional wave euation Today we look at the
More informationStrauss PDEs 2e: Section Exercise 3 Page 1 of 13. u tt c 2 u xx = cos x. ( 2 t c 2 2 x)u = cos x. v = ( t c x )u
Strauss PDEs e: Setion 3.4 - Exerise 3 Page 1 of 13 Exerise 3 Solve u tt = u xx + os x, u(x, ) = sin x, u t (x, ) = 1 + x. Solution Solution by Operator Fatorization Bring u xx to the other side. Write
More informationMA 201: Method of Separation of Variables Finite Vibrating String Problem Lecture - 11 MA201(2016): PDE
MA 201: Method of Separation of Variables Finite Vibrating String Problem ecture - 11 IBVP for Vibrating string with no external forces We consider the problem in a computational domain (x,t) [0,] [0,
More informationLecture 10: Whitham Modulation Theory
Lecture 10: Whitham Modulation Theory Lecturer: Roger Grimshaw. Write-up: Andong He June 19, 2009 1 Introduction The Whitham modulation theory provides an asymptotic method for studying slowly varying
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationIntermediate Algebra Summary - Part II
Intermediate Algebra Summary - Part II This is an overview of the key ideas we have discussed during the middle part of this course. You may find this summary useful as a study aid, but remember that the
More informationComputer Problems for Taylor Series and Series Convergence
Computer Problems for Taylor Series and Series Convergence The two problems below are a set; the first should be done without a computer and the second is a computer-based follow up. 1. The drawing below
More information